Integrand size = 24, antiderivative size = 295 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=-\frac {b c^3}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}-\frac {b c}{6 d^3 x^2 \left (1+c^2 x^2\right )^{3/2}}+\frac {29 b c^3}{24 d^3 \sqrt {1+c^2 x^2}}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (1+c^2 x^2\right )^2}+\frac {7 c^2 (a+b \text {arcsinh}(c x))}{3 d^3 x \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{12 d^3 \left (1+c^2 x^2\right )^2}+\frac {35 c^4 x (a+b \text {arcsinh}(c x))}{8 d^3 \left (1+c^2 x^2\right )}+\frac {35 c^3 (a+b \text {arcsinh}(c x)) \arctan \left (e^{\text {arcsinh}(c x)}\right )}{4 d^3}+\frac {19 b c^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )}{6 d^3}-\frac {35 i b c^3 \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )}{8 d^3}+\frac {35 i b c^3 \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{8 d^3} \]
-1/12*b*c^3/d^3/(c^2*x^2+1)^(3/2)-1/6*b*c/d^3/x^2/(c^2*x^2+1)^(3/2)+1/3*(- a-b*arcsinh(c*x))/d^3/x^3/(c^2*x^2+1)^2+7/3*c^2*(a+b*arcsinh(c*x))/d^3/x/( c^2*x^2+1)^2+35/12*c^4*x*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)^2+35/8*c^4*x*( a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)+35/4*c^3*(a+b*arcsinh(c*x))*arctan(c*x+( c^2*x^2+1)^(1/2))/d^3+19/6*b*c^3*arctanh((c^2*x^2+1)^(1/2))/d^3-35/8*I*b*c ^3*polylog(2,-I*(c*x+(c^2*x^2+1)^(1/2)))/d^3+35/8*I*b*c^3*polylog(2,I*(c*x +(c^2*x^2+1)^(1/2)))/d^3+29/24*b*c^3/d^3/(c^2*x^2+1)^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 1.06 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.27 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\frac {\frac {3 (a+b \text {arcsinh}(c x))}{x^3 \left (1+c^2 x^2\right )^2}+\frac {21 (a+b \text {arcsinh}(c x))}{2 \left (x^3+c^2 x^5\right )}+\frac {b c^3 \operatorname {Hypergeometric2F1}\left (-\frac {3}{2},2,-\frac {1}{2},1+c^2 x^2\right )}{\left (1+c^2 x^2\right )^{3/2}}+\frac {21 b c^3 \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2,\frac {1}{2},1+c^2 x^2\right )}{2 \sqrt {1+c^2 x^2}}+\frac {35 \left (-2 a+6 a c^2 x^2-b c x \sqrt {1+c^2 x^2}-2 b \text {arcsinh}(c x)+6 b c^2 x^2 \text {arcsinh}(c x)+6 a c^3 x^3 \arctan (c x)+7 b c^3 x^3 \text {arctanh}\left (\sqrt {1+c^2 x^2}\right )-6 b \left (-c^2\right )^{3/2} x^3 \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+6 b \left (-c^2\right )^{3/2} x^3 \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+6 b \left (-c^2\right )^{3/2} x^3 \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )-6 b \left (-c^2\right )^{3/2} x^3 \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )\right )}{4 x^3}}{12 d^3} \]
((3*(a + b*ArcSinh[c*x]))/(x^3*(1 + c^2*x^2)^2) + (21*(a + b*ArcSinh[c*x]) )/(2*(x^3 + c^2*x^5)) + (b*c^3*Hypergeometric2F1[-3/2, 2, -1/2, 1 + c^2*x^ 2])/(1 + c^2*x^2)^(3/2) + (21*b*c^3*Hypergeometric2F1[-1/2, 2, 1/2, 1 + c^ 2*x^2])/(2*Sqrt[1 + c^2*x^2]) + (35*(-2*a + 6*a*c^2*x^2 - b*c*x*Sqrt[1 + c ^2*x^2] - 2*b*ArcSinh[c*x] + 6*b*c^2*x^2*ArcSinh[c*x] + 6*a*c^3*x^3*ArcTan [c*x] + 7*b*c^3*x^3*ArcTanh[Sqrt[1 + c^2*x^2]] - 6*b*(-c^2)^(3/2)*x^3*ArcS inh[c*x]*Log[1 + (c*E^ArcSinh[c*x])/Sqrt[-c^2]] + 6*b*(-c^2)^(3/2)*x^3*Arc Sinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c] + 6*b*(-c^2)^(3/2)*x^3*Po lyLog[2, (c*E^ArcSinh[c*x])/Sqrt[-c^2]] - 6*b*(-c^2)^(3/2)*x^3*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c]))/(4*x^3))/(12*d^3)
Time = 1.60 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.23, number of steps used = 24, number of rules used = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.958, Rules used = {6224, 27, 243, 52, 61, 61, 73, 221, 6224, 243, 61, 61, 73, 221, 6203, 241, 6203, 241, 6204, 3042, 4668, 2715, 2838}
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.
