Integrand size = 24, antiderivative size = 216 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\frac {b c^3 x}{12 d^3 \left (1+c^2 x^2\right )^{3/2}}+\frac {7 b c^3 x}{6 d^3 \sqrt {1+c^2 x^2}}-\frac {b c \sqrt {1+c^2 x^2}}{2 d^3 x}+\frac {b c^2 \text {arcsinh}(c x)}{d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2}-\frac {c^2 \left (3+2 c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}{4 d^3 \left (1+c^2 x^2\right )^2}+\frac {6 c^2 (a+b \text {arcsinh}(c x)) \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right )}{d^3}+\frac {3 b c^2 \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )}{2 d^3}-\frac {3 b c^2 \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )}{2 d^3} \] Output:
1/12*b*c^3*x/d^3/(c^2*x^2+1)^(3/2)+7/6*b*c^3*x/d^3/(c^2*x^2+1)^(1/2)-1/2*b *c*(c^2*x^2+1)^(1/2)/d^3/x+b*c^2*arcsinh(c*x)/d^3-1/2*(a+b*arcsinh(c*x))/d ^3/x^2-1/4*c^2*(2*c^2*x^2+3)^2*(a+b*arcsinh(c*x))/d^3/(c^2*x^2+1)^2+6*c^2* (a+b*arcsinh(c*x))*arctanh((c*x+(c^2*x^2+1)^(1/2))^2)/d^3+3/2*b*c^2*polylo g(2,-(c*x+(c^2*x^2+1)^(1/2))^2)/d^3-3/2*b*c^2*polylog(2,(c*x+(c^2*x^2+1)^( 1/2))^2)/d^3
Time = 0.56 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.63 \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\frac {-\frac {18 b c \sqrt {1+c^2 x^2}}{x}+\frac {9 b c \left (1+2 c^2 x^2\right )}{x \sqrt {1+c^2 x^2}}+\frac {b c \left (3+12 c^2 x^2+8 c^4 x^4\right )}{x \left (1+c^2 x^2\right )^{3/2}}-18 b c^2 \text {arcsinh}(c x)^2-\frac {18 (a+b \text {arcsinh}(c x))}{x^2}+\frac {3 (a+b \text {arcsinh}(c x))}{\left (x+c^2 x^3\right )^2}+\frac {9 (a+b \text {arcsinh}(c x))}{x^2+c^2 x^4}+\frac {18 c^2 (a+b \text {arcsinh}(c x))^2}{b}+36 b c^2 \text {arcsinh}(c x) \log \left (1+\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+36 b c^2 \text {arcsinh}(c x) \log \left (1+\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )+18 a c^2 \log \left (1+c^2 x^2\right )+36 b c^2 \operatorname {PolyLog}\left (2,\frac {c e^{\text {arcsinh}(c x)}}{\sqrt {-c^2}}\right )+36 b c^2 \operatorname {PolyLog}\left (2,\frac {\sqrt {-c^2} e^{\text {arcsinh}(c x)}}{c}\right )-18 c^2 \left (2 (a+b \text {arcsinh}(c x)) \log \left (1-e^{2 \text {arcsinh}(c x)}\right )+b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )}{12 d^3} \] Input:
Integrate[(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)^3),x]
Output:
((-18*b*c*Sqrt[1 + c^2*x^2])/x + (9*b*c*(1 + 2*c^2*x^2))/(x*Sqrt[1 + c^2*x ^2]) + (b*c*(3 + 12*c^2*x^2 + 8*c^4*x^4))/(x*(1 + c^2*x^2)^(3/2)) - 18*b*c ^2*ArcSinh[c*x]^2 - (18*(a + b*ArcSinh[c*x]))/x^2 + (3*(a + b*ArcSinh[c*x] ))/(x + c^2*x^3)^2 + (9*(a + b*ArcSinh[c*x]))/(x^2 + c^2*x^4) + (18*c^2*(a + b*ArcSinh[c*x])^2)/b + 36*b*c^2*ArcSinh[c*x]*Log[1 + (c*E^ArcSinh[c*x]) /Sqrt[-c^2]] + 36*b*c^2*ArcSinh[c*x]*Log[1 + (Sqrt[-c^2]*E^ArcSinh[c*x])/c ] + 18*a*c^2*Log[1 + c^2*x^2] + 36*b*c^2*PolyLog[2, (c*E^ArcSinh[c*x])/Sqr t[-c^2]] + 36*b*c^2*PolyLog[2, (Sqrt[-c^2]*E^ArcSinh[c*x])/c] - 18*c^2*(2* (a + b*ArcSinh[c*x])*Log[1 - E^(2*ArcSinh[c*x])] + b*PolyLog[2, E^(2*ArcSi nh[c*x])]))/(12*d^3)
Result contains complex when optimal does not.
