Integrand size = 28, antiderivative size = 676 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\frac {40}{9} b^2 c^2 d^2 \sqrt {d+c^2 d x^2}-\frac {5 a b c^3 d^2 x \sqrt {d+c^2 d x^2}}{\sqrt {1+c^2 x^2}}+\frac {2}{27} b^2 c^2 d \left (d+c^2 d x^2\right )^{3/2}-\frac {5 b^2 c^3 d^2 x \sqrt {d+c^2 d x^2} \text {arcsinh}(c x)}{\sqrt {1+c^2 x^2}}-\frac {b c d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{x \sqrt {1+c^2 x^2}}+\frac {b c^3 d^2 x \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{3 \sqrt {1+c^2 x^2}}-\frac {2 b c^5 d^2 x^3 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))}{9 \sqrt {1+c^2 x^2}}+\frac {5}{2} c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2+\frac {5}{6} c^2 d \left (d+c^2 d x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}-\frac {5 c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x))^2 \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {b^2 c^2 d^2 \sqrt {d+c^2 d x^2} \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{\sqrt {1+c^2 x^2}}-\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {5 b c^2 d^2 \sqrt {d+c^2 d x^2} (a+b \text {arcsinh}(c x)) \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}+\frac {5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}}-\frac {5 b^2 c^2 d^2 \sqrt {d+c^2 d x^2} \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )}{\sqrt {1+c^2 x^2}} \] Output:
40/9*b^2*c^2*d^2*(c^2*d*x^2+d)^(1/2)-5*a*b*c^3*d^2*x*(c^2*d*x^2+d)^(1/2)/( c^2*x^2+1)^(1/2)+2/27*b^2*c^2*d*(c^2*d*x^2+d)^(3/2)-5*b^2*c^3*d^2*x*(c^2*d *x^2+d)^(1/2)*arcsinh(c*x)/(c^2*x^2+1)^(1/2)-b*c*d^2*(c^2*d*x^2+d)^(1/2)*( a+b*arcsinh(c*x))/x/(c^2*x^2+1)^(1/2)+1/3*b*c^3*d^2*x*(c^2*d*x^2+d)^(1/2)* (a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)-2/9*b*c^5*d^2*x^3*(c^2*d*x^2+d)^(1/2) *(a+b*arcsinh(c*x))/(c^2*x^2+1)^(1/2)+5/2*c^2*d^2*(c^2*d*x^2+d)^(1/2)*(a+b *arcsinh(c*x))^2+5/6*c^2*d*(c^2*d*x^2+d)^(3/2)*(a+b*arcsinh(c*x))^2-1/2*(c ^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^2-5*c^2*d^2*(c^2*d*x^2+d)^(1/2)*( a+b*arcsinh(c*x))^2*arctanh(c*x+(c^2*x^2+1)^(1/2))/(c^2*x^2+1)^(1/2)-b^2*c ^2*d^2*(c^2*d*x^2+d)^(1/2)*arctanh((c^2*x^2+1)^(1/2))/(c^2*x^2+1)^(1/2)-5* b*c^2*d^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c*x))*polylog(2,-c*x-(c^2*x^2+1 )^(1/2))/(c^2*x^2+1)^(1/2)+5*b*c^2*d^2*(c^2*d*x^2+d)^(1/2)*(a+b*arcsinh(c* x))*polylog(2,c*x+(c^2*x^2+1)^(1/2))/(c^2*x^2+1)^(1/2)+5*b^2*c^2*d^2*(c^2* d*x^2+d)^(1/2)*polylog(3,-c*x-(c^2*x^2+1)^(1/2))/(c^2*x^2+1)^(1/2)-5*b^2*c ^2*d^2*(c^2*d*x^2+d)^(1/2)*polylog(3,c*x+(c^2*x^2+1)^(1/2))/(c^2*x^2+1)^(1 /2)
Time = 7.94 (sec) , antiderivative size = 990, normalized size of antiderivative = 1.46 \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx =\text {Too large to display} \] Input:
Integrate[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^3,x]
Output:
Sqrt[d*(1 + c^2*x^2)]*((7*a^2*c^2*d^2)/3 - (a^2*d^2)/(2*x^2) + (a^2*c^4*d^ 2*x^2)/3) + 2*a*b*c^2*d^2*(-1/9*(c*x*Sqrt[d*(1 + c^2*x^2)]*(3 + c^2*x^2))/ Sqrt[1 + c^2*x^2] + ((1 + c^2*x^2)*Sqrt[d*(1 + c^2*x^2)]*ArcSinh[c*x])/3) + (5*a^2*c^2*d^(5/2)*Log[x])/2 - (5*a^2*c^2*d^(5/2)*Log[d + Sqrt[d]*Sqrt[d *(1 + c^2*x^2)]])/2 + (4*a*b*c^2*d^2*Sqrt[d*(1 + c^2*x^2)]*(-(c*x) + Sqrt[ 1 + c^2*x^2]*ArcSinh[c*x] + ArcSinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - ArcS inh[c*x]*Log[1 + E^(-ArcSinh[c*x])] + PolyLog[2, -E^(-ArcSinh[c*x])] - Pol yLog[2, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + 2*b^2*c^2*d^2*Sqrt[d*(1 + c^2*x^2)]*(2 - (2*c*x*ArcSinh[c*x])/Sqrt[1 + c^2*x^2] + ArcSinh[c*x]^2 + (ArcSinh[c*x]^2*(Log[1 - E^(-ArcSinh[c*x])] - Log[1 + E^(-ArcSinh[c*x])])) /Sqrt[1 + c^2*x^2] + (2*ArcSinh[c*x]*(PolyLog[2, -E^(-ArcSinh[c*x])] - Pol yLog[2, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2] + (2*(PolyLog[3, -E^(-ArcSi nh[c*x])] - PolyLog[3, E^(-ArcSinh[c*x])]))/Sqrt[1 + c^2*x^2]) + (b^2*c^2* d^2*Sqrt[d*(1 + c^2*x^2)]*(27*Sqrt[1 + c^2*x^2]*(2 + ArcSinh[c*x]^2) + (2 + 9*ArcSinh[c*x]^2)*Cosh[3*ArcSinh[c*x]] - 6*ArcSinh[c*x]*(9*c*x + Sinh[3* ArcSinh[c*x]])))/(108*Sqrt[1 + c^2*x^2]) + (a*b*c^2*d^2*Sqrt[d*(1 + c^2*x^ 2)]*(-2*Coth[ArcSinh[c*x]/2] - ArcSinh[c*x]*Csch[ArcSinh[c*x]/2]^2 + 4*Arc Sinh[c*x]*Log[1 - E^(-ArcSinh[c*x])] - 4*ArcSinh[c*x]*Log[1 + E^(-ArcSinh[ c*x])] + 4*PolyLog[2, -E^(-ArcSinh[c*x])] - 4*PolyLog[2, E^(-ArcSinh[c*x]) ] - ArcSinh[c*x]*Sech[ArcSinh[c*x]/2]^2 + 2*Tanh[ArcSinh[c*x]/2]))/(4*S...
Result contains complex when optimal does not.