| \(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (c^2 d x^2+d\right )^3} \, dx\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle -\frac {7}{3} c^2 \int \frac {a+b \text {arcsinh}(c x)}{d^3 x^2 \left (c^2 x^2+1\right )^3}dx+\frac {b c \int \frac {1}{x^3 \left (c^2 x^2+1\right )^{5/2}}dx}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {7 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^3}dx}{3 d^3}+\frac {b c \int \frac {1}{x^3 \left (c^2 x^2+1\right )^{5/2}}dx}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {7 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^3}dx}{3 d^3}+\frac {b c \int \frac {1}{x^4 \left (c^2 x^2+1\right )^{5/2}}dx^2}{6 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {7 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^3}dx}{3 d^3}+\frac {b c \left (-\frac {5}{2} c^2 \int \frac {1}{x^2 \left (c^2 x^2+1\right )^{5/2}}dx^2-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {7 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^3}dx}{3 d^3}+\frac {b c \left (-\frac {5}{2} c^2 \left (\int \frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}dx^2+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {7 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^3}dx}{3 d^3}+\frac {b c \left (-\frac {5}{2} c^2 \left (\int \frac {1}{x^2 \sqrt {c^2 x^2+1}}dx^2+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {7 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^3}dx}{3 d^3}+\frac {b c \left (-\frac {5}{2} c^2 \left (\frac {2 \int \frac {1}{\frac {x^4}{c^2}-\frac {1}{c^2}}d\sqrt {c^2 x^2+1}}{c^2}+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {7 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x^2 \left (c^2 x^2+1\right )^3}dx}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^3}dx+b c \int \frac {1}{x \left (c^2 x^2+1\right )^{5/2}}dx-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 243 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^3}dx+\frac {1}{2} b c \int \frac {1}{x^2 \left (c^2 x^2+1\right )^{5/2}}dx^2-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^3}dx+\frac {1}{2} b c \left (\int \frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}dx^2+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^3}dx+\frac {1}{2} b c \left (\int \frac {1}{x^2 \sqrt {c^2 x^2+1}}dx^2+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^3}dx+\frac {1}{2} b c \left (\frac {2 \int \frac {1}{\frac {x^4}{c^2}-\frac {1}{c^2}}d\sqrt {c^2 x^2+1}}{c^2}+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^3}dx-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 6203 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx-\frac {1}{4} b c \int \frac {x}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \int \frac {a+b \text {arcsinh}(c x)}{\left (c^2 x^2+1\right )^2}dx+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 6203 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx-\frac {1}{2} b c \int \frac {x}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}\right )+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 241 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {1}{2} \int \frac {a+b \text {arcsinh}(c x)}{c^2 x^2+1}dx+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 6204 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {\int \frac {a+b \text {arcsinh}(c x)}{\sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {\int (a+b \text {arcsinh}(c x)) \csc \left (i \text {arcsinh}(c x)+\frac {\pi }{2}\right )d\text {arcsinh}(c x)}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 4668 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {-i b \int \log \left (1-i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i b \int \log \left (1+i e^{\text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {-i b \int e^{-\text {arcsinh}(c x)} \log \left (1-i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+i b \int e^{-\text {arcsinh}(c x)} \log \left (1+i e^{\text {arcsinh}(c x)}\right )de^{\text {arcsinh}(c x)}+2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {7 c^2 \left (-5 c^2 \left (\frac {3}{4} \left (\frac {2 \arctan \left (e^{\text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))-i b \operatorname {PolyLog}\left (2,-i e^{\text {arcsinh}(c x)}\right )+i b \operatorname {PolyLog}\left (2,i e^{\text {arcsinh}(c x)}\right )}{2 c}+\frac {x (a+b \text {arcsinh}(c x))}{2 \left (c^2 x^2+1\right )}+\frac {b}{2 c \sqrt {c^2 x^2+1}}\right )+\frac {x (a+b \text {arcsinh}(c x))}{4 \left (c^2 x^2+1\right )^2}+\frac {b}{12 c \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}+\frac {1}{2} b c \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{3 d^3}-\frac {a+b \text {arcsinh}(c x)}{3 d^3 x^3 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-\frac {5}{2} c^2 \left (-2 \text {arctanh}\left (\sqrt {c^2 x^2+1}\right )+\frac {2}{\sqrt {c^2 x^2+1}}+\frac {2}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x^2 \left (c^2 x^2+1\right )^{3/2}}\right )}{6 d^3}\) |
-1/3*(a + b*ArcSinh[c*x])/(d^3*x^3*(1 + c^2*x^2)^2) + (b*c*(-(1/(x^2*(1 + c^2*x^2)^(3/2))) - (5*c^2*(2/(3*(1 + c^2*x^2)^(3/2)) + 2/Sqrt[1 + c^2*x^2] - 2*ArcTanh[Sqrt[1 + c^2*x^2]]))/2))/(6*d^3) - (7*c^2*(-((a + b*ArcSinh[c *x])/(x*(1 + c^2*x^2)^2)) + (b*c*(2/(3*(1 + c^2*x^2)^(3/2)) + 2/Sqrt[1 + c ^2*x^2] - 2*ArcTanh[Sqrt[1 + c^2*x^2]]))/2 - 5*c^2*(b/(12*c*(1 + c^2*x^2)^ (3/2)) + (x*(a + b*ArcSinh[c*x]))/(4*(1 + c^2*x^2)^2) + (3*(b/(2*c*Sqrt[1 + c^2*x^2]) + (x*(a + b*ArcSinh[c*x]))/(2*(1 + c^2*x^2)) + (2*(a + b*ArcSi nh[c*x])*ArcTan[E^ArcSinh[c*x]] - I*b*PolyLog[2, (-I)*E^ArcSinh[c*x]] + I* b*PolyLog[2, I*E^ArcSinh[c*x]])/(2*c)))/4)))/(3*d^3)
3.1.54.3.1 Defintions of rubi rules used
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] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[(x_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(a + b*x^2)^(p + 1)/ (2*b*(p + 1)), x] /; FreeQ[{a, b, p}, x] && NeQ[p, -1]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[1/2 Subst[In t[x^((m - 1)/2)*(a + b*x)^p, x], x, x^2], x] /; FreeQ[{a, b, m, p}, x] && I ntegerQ[(m - 1)/2]
Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Simp[1/(d*e*n*Log[F]) Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) ))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 , (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
Int[csc[(e_.) + Pi*(k_.) + (Complex[0, fz_])*(f_.)*(x_)]*((c_.) + (d_.)*(x_ ))^(m_.), x_Symbol] :> Simp[-2*(c + d*x)^m*(ArcTanh[E^((-I)*e + f*fz*x)/E^( I*k*Pi)]/(f*fz*I)), x] + (-Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[ 1 - E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x] + Simp[d*(m/(f*fz*I)) Int[(c + d*x)^(m - 1)*Log[1 + E^((-I)*e + f*fz*x)/E^(I*k*Pi)], x], x]) /; FreeQ[{c , d, e, f, fz}, x] && IntegerQ[2*k] && IGtQ[m, 0]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)*((d_) + (e_.)*(x_)^2)^(p_), x _Symbol] :> Simp[(-x)*(d + e*x^2)^(p + 1)*((a + b*ArcSinh[c*x])^n/(2*d*(p + 1))), x] + (Simp[(2*p + 3)/(2*d*(p + 1)) Int[(d + e*x^2)^(p + 1)*(a + b* ArcSinh[c*x])^n, x], x] + Simp[b*c*(n/(2*(p + 1)))*Simp[(d + e*x^2)^p/(1 + c^2*x^2)^p] Int[x*(1 + c^2*x^2)^(p + 1/2)*(a + b*ArcSinh[c*x])^(n - 1), x ], x]) /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && LtQ[p, -1] && NeQ[p, -3/2]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((d_) + (e_.)*(x_)^2), x_Symb ol] :> Simp[1/(c*d) Subst[Int[(a + b*x)^n*Sech[x], x], x, ArcSinh[c*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[e, c^2*d] && IGtQ[n, 0]
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 + 1)*((a + b*ArcSinh[c*x])^n/(d*f*(m + 1))), x] + (-Simp[c^2*((m + 2*p + 3)/(f^2*(m + 1))) Int[(f*x)^(m + 2)*(d + e*x^2)^p*(a + b*ArcSinh[c*x])^n, x], x] - Sim p[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, p}, x] && EqQ[e, c^2*d] && GtQ[n, 0] && ILtQ[m, -1]
Time = 0.24 (sec) , antiderivative size = 333, normalized size of antiderivative = 1.13
| method | result | size |
| derivativedivides | \(c^{3} \left (\frac {a \left (-\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}+\frac {\frac {11}{8} c^{3} x^{3}+\frac {13}{8} c x}{\left (c^{2} x^{2}+1\right )^{2}}+\frac {35 \arctan \left (c x \right )}{8}\right )}{d^{3}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}+\frac {11 c^{3} x^{3} \operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {13 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {35 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {103}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {1}{6 c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {19}{6 \sqrt {c^{2} x^{2}+1}}+\frac {19 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}+\frac {35 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {35 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {35 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}\right )}{d^{3}}\right )\) | \(333\) |