Time = 1.75 (sec) , antiderivative size = 280, normalized size of antiderivative = 1.30, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.708, Rules used = {6224, 27, 245, 209, 208, 6226, 209, 208, 6226, 208, 6214, 5984, 3042, 26, 4670, 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.
| \(\displaystyle \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (c^2 d x^2+d\right )^3} \, dx\) |
| \(\Big \downarrow \) 6224 |
| \(\displaystyle -3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{d^3 x \left (c^2 x^2+1\right )^3}dx+\frac {b c \int \frac {1}{x^2 \left (c^2 x^2+1\right )^{5/2}}dx}{2 d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^3}dx}{d^3}+\frac {b c \int \frac {1}{x^2 \left (c^2 x^2+1\right )^{5/2}}dx}{2 d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 245 |
| \(\displaystyle -\frac {3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^3}dx}{d^3}+\frac {b c \left (-4 c^2 \int \frac {1}{\left (c^2 x^2+1\right )^{5/2}}dx-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^3}dx}{d^3}+\frac {b c \left (-4 c^2 \left (\frac {2}{3} \int \frac {1}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -\frac {3 c^2 \int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^3}dx}{d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-4 c^2 \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 6226 |
| \(\displaystyle -\frac {3 c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx-\frac {1}{4} b c \int \frac {1}{\left (c^2 x^2+1\right )^{5/2}}dx+\frac {a+b \text {arcsinh}(c x)}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-4 c^2 \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 209 |
| \(\displaystyle -\frac {3 c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx-\frac {1}{4} b c \left (\frac {2}{3} \int \frac {1}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )+\frac {a+b \text {arcsinh}(c x)}{4 \left (c^2 x^2+1\right )^2}\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-4 c^2 \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -\frac {3 c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )^2}dx+\frac {a+b \text {arcsinh}(c x)}{4 \left (c^2 x^2+1\right )^2}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-4 c^2 \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 6226 |
| \(\displaystyle -\frac {3 c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}dx-\frac {1}{2} b c \int \frac {1}{\left (c^2 x^2+1\right )^{3/2}}dx+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}+\frac {a+b \text {arcsinh}(c x)}{4 \left (c^2 x^2+1\right )^2}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-4 c^2 \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 208 |
| \(\displaystyle -\frac {3 c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{x \left (c^2 x^2+1\right )}dx+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}+\frac {a+b \text {arcsinh}(c x)}{4 \left (c^2 x^2+1\right )^2}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-4 c^2 \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 6214 |