Time = 3.46 (sec) , antiderivative size = 526, normalized size of antiderivative = 0.78, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.821, Rules used = {6222, 6218, 27, 1578, 1192, 25, 1467, 2009, 6223, 6199, 27, 353, 53, 2009, 6221, 2009, 6231, 3042, 26, 4670, 3011, 2720, 7143}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx\) |
| \(\Big \downarrow \) 6222 |
| \(\displaystyle \frac {b c d^2 \sqrt {c^2 d x^2+d} \int \frac {\left (c^2 x^2+1\right )^2 (a+b \text {arcsinh}(c x))}{x^2}dx}{\sqrt {c^2 x^2+1}}+\frac {5}{2} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6218 |
| \(\displaystyle \frac {5}{2} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (-b c \int -\frac {-c^4 x^4-6 c^2 x^2+3}{3 x \sqrt {c^2 x^2+1}}dx+\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{2} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} b c \int \frac {-c^4 x^4-6 c^2 x^2+3}{x \sqrt {c^2 x^2+1}}dx+\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 1578 |
| \(\displaystyle \frac {5}{2} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{6} b c \int \frac {-c^4 x^4-6 c^2 x^2+3}{x^2 \sqrt {c^2 x^2+1}}dx^2+\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 1192 |
| \(\displaystyle \frac {5}{2} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {b \int -\frac {-c^4 x^8-4 c^4 x^4+8 c^4}{1-x^4}d\sqrt {c^2 x^2+1}}{3 c^3}+\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {5}{2} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (-\frac {b \int \frac {-c^4 x^8-4 c^4 x^4+8 c^4}{1-x^4}d\sqrt {c^2 x^2+1}}{3 c^3}+\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 1467 |
| \(\displaystyle \frac {5}{2} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (-\frac {b \int \left (x^4 c^4+\frac {3 c^4}{1-x^4}+5 c^4\right )d\sqrt {c^2 x^2+1}}{3 c^3}+\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{2} c^2 d \int \frac {\left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6223 |
| \(\displaystyle \frac {5}{2} c^2 d \left (-\frac {2 b c d \sqrt {c^2 d x^2+d} \int \left (c^2 x^2+1\right ) (a+b \text {arcsinh}(c x))dx}{3 \sqrt {c^2 x^2+1}}+d \int \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6199 |
| \(\displaystyle \frac {5}{2} c^2 d \left (d \int \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-b c \int \frac {x \left (c^2 x^2+3\right )}{3 \sqrt {c^2 x^2+1}}dx+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {5}{2} c^2 d \left (d \int \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{3} b c \int \frac {x \left (c^2 x^2+3\right )}{\sqrt {c^2 x^2+1}}dx+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 353 |
| \(\displaystyle \frac {5}{2} c^2 d \left (d \int \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{6} b c \int \frac {c^2 x^2+3}{\sqrt {c^2 x^2+1}}dx^2+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 53 |
| \(\displaystyle \frac {5}{2} c^2 d \left (d \int \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{x}dx-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (-\frac {1}{6} b c \int \left (\sqrt {c^2 x^2+1}+\frac {2}{\sqrt {c^2 x^2+1}}\right )dx^2+\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))\right )}{3 \sqrt {c^2 x^2+1}}+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{2} c^2 d \left (d \int \frac {\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2}{x}dx+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6221 |