| default | \(c^{3} \left (\frac {a \left (-\frac {1}{3 c^{3} x^{3}}+\frac {3}{c x}+\frac {\frac {11}{8} c^{3} x^{3}+\frac {13}{8} c x}{\left (c^{2} x^{2}+1\right )^{2}}+\frac {35 \arctan \left (c x \right )}{8}\right )}{d^{3}}+\frac {b \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}+\frac {11 c^{3} x^{3} \operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {13 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {35 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {103}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {1}{6 c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {19}{6 \sqrt {c^{2} x^{2}+1}}+\frac {19 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}+\frac {35 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {35 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {35 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}\right )}{d^{3}}\right )\) | \(333\) |
| parts | \(\frac {a \left (c^{4} \left (\frac {\frac {11}{8} x^{3} c^{2}+\frac {13}{8} x}{\left (c^{2} x^{2}+1\right )^{2}}+\frac {35 \arctan \left (c x \right )}{8 c}\right )-\frac {1}{3 x^{3}}+\frac {3 c^{2}}{x}\right )}{d^{3}}+\frac {b \,c^{3} \left (-\frac {\operatorname {arcsinh}\left (c x \right )}{3 c^{3} x^{3}}+\frac {3 \,\operatorname {arcsinh}\left (c x \right )}{c x}+\frac {11 c^{3} x^{3} \operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {13 c x \,\operatorname {arcsinh}\left (c x \right )}{8 \left (c^{2} x^{2}+1\right )^{2}}+\frac {35 \,\operatorname {arcsinh}\left (c x \right ) \arctan \left (c x \right )}{8}+\frac {103}{24 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {1}{6 c^{2} x^{2} \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}-\frac {19}{6 \sqrt {c^{2} x^{2}+1}}+\frac {19 \,\operatorname {arctanh}\left (\frac {1}{\sqrt {c^{2} x^{2}+1}}\right )}{6}+\frac {35 c^{2} x^{2}}{8 \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}+\frac {35 \arctan \left (c x \right ) \ln \left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {35 \arctan \left (c x \right ) \ln \left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}-\frac {35 i \operatorname {dilog}\left (1+\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}+\frac {35 i \operatorname {dilog}\left (1-\frac {i \left (i c x +1\right )}{\sqrt {c^{2} x^{2}+1}}\right )}{8}\right )}{d^{3}}\) | \(336\) |
c^3*(a/d^3*(-1/3/c^3/x^3+3/c/x+(11/8*c^3*x^3+13/8*c*x)/(c^2*x^2+1)^2+35/8* arctan(c*x))+b/d^3*(-1/3*arcsinh(c*x)/c^3/x^3+3*arcsinh(c*x)/c/x+11/8*c^3* x^3/(c^2*x^2+1)^2*arcsinh(c*x)+13/8*c*x/(c^2*x^2+1)^2*arcsinh(c*x)+35/8*ar csinh(c*x)*arctan(c*x)+103/24/(c^2*x^2+1)^(3/2)-1/6/c^2/x^2/(c^2*x^2+1)^(3 /2)-19/6/(c^2*x^2+1)^(1/2)+19/6*arctanh(1/(c^2*x^2+1)^(1/2))+35/8*c^2*x^2/ (c^2*x^2+1)^(3/2)+35/8*arctan(c*x)*ln(1+I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-35/ 8*arctan(c*x)*ln(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))-35/8*I*dilog(1+I*(1+I*c* x)/(c^2*x^2+1)^(1/2))+35/8*I*dilog(1-I*(1+I*c*x)/(c^2*x^2+1)^(1/2))))
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{4}} \,d x } \]
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a}{c^{6} x^{10} + 3 c^{4} x^{8} + 3 c^{2} x^{6} + x^{4}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{10} + 3 c^{4} x^{8} + 3 c^{2} x^{6} + x^{4}}\, dx}{d^{3}} \]
(Integral(a/(c**6*x**10 + 3*c**4*x**8 + 3*c**2*x**6 + x**4), x) + Integral (b*asinh(c*x)/(c**6*x**10 + 3*c**4*x**8 + 3*c**2*x**6 + x**4), x))/d**3
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{4}} \,d x } \]
1/24*a*(105*c^3*arctan(c*x)/d^3 + (105*c^6*x^6 + 175*c^4*x^4 + 56*c^2*x^2 - 8)/(c^4*d^3*x^7 + 2*c^2*d^3*x^5 + d^3*x^3)) + b*integrate(log(c*x + sqrt (c^2*x^2 + 1))/(c^6*d^3*x^10 + 3*c^4*d^3*x^8 + 3*c^2*d^3*x^6 + d^3*x^4), x )
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{4}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^4 \left (d+c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^4\,{\left (d\,c^2\,x^2+d\right )}^3} \,d x \]