| \(\displaystyle -\frac {3 c^2 \left (\int \frac {a+b \text {arcsinh}(c x)}{c x \sqrt {c^2 x^2+1}}d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}+\frac {a+b \text {arcsinh}(c x)}{4 \left (c^2 x^2+1\right )^2}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-4 c^2 \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 5984 |
| \(\displaystyle -\frac {3 c^2 \left (2 \int (a+b \text {arcsinh}(c x)) \text {csch}(2 \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}+\frac {a+b \text {arcsinh}(c x)}{4 \left (c^2 x^2+1\right )^2}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-4 c^2 \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {3 c^2 \left (2 \int i (a+b \text {arcsinh}(c x)) \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}+\frac {a+b \text {arcsinh}(c x)}{4 \left (c^2 x^2+1\right )^2}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-4 c^2 \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {3 c^2 \left (2 i \int (a+b \text {arcsinh}(c x)) \csc (2 i \text {arcsinh}(c x))d\text {arcsinh}(c x)+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}+\frac {a+b \text {arcsinh}(c x)}{4 \left (c^2 x^2+1\right )^2}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-4 c^2 \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 4670 |
| \(\displaystyle -\frac {3 c^2 \left (2 i \left (\frac {1}{2} i b \int \log \left (1-e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)-\frac {1}{2} i b \int \log \left (1+e^{2 \text {arcsinh}(c x)}\right )d\text {arcsinh}(c x)+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}+\frac {a+b \text {arcsinh}(c x)}{4 \left (c^2 x^2+1\right )^2}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-4 c^2 \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 2715 |
| \(\displaystyle -\frac {3 c^2 \left (2 i \left (\frac {1}{4} i b \int e^{-2 \text {arcsinh}(c x)} \log \left (1-e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}-\frac {1}{4} i b \int e^{-2 \text {arcsinh}(c x)} \log \left (1+e^{2 \text {arcsinh}(c x)}\right )de^{2 \text {arcsinh}(c x)}+i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))\right )+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}+\frac {a+b \text {arcsinh}(c x)}{4 \left (c^2 x^2+1\right )^2}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-4 c^2 \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\) |
| \(\Big \downarrow \) 2838 |
| \(\displaystyle -\frac {3 c^2 \left (2 i \left (i \text {arctanh}\left (e^{2 \text {arcsinh}(c x)}\right ) (a+b \text {arcsinh}(c x))+\frac {1}{4} i b \operatorname {PolyLog}\left (2,-e^{2 \text {arcsinh}(c x)}\right )-\frac {1}{4} i b \operatorname {PolyLog}\left (2,e^{2 \text {arcsinh}(c x)}\right )\right )+\frac {a+b \text {arcsinh}(c x)}{2 \left (c^2 x^2+1\right )}+\frac {a+b \text {arcsinh}(c x)}{4 \left (c^2 x^2+1\right )^2}-\frac {b c x}{2 \sqrt {c^2 x^2+1}}-\frac {1}{4} b c \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )\right )}{d^3}-\frac {a+b \text {arcsinh}(c x)}{2 d^3 x^2 \left (c^2 x^2+1\right )^2}+\frac {b c \left (-4 c^2 \left (\frac {2 x}{3 \sqrt {c^2 x^2+1}}+\frac {x}{3 \left (c^2 x^2+1\right )^{3/2}}\right )-\frac {1}{x \left (c^2 x^2+1\right )^{3/2}}\right )}{2 d^3}\) |