| \(\displaystyle \frac {5}{2} c^2 d \left (d \left (-\frac {2 b c \sqrt {c^2 d x^2+d} \int (a+b \text {arcsinh}(c x))dx}{\sqrt {c^2 x^2+1}}+\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {c^2 x^2+1}}dx}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{x \sqrt {c^2 x^2+1}}dx}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 6231 |
| \(\displaystyle \frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {c^2 d x^2+d} \int \frac {(a+b \text {arcsinh}(c x))^2}{c x}d\text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {5}{2} c^2 d \left (d \left (\frac {\sqrt {c^2 d x^2+d} \int i (a+b \text {arcsinh}(c x))^2 \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {5}{2} c^2 d \left (d \left (\frac {i \sqrt {c^2 d x^2+d} \int (a+b \text {arcsinh}(c x))^2 \csc (i \text {arcsinh}(c x))d\text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle \frac {5}{2} c^2 d \left (d \left (\frac {i \sqrt {c^2 d x^2+d} \left (2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-2 i b \int (a+b \text {arcsinh}(c x)) \log \left (1+e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 3011 |
| \(\displaystyle \frac {5}{2} c^2 d \left (d \left (\frac {i \sqrt {c^2 d x^2+d} \left (-2 i b \left (b \int \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 i b \left (b \int \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 2720 |
| \(\displaystyle \frac {5}{2} c^2 d \left (d \left (\frac {i \sqrt {c^2 d x^2+d} \left (-2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 i b \left (b \int e^{-\text {arcsinh}(c x)} \operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2\right )}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
| \(\Big \downarrow \) 7143 |
| \(\displaystyle \frac {5}{2} c^2 d \left (d \left (\frac {i \sqrt {c^2 d x^2+d} \left (2 i \text {arctanh}\left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))^2-2 i b \left (b \operatorname {PolyLog}\left (3,-e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,-e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+2 i b \left (b \operatorname {PolyLog}\left (3,e^{\text {arcsinh}(c x)}\right )-\operatorname {PolyLog}\left (2,e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )\right )}{\sqrt {c^2 x^2+1}}+\sqrt {c^2 d x^2+d} (a+b \text {arcsinh}(c x))^2-\frac {2 b c \sqrt {c^2 d x^2+d} \left (a x+b x \text {arcsinh}(c x)-\frac {b \sqrt {c^2 x^2+1}}{c}\right )}{\sqrt {c^2 x^2+1}}\right )+\frac {1}{3} \left (c^2 d x^2+d\right )^{3/2} (a+b \text {arcsinh}(c x))^2-\frac {2 b c d \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^2 x^3 (a+b \text {arcsinh}(c x))+x (a+b \text {arcsinh}(c x))-\frac {1}{6} b c \left (\frac {2 \left (c^2 x^2+1\right )^{3/2}}{3 c^2}+\frac {4 \sqrt {c^2 x^2+1}}{c^2}\right )\right )}{3 \sqrt {c^2 x^2+1}}\right )+\frac {b c d^2 \sqrt {c^2 d x^2+d} \left (\frac {1}{3} c^4 x^3 (a+b \text {arcsinh}(c x))+2 c^2 x (a+b \text {arcsinh}(c x))-\frac {a+b \text {arcsinh}(c x)}{x}+\frac {b \left (-3 c^4 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )-\frac {1}{3} c^4 x^6-5 c^4 \sqrt {c^2 x^2+1}\right )}{3 c^3}\right )}{\sqrt {c^2 x^2+1}}-\frac {\left (c^2 d x^2+d\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{2 x^2}\) |
Input:
Int[((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^3,x]
Output:
-1/2*((d + c^2*d*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2)/x^2 + (b*c*d^2*Sqrt[d + c^2*d*x^2]*(-((a + b*ArcSinh[c*x])/x) + 2*c^2*x*(a + b*ArcSinh[c*x]) + ( c^4*x^3*(a + b*ArcSinh[c*x]))/3 + (b*(-1/3*(c^4*x^6) - 5*c^4*Sqrt[1 + c^2* x^2] - 3*c^4*ArcTanh[Sqrt[1 + c^2*x^2]]))/(3*c^3)))/Sqrt[1 + c^2*x^2] + (5 *c^2*d*(((d + c^2*d*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2)/3 - (2*b*c*d*Sqrt[d + c^2*d*x^2]*(-1/6*(b*c*((4*Sqrt[1 + c^2*x^2])/c^2 + (2*(1 + c^2*x^2)^(3/ 2))/(3*c^2))) + x*(a + b*ArcSinh[c*x]) + (c^2*x^3*(a + b*ArcSinh[c*x]))/3) )/(3*Sqrt[1 + c^2*x^2]) + d*(Sqrt[d + c^2*d*x^2]*(a + b*ArcSinh[c*x])^2 - (2*b*c*Sqrt[d + c^2*d*x^2]*(a*x - (b*Sqrt[1 + c^2*x^2])/c + b*x*ArcSinh[c* x]))/Sqrt[1 + c^2*x^2] + (I*Sqrt[d + c^2*d*x^2]*((2*I)*(a + b*ArcSinh[c*x] )^2*ArcTanh[E^ArcSinh[c*x]] - (2*I)*b*(-((a + b*ArcSinh[c*x])*PolyLog[2, - E^ArcSinh[c*x]]) + b*PolyLog[3, -E^ArcSinh[c*x]]) + (2*I)*b*(-((a + b*ArcS inh[c*x])*PolyLog[2, E^ArcSinh[c*x]]) + b*PolyLog[3, E^ArcSinh[c*x]])))/Sq rt[1 + c^2*x^2])))/2
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int [ExpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0] && LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Simp[1/2 Subst[Int[(a + b*x)^p*(c + d*x)^q, x], x, x^2], x] /; FreeQ[ {a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[2/e^(n + 2*p + 1) Subst[Int[x^( 2*m + 1)*(e*f - d*g + g*x^2)^n*(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4)^p, x], x, Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ[p, 0] && ILtQ[n, 0] && IntegerQ[m + 1/2]
Int[((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x^2)^q*(a + b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[p, 0] && IGtQ[q, -2]
Int[(x_)^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_ )^4)^(p_.), x_Symbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(d + e*x)^q*(a + b*x + c*x^2)^p, x], x, x^2], x] /; FreeQ[{a, b, c, d, e, p, q}, x] && Int egerQ[(m - 1)/2]
Int[u_, x_Symbol] :> With[{v = FunctionOfExponential[u, x]}, Simp[v/D[v, x] Subst[Int[FunctionOfExponentialFunction[u, x]/x, x], x, v], x]] /; Funct ionOfExponentialQ[u, x] && !MatchQ[u, (w_)*((a_.)*(v_)^(n_))^(m_) /; FreeQ [{a, m, n}, x] && IntegerQ[m*n]] && !MatchQ[u, E^((c_.)*((a_.) + (b_.)*x)) *(F_)[v_] /; FreeQ[{a, b, c}, x] && InverseFunctionQ[F[x]]]
Int[Log[1 + (e_.)*((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.)]*((f_.) + (g_.) *(x_))^(m_.), x_Symbol] :> Simp[(-(f + g*x)^m)*(PolyLog[2, (-e)*(F^(c*(a + b*x)))^n]/(b*c*n*Log[F])), x] + Simp[g*(m/(b*c*n*Log[F])) Int[(f + g*x)^( m - 1)*PolyLog[2, (-e)*(F^(c*(a + b*x)))^n], x], x] /; FreeQ[{F, a, b, c, e , f, g, n}, x] && GtQ[m, 0]
Int[csc[(e_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_))^(m_.), x _Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 - E^((-I)*e + f*fz*x )], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)], x], x]) /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((d_) + (e_.)*(x_)^2)^(p_.), x_Symb ol] :> With[{u = IntHide[(d + e*x^2)^p, x]}, Simp[(a + b*ArcSinh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))*((f_.)*(x_))^(m_)*((d_) + (e_.)*(x_ )^2)^(p_.), x_Symbol] :> With[{u = IntHide[(f*x)^m*(d + e*x^2)^p, x]}, Simp [(a + b*ArcSinh[c*x]) u, x] - Simp[b*c Int[SimplifyIntegrand[u/Sqrt[1 + c^2*x^2], x], x], x]] /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[p, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*Sqrt[(d_) + (e_.)*(x_)^2], x_Symbol] :> Simp[(f*x)^(m + 1)*Sqrt[d + e*x^2]*((a + b*Arc Sinh[c*x])^n/(f*(m + 2))), x] + (Simp[(1/(m + 2))*Simp[Sqrt[d + e*x^2]/Sqrt [1 + c^2*x^2]] Int[(f*x)^m*((a + b*ArcSinh[c*x])^n/Sqrt[1 + c^2*x^2]), x] , x] - Simp[b*c*(n/(f*(m + 2)))*Simp[Sqrt[d + e*x^2]/Sqrt[1 + c^2*x^2]] I nt[(f*x)^(m + 1)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d , e, f, m}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && (IGtQ[m, -2] || EqQ[n, 1])