Input:
Int[(a + b*ArcSinh[c*x])/(x^3*(d + c^2*d*x^2)^3),x]
Output:
(b*c*(-(1/(x*(1 + c^2*x^2)^(3/2))) - 4*c^2*(x/(3*(1 + c^2*x^2)^(3/2)) + (2 *x)/(3*Sqrt[1 + c^2*x^2]))))/(2*d^3) - (a + b*ArcSinh[c*x])/(2*d^3*x^2*(1 + c^2*x^2)^2) - (3*c^2*(-1/2*(b*c*x)/Sqrt[1 + c^2*x^2] - (b*c*(x/(3*(1 + c ^2*x^2)^(3/2)) + (2*x)/(3*Sqrt[1 + c^2*x^2])))/4 + (a + b*ArcSinh[c*x])/(4 *(1 + c^2*x^2)^2) + (a + b*ArcSinh[c*x])/(2*(1 + c^2*x^2)) + (2*I)*(I*(a + b*ArcSinh[c*x])*ArcTanh[E^(2*ArcSinh[c*x])] + (I/4)*b*PolyLog[2, -E^(2*Ar cSinh[c*x])] - (I/4)*b*PolyLog[2, E^(2*ArcSinh[c*x])])))/d^3
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_)^2)^(-3/2), x_Symbol] :> Simp[x/(a*Sqrt[a + b*x^2]), x] /; FreeQ[{a, b}, x]
Int[((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-x)*((a + b*x^2)^(p + 1) /(2*a*(p + 1))), x] + Simp[(2*p + 3)/(2*a*(p + 1)) Int[(a + b*x^2)^(p + 1 ), x], x] /; FreeQ[{a, b}, x] && ILtQ[p + 3/2, 0]
Int[(x_)^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> Simp[x^(m + 1)*((a + b*x^2)^(p + 1)/(a*(m + 1))), x] - Simp[b*((m + 2*(p + 1) + 1)/(a*(m + 1))) Int[x^(m + 2)*(a + b*x^2)^p, x], x] /; FreeQ[{a, b, m, p}, x] && ILtQ[Si mplify[(m + 1)/2 + p + 1], 0] && NeQ[m, -1]
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_.) + (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[Csch[(a_.) + (b_.)*(x_)]^(n_.)*((c_.) + (d_.)*(x_))^(m_.)*Sech[(a_.) + (b_.)*(x_)]^(n_.), x_Symbol] :> Simp[2^n Int[(c + d*x)^m*Csch[2*a + 2*b*x ]^n, x], x] /; FreeQ[{a, b, c, d}, x] && RationalQ[m] && IntegerQ[n]
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_.)/((x_)*((d_) + (e_.)*(x_)^2)), x_Symbol] :> Simp[1/d Subst[Int[(a + b*x)^n/(Cosh[x]*Sinh[x]), x], x, Ar cSinh[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]
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/(2*d*f*(p + 1))), x] + (Simp[(m + 2*p + 3)/(2*d*(p + 1 )) Int[(f*x)^m*(d + e*x^2)^(p + 1)*(a + b*ArcSinh[c*x])^n, x], x] + Simp[ b*c*(n/(2*f*(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] && LtQ[p, -1] && !G tQ[m, 1] && (IntegerQ[m] || IntegerQ[p] || EqQ[n, 1])
Time = 1.20 (sec) , antiderivative size = 344, normalized size of antiderivative = 1.59
| method | result | size |
| derivativedivides | \(c^{2} \left (\frac {a \left (-\frac {1}{2 c^{2} x^{2}}-3 \ln \left (x c \right )-\frac {1}{c^{2} x^{2}+1}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}\right )}{d^{3}}+\frac {b \left (-\frac {-8 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+8 c^{6} x^{6}+18 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-3 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+16 c^{4} x^{4}+27 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+6 \sqrt {c^{2} x^{2}+1}\, x c +8 c^{2} x^{2}+6 \,\operatorname {arcsinh}\left (x c \right )}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) x^{2} c^{2}}-3 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-3 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+3 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{3}}\right )\) | \(344\) |