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Sinh[c*x])^n/(f*(m + 1))), x] + (-Simp[2*e*(p/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*( m + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x ^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e , f}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && LtQ[m, -1]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d_) + (e_ .)*(x_)^2)^(p_.), x_Symbol] :> Simp[(f*x)^(m + 1)*(d + e*x^2)^p*((a + b*Arc Sinh[c*x])^n/(f*(m + 2*p + 1))), x] + (Simp[2*d*(p/(m + 2*p + 1)) Int[(f* x)^m*(d + e*x^2)^(p - 1)*(a + b*ArcSinh[c*x])^n, x], x] - Simp[b*c*(n/(f*(m + 2*p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[(f*x)^(m + 1)*(1 + c^2*x^2)^(p - 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x], x]) /; FreeQ[{a, b, c, d, e, f, m}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && GtQ[p, 0] && !LtQ[m, -1]
Int[(((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)^(m_))/Sqrt[(d_) + (e_.) *(x_)^2], x_Symbol] :> Simp[(1/c^(m + 1))*Simp[Sqrt[1 + c^2*x^2]/Sqrt[d + e *x^2]] Subst[Int[(a + b*x)^n*Sinh[x]^m, x], x, ArcSinh[c*x]], x] /; FreeQ [{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0] && IntegerQ[m]
Int[PolyLog[n_, (c_.)*((a_.) + (b_.)*(x_))^(p_.)]/((d_.) + (e_.)*(x_)), x_S ymbol] :> Simp[PolyLog[n + 1, c*(a + b*x)^p]/(e*p), x] /; FreeQ[{a, b, c, d , e, n, p}, x] && EqQ[b*d, a*e]
Time = 1.41 (sec) , antiderivative size = 720, normalized size of antiderivative = 1.07
| method | result | size |
| default | \(a^{2} \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{2 d \,x^{2}}+\frac {5 c^{2} \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5}+d \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )\right )\right )}{2}\right )-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-18 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2} x^{4} c^{4}+12 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}-4 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}-135 \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+135 \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}-126 \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+252 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}+270 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-270 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}+108 \,\operatorname {arctanh}\left (x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+270 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-270 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-244 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+27 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2}+54 x c \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{54 \sqrt {c^{2} x^{2}+1}\, x^{2}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-6 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}+2 x^{5} c^{5}-45 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+45 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-42 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}+42 x^{3} c^{3}+45 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-45 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+9 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}+9 x c \right ) d^{2}}{9 \sqrt {c^{2} x^{2}+1}\, x^{2}}\) | \(720\) |
| parts | \(a^{2} \left (-\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {7}{2}}}{2 d \,x^{2}}+\frac {5 c^{2} \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {5}{2}}}{5}+d \left (\frac {\left (c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}{3}+d \left (\sqrt {c^{2} d \,x^{2}+d}-\sqrt {d}\, \ln \left (\frac {2 d +2 \sqrt {d}\, \sqrt {c^{2} d \,x^{2}+d}}{x}\right )\right )\right )\right )}{2}\right )-\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-18 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2} x^{4} c^{4}+12 \,\operatorname {arcsinh}\left (x c \right ) x^{5} c^{5}-4 x^{4} c^{4} \sqrt {c^{2} x^{2}+1}-135 \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}+135 \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) \operatorname {arcsinh}\left (x c \right )^{2} x^{2} c^{2}-126 \operatorname {arcsinh}\left (x c \right )^{2} \sqrt {c^{2} x^{2}+1}\, x^{2} c^{2}+252 \,\operatorname {arcsinh}\left (x c \right ) x^{3} c^{3}+270 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}-270 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}+108 \,\operatorname {arctanh}\left (x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+270 \operatorname {polylog}\left (3, x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-270 \operatorname {polylog}\left (3, -x c -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-244 x^{2} c^{2} \sqrt {c^{2} x^{2}+1}+27 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right )^{2}+54 x c \,\operatorname {arcsinh}\left (x c \right )\right ) d^{2}}{54 \sqrt {c^{2} x^{2}+1}\, x^{2}}-\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \left (-6 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}\, x^{4} c^{4}+2 x^{5} c^{5}-45 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+45 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-42 \sqrt {c^{2} x^{2}+1}\, \operatorname {arcsinh}\left (x c \right ) x^{2} c^{2}+42 x^{3} c^{3}+45 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}-45 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right ) x^{2} c^{2}+9 \,\operatorname {arcsinh}\left (x c \right ) \sqrt {c^{2} x^{2}+1}+9 x c \right ) d^{2}}{9 \sqrt {c^{2} x^{2}+1}\, x^{2}}\) | \(720\) |
Input:
int((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(x*c))^2/x^3,x,method=_RETURNVERBOSE)
Output:
a^2*(-1/2/d/x^2*(c^2*d*x^2+d)^(7/2)+5/2*c^2*(1/5*(c^2*d*x^2+d)^(5/2)+d*(1/ 3*(c^2*d*x^2+d)^(3/2)+d*((c^2*d*x^2+d)^(1/2)-d^(1/2)*ln((2*d+2*d^(1/2)*(c^ 2*d*x^2+d)^(1/2))/x)))))-1/54*b^2*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^(1/2)/ x^2*(-18*(c^2*x^2+1)^(1/2)*arcsinh(x*c)^2*x^4*c^4+12*arcsinh(x*c)*x^5*c^5- 4*x^4*c^4*(c^2*x^2+1)^(1/2)-135*ln(1-x*c-(c^2*x^2+1)^(1/2))*arcsinh(x*c)^2 *x^2*c^2+135*ln(1+x*c+(c^2*x^2+1)^(1/2))*arcsinh(x*c)^2*x^2*c^2-126*arcsin h(x*c)^2*(c^2*x^2+1)^(1/2)*x^2*c^2+252*arcsinh(x*c)*x^3*c^3+270*polylog(2, -x*c-(c^2*x^2+1)^(1/2))*arcsinh(x*c)*x^2*c^2-270*polylog(2,x*c+(c^2*x^2+1) ^(1/2))*arcsinh(x*c)*x^2*c^2+108*arctanh(x*c+(c^2*x^2+1)^(1/2))*x^2*c^2+27 0*polylog(3,x*c+(c^2*x^2+1)^(1/2))*x^2*c^2-270*polylog(3,-x*c-(c^2*x^2+1)^ (1/2))*x^2*c^2-244*x^2*c^2*(c^2*x^2+1)^(1/2)+27*(c^2*x^2+1)^(1/2)*arcsinh( x*c)^2+54*x*c*arcsinh(x*c))*d^2-1/9*a*b*(d*(c^2*x^2+1))^(1/2)/(c^2*x^2+1)^ (1/2)/x^2*(-6*arcsinh(x*c)*(c^2*x^2+1)^(1/2)*x^4*c^4+2*x^5*c^5-45*arcsinh( x*c)*ln(1-x*c-(c^2*x^2+1)^(1/2))*x^2*c^2+45*arcsinh(x*c)*ln(1+x*c+(c^2*x^2 +1)^(1/2))*x^2*c^2-42*(c^2*x^2+1)^(1/2)*arcsinh(x*c)*x^2*c^2+42*x^3*c^3+45 *polylog(2,-x*c-(c^2*x^2+1)^(1/2))*x^2*c^2-45*polylog(2,x*c+(c^2*x^2+1)^(1 /2))*x^2*c^2+9*arcsinh(x*c)*(c^2*x^2+1)^(1/2)+9*x*c)*d^2