| default | \(c^{2} \left (\frac {a \left (-\frac {1}{2 c^{2} x^{2}}-3 \ln \left (x c \right )-\frac {1}{c^{2} x^{2}+1}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{2}-\frac {1}{4 \left (c^{2} x^{2}+1\right )^{2}}\right )}{d^{3}}+\frac {b \left (-\frac {-8 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+8 c^{6} x^{6}+18 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-3 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+16 c^{4} x^{4}+27 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+6 \sqrt {c^{2} x^{2}+1}\, x c +8 c^{2} x^{2}+6 \,\operatorname {arcsinh}\left (x c \right )}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) x^{2} c^{2}}-3 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-3 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+3 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{3}}\right )\) | \(344\) |
| parts | \(\frac {a \left (-\frac {1}{2 x^{2}}-3 c^{2} \ln \left (x \right )+\frac {c^{4} \left (-\frac {2}{c^{2} \left (c^{2} x^{2}+1\right )}+\frac {3 \ln \left (c^{2} x^{2}+1\right )}{c^{2}}-\frac {1}{2 c^{2} \left (c^{2} x^{2}+1\right )^{2}}\right )}{2}\right )}{d^{3}}+\frac {b \,c^{2} \left (-\frac {-8 \sqrt {c^{2} x^{2}+1}\, x^{5} c^{5}+8 c^{6} x^{6}+18 \,\operatorname {arcsinh}\left (x c \right ) c^{4} x^{4}-3 \sqrt {c^{2} x^{2}+1}\, c^{3} x^{3}+16 c^{4} x^{4}+27 \,\operatorname {arcsinh}\left (x c \right ) c^{2} x^{2}+6 \sqrt {c^{2} x^{2}+1}\, x c +8 c^{2} x^{2}+6 \,\operatorname {arcsinh}\left (x c \right )}{12 \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right ) x^{2} c^{2}}-3 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1-x c -\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, x c +\sqrt {c^{2} x^{2}+1}\right )-3 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+x c +\sqrt {c^{2} x^{2}+1}\right )-3 \operatorname {polylog}\left (2, -x c -\sqrt {c^{2} x^{2}+1}\right )+3 \,\operatorname {arcsinh}\left (x c \right ) \ln \left (1+\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )+\frac {3 \operatorname {polylog}\left (2, -\left (x c +\sqrt {c^{2} x^{2}+1}\right )^{2}\right )}{2}\right )}{d^{3}}\) | \(356\) |
Input:
int((a+b*arcsinh(x*c))/x^3/(c^2*d*x^2+d)^3,x,method=_RETURNVERBOSE)
Output:
c^2*(a/d^3*(-1/2/c^2/x^2-3*ln(x*c)-1/(c^2*x^2+1)+3/2*ln(c^2*x^2+1)-1/4/(c^ 2*x^2+1)^2)+b/d^3*(-1/12/(c^4*x^4+2*c^2*x^2+1)/x^2/c^2*(-8*(c^2*x^2+1)^(1/ 2)*x^5*c^5+8*c^6*x^6+18*arcsinh(x*c)*c^4*x^4-3*(c^2*x^2+1)^(1/2)*c^3*x^3+1 6*c^4*x^4+27*arcsinh(x*c)*c^2*x^2+6*(c^2*x^2+1)^(1/2)*x*c+8*c^2*x^2+6*arcs inh(x*c))-3*arcsinh(x*c)*ln(1-x*c-(c^2*x^2+1)^(1/2))-3*polylog(2,x*c+(c^2* x^2+1)^(1/2))-3*arcsinh(x*c)*ln(1+x*c+(c^2*x^2+1)^(1/2))-3*polylog(2,-x*c- (c^2*x^2+1)^(1/2))+3*arcsinh(x*c)*ln(1+(x*c+(c^2*x^2+1)^(1/2))^2)+3/2*poly log(2,-(x*c+(c^2*x^2+1)^(1/2))^2)))
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{3}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d)^3,x, algorithm="fricas")
Output:
integral((b*arcsinh(c*x) + a)/(c^6*d^3*x^9 + 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 + d^3*x^3), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\frac {\int \frac {a}{c^{6} x^{9} + 3 c^{4} x^{7} + 3 c^{2} x^{5} + x^{3}}\, dx + \int \frac {b \operatorname {asinh}{\left (c x \right )}}{c^{6} x^{9} + 3 c^{4} x^{7} + 3 c^{2} x^{5} + x^{3}}\, dx}{d^{3}} \] Input:
integrate((a+b*asinh(c*x))/x**3/(c**2*d*x**2+d)**3,x)
Output:
(Integral(a/(c**6*x**9 + 3*c**4*x**7 + 3*c**2*x**5 + x**3), x) + Integral( b*asinh(c*x)/(c**6*x**9 + 3*c**4*x**7 + 3*c**2*x**5 + x**3), x))/d**3
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{3}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d)^3,x, algorithm="maxima")
Output:
-1/4*a*((6*c^4*x^4 + 9*c^2*x^2 + 2)/(c^4*d^3*x^6 + 2*c^2*d^3*x^4 + d^3*x^2 ) - 6*c^2*log(c^2*x^2 + 1)/d^3 + 12*c^2*log(x)/d^3) + b*integrate(log(c*x + sqrt(c^2*x^2 + 1))/(c^6*d^3*x^9 + 3*c^4*d^3*x^7 + 3*c^2*d^3*x^5 + d^3*x^ 3), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\int { \frac {b \operatorname {arsinh}\left (c x\right ) + a}{{\left (c^{2} d x^{2} + d\right )}^{3} x^{3}} \,d x } \] Input:
integrate((a+b*arcsinh(c*x))/x^3/(c^2*d*x^2+d)^3,x, algorithm="giac")
Output:
integrate((b*arcsinh(c*x) + a)/((c^2*d*x^2 + d)^3*x^3), x)
Timed out. \[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (c\,x\right )}{x^3\,{\left (d\,c^2\,x^2+d\right )}^3} \,d x \] Input:
int((a + b*asinh(c*x))/(x^3*(d + c^2*d*x^2)^3),x)
Output:
int((a + b*asinh(c*x))/(x^3*(d + c^2*d*x^2)^3), x)
\[ \int \frac {a+b \text {arcsinh}(c x)}{x^3 \left (d+c^2 d x^2\right )^3} \, dx=\frac {4 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{9}+3 c^{4} x^{7}+3 c^{2} x^{5}+x^{3}}d x \right ) b \,c^{4} x^{6}+8 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{9}+3 c^{4} x^{7}+3 c^{2} x^{5}+x^{3}}d x \right ) b \,c^{2} x^{4}+4 \left (\int \frac {\mathit {asinh} \left (c x \right )}{c^{6} x^{9}+3 c^{4} x^{7}+3 c^{2} x^{5}+x^{3}}d x \right ) b \,x^{2}+6 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a \,c^{6} x^{6}+12 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a \,c^{4} x^{4}+6 \,\mathrm {log}\left (c^{2} x^{2}+1\right ) a \,c^{2} x^{2}-12 \,\mathrm {log}\left (x \right ) a \,c^{6} x^{6}-24 \,\mathrm {log}\left (x \right ) a \,c^{4} x^{4}-12 \,\mathrm {log}\left (x \right ) a \,c^{2} x^{2}+3 a \,c^{6} x^{6}-6 a \,c^{2} x^{2}-2 a}{4 d^{3} x^{2} \left (c^{4} x^{4}+2 c^{2} x^{2}+1\right )} \] Input:
int((a+b*asinh(c*x))/x^3/(c^2*d*x^2+d)^3,x)
Output:
(4*int(asinh(c*x)/(c**6*x**9 + 3*c**4*x**7 + 3*c**2*x**5 + x**3),x)*b*c**4 *x**6 + 8*int(asinh(c*x)/(c**6*x**9 + 3*c**4*x**7 + 3*c**2*x**5 + x**3),x) *b*c**2*x**4 + 4*int(asinh(c*x)/(c**6*x**9 + 3*c**4*x**7 + 3*c**2*x**5 + x **3),x)*b*x**2 + 6*log(c**2*x**2 + 1)*a*c**6*x**6 + 12*log(c**2*x**2 + 1)* a*c**4*x**4 + 6*log(c**2*x**2 + 1)*a*c**2*x**2 - 12*log(x)*a*c**6*x**6 - 2 4*log(x)*a*c**4*x**4 - 12*log(x)*a*c**2*x**2 + 3*a*c**6*x**6 - 6*a*c**2*x* *2 - 2*a)/(4*d**3*x**2*(c**4*x**4 + 2*c**2*x**2 + 1))