\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:
integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="frica s")
Output:
integral((a^2*c^4*d^2*x^4 + 2*a^2*c^2*d^2*x^2 + a^2*d^2 + (b^2*c^4*d^2*x^4 + 2*b^2*c^2*d^2*x^2 + b^2*d^2)*arcsinh(c*x)^2 + 2*(a*b*c^4*d^2*x^4 + 2*a* b*c^2*d^2*x^2 + a*b*d^2)*arcsinh(c*x))*sqrt(c^2*d*x^2 + d)/x^3, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int \frac {\left (d \left (c^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{x^{3}}\, dx \] Input:
integrate((c**2*d*x**2+d)**(5/2)*(a+b*asinh(c*x))**2/x**3,x)
Output:
Integral((d*(c**2*x**2 + 1))**(5/2)*(a + b*asinh(c*x))**2/x**3, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int { \frac {{\left (c^{2} d x^{2} + d\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}}{x^{3}} \,d x } \] Input:
integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="maxim a")
Output:
-1/6*(15*c^2*d^(5/2)*arcsinh(1/(c*abs(x))) - 3*(c^2*d*x^2 + d)^(5/2)*c^2 - 5*(c^2*d*x^2 + d)^(3/2)*c^2*d - 15*sqrt(c^2*d*x^2 + d)*c^2*d^2 + 3*(c^2*d *x^2 + d)^(7/2)/(d*x^2))*a^2 + integrate((c^2*d*x^2 + d)^(5/2)*b^2*log(c*x + sqrt(c^2*x^2 + 1))^2/x^3 + 2*(c^2*d*x^2 + d)^(5/2)*a*b*log(c*x + sqrt(c ^2*x^2 + 1))/x^3, x)
Exception generated. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate((c^2*d*x^2+d)^(5/2)*(a+b*arcsinh(c*x))^2/x^3,x, algorithm="giac" )
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value
Timed out. \[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^{5/2}}{x^3} \,d x \] Input:
int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2))/x^3,x)
Output:
int(((a + b*asinh(c*x))^2*(d + c^2*d*x^2)^(5/2))/x^3, x)
\[ \int \frac {\left (d+c^2 d x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2}{x^3} \, dx=\frac {\sqrt {d}\, d^{2} \left (2 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{4} x^{4}+14 \sqrt {c^{2} x^{2}+1}\, a^{2} c^{2} x^{2}-3 \sqrt {c^{2} x^{2}+1}\, a^{2}+12 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x^{3}}d x \right ) a b \,x^{2}+24 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )}{x}d x \right ) a b \,c^{2} x^{2}+6 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2}}{x^{3}}d x \right ) b^{2} x^{2}+12 \left (\int \frac {\sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2}}{x}d x \right ) b^{2} c^{2} x^{2}+12 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right ) x d x \right ) a b \,c^{4} x^{2}+6 \left (\int \sqrt {c^{2} x^{2}+1}\, \mathit {asinh} \left (c x \right )^{2} x d x \right ) b^{2} c^{4} x^{2}+15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x -1\right ) a^{2} c^{2} x^{2}-15 \,\mathrm {log}\left (\sqrt {c^{2} x^{2}+1}+c x +1\right ) a^{2} c^{2} x^{2}\right )}{6 x^{2}} \] Input:
int((c^2*d*x^2+d)^(5/2)*(a+b*asinh(c*x))^2/x^3,x)
Output:
(sqrt(d)*d**2*(2*sqrt(c**2*x**2 + 1)*a**2*c**4*x**4 + 14*sqrt(c**2*x**2 + 1)*a**2*c**2*x**2 - 3*sqrt(c**2*x**2 + 1)*a**2 + 12*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/x**3,x)*a*b*x**2 + 24*int((sqrt(c**2*x**2 + 1)*asinh(c*x))/ x,x)*a*b*c**2*x**2 + 6*int((sqrt(c**2*x**2 + 1)*asinh(c*x)**2)/x**3,x)*b** 2*x**2 + 12*int((sqrt(c**2*x**2 + 1)*asinh(c*x)**2)/x,x)*b**2*c**2*x**2 + 12*int(sqrt(c**2*x**2 + 1)*asinh(c*x)*x,x)*a*b*c**4*x**2 + 6*int(sqrt(c**2 *x**2 + 1)*asinh(c*x)**2*x,x)*b**2*c**4*x**2 + 15*log(sqrt(c**2*x**2 + 1) + c*x - 1)*a**2*c**2*x**2 - 15*log(sqrt(c**2*x**2 + 1) + c*x + 1)*a**2*c** 2*x**2))/(6*x**2)