Integrand size = 27, antiderivative size = 324 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {3 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c \text {arctanh}(c x)}{d^2}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2}+\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 b^2 c \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d^2}+\frac {3 b^2 c \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d^2} \]
[Out]
Time = 0.36 (sec) , antiderivative size = 324, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.519, Rules used = {4789, 4747, 4749, 4266, 2611, 2320, 6724, 4767, 212, 4793, 4803, 4268, 2317, 2438} \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {3 i c \arctan \left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))^2}{d^2}-\frac {4 b c \text {arctanh}\left (e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^2}-\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 i b c \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^2}-\frac {3 i b c \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right ) (a+b \arcsin (c x))}{d^2}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 b^2 c \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d^2}+\frac {3 b^2 c \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c \text {arctanh}(c x)}{d^2} \]
[In]
[Out]
Rule 212
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4266
Rule 4268
Rule 4747
Rule 4749
Rule 4767
Rule 4789
Rule 4793
Rule 4803
Rule 6724
Rubi steps \begin{align*} \text {integral}& = -\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\left (3 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{\left (d-c^2 d x^2\right )^2} \, dx+\frac {(2 b c) \int \frac {a+b \arcsin (c x)}{x \left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2} \\ & = \frac {2 b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}+\frac {(2 b c) \int \frac {a+b \arcsin (c x)}{x \sqrt {1-c^2 x^2}} \, dx}{d^2}-\frac {\left (2 b^2 c^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{d^2}-\frac {\left (3 b c^3\right ) \int \frac {x (a+b \arcsin (c x))}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^2}+\frac {\left (3 c^2\right ) \int \frac {(a+b \arcsin (c x))^2}{d-c^2 d x^2} \, dx}{2 d} \\ & = -\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {2 b^2 c \text {arctanh}(c x)}{d^2}+\frac {(3 c) \text {Subst}\left (\int (a+b x)^2 \sec (x) \, dx,x,\arcsin (c x)\right )}{2 d^2}+\frac {(2 b c) \text {Subst}(\int (a+b x) \csc (x) \, dx,x,\arcsin (c x))}{d^2}+\frac {\left (3 b^2 c^2\right ) \int \frac {1}{1-c^2 x^2} \, dx}{d^2} \\ & = -\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {3 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c \text {arctanh}(c x)}{d^2}-\frac {(3 b c) \text {Subst}\left (\int (a+b x) \log \left (1-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2}+\frac {(3 b c) \text {Subst}\left (\int (a+b x) \log \left (1+i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2}-\frac {\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1-e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2}+\frac {\left (2 b^2 c\right ) \text {Subst}\left (\int \log \left (1+e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2} \\ & = -\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {3 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c \text {arctanh}(c x)}{d^2}+\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}+\frac {\left (2 i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1-x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2}-\frac {\left (2 i b^2 c\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2}-\frac {\left (3 i b^2 c\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,-i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2}+\frac {\left (3 i b^2 c\right ) \text {Subst}\left (\int \operatorname {PolyLog}\left (2,i e^{i x}\right ) \, dx,x,\arcsin (c x)\right )}{d^2} \\ & = -\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {3 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c \text {arctanh}(c x)}{d^2}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2}+\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2}-\frac {\left (3 b^2 c\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,-i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2}+\frac {\left (3 b^2 c\right ) \text {Subst}\left (\int \frac {\operatorname {PolyLog}(2,i x)}{x} \, dx,x,e^{i \arcsin (c x)}\right )}{d^2} \\ & = -\frac {b c (a+b \arcsin (c x))}{d^2 \sqrt {1-c^2 x^2}}-\frac {(a+b \arcsin (c x))^2}{d^2 x \left (1-c^2 x^2\right )}+\frac {3 c^2 x (a+b \arcsin (c x))^2}{2 d^2 \left (1-c^2 x^2\right )}-\frac {3 i c (a+b \arcsin (c x))^2 \arctan \left (e^{i \arcsin (c x)}\right )}{d^2}-\frac {4 b c (a+b \arcsin (c x)) \text {arctanh}\left (e^{i \arcsin (c x)}\right )}{d^2}+\frac {b^2 c \text {arctanh}(c x)}{d^2}+\frac {2 i b^2 c \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )}{d^2}+\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 i b c (a+b \arcsin (c x)) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )}{d^2}-\frac {2 i b^2 c \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )}{d^2}-\frac {3 b^2 c \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )}{d^2}+\frac {3 b^2 c \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )}{d^2} \\ \end{align*}
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1161\) vs. \(2(324)=648\).
Time = 10.44 (sec) , antiderivative size = 1161, normalized size of antiderivative = 3.58 \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=-\frac {a^2}{d^2 x}-\frac {a^2 c^2 x}{2 d^2 \left (-1+c^2 x^2\right )}-\frac {3 a^2 c \log (1-c x)}{4 d^2}+\frac {3 a^2 c \log (1+c x)}{4 d^2}+\frac {2 a b c \left (\frac {\sqrt {1-c^2 x^2}-\arcsin (c x)}{4 (-1+c x)}-\frac {\arcsin (c x)}{c x}-\frac {\sqrt {1-c^2 x^2}+\arcsin (c x)}{4 (1+c x)}-\text {arctanh}\left (\sqrt {1-c^2 x^2}\right )-\frac {3}{4} \left (\frac {3}{2} i \pi \arcsin (c x)-\frac {1}{2} i \arcsin (c x)^2+2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )-\pi \log \left (1+i e^{i \arcsin (c x)}\right )+2 \arcsin (c x) \log \left (1+i e^{i \arcsin (c x)}\right )-2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )+\pi \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-2 i \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )\right )+\frac {3}{4} \left (\frac {1}{2} i \pi \arcsin (c x)-\frac {1}{2} i \arcsin (c x)^2+2 \pi \log \left (1+e^{-i \arcsin (c x)}\right )+\pi \log \left (1-i e^{i \arcsin (c x)}\right )+2 \arcsin (c x) \log \left (1-i e^{i \arcsin (c x)}\right )-2 \pi \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )\right )-\pi \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-2 i \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )\right )\right )}{d^2}+\frac {b^2 c \left (-4 \arcsin (c x)-2 \arcsin (c x)^2 \cot \left (\frac {1}{2} \arcsin (c x)\right )+8 \arcsin (c x) \log \left (1-e^{i \arcsin (c x)}\right )+6 \arcsin (c x)^2 \log \left (1-i e^{i \arcsin (c x)}\right )+6 \pi \arcsin (c x) \log \left (\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \arcsin (c x)} \left (1-i e^{i \arcsin (c x)}\right )\right )-6 \arcsin (c x)^2 \log \left (1+i e^{i \arcsin (c x)}\right )-6 \arcsin (c x)^2 \log \left (\left (\frac {1}{2}+\frac {i}{2}\right ) e^{-\frac {1}{2} i \arcsin (c x)} \left (-i+e^{i \arcsin (c x)}\right )\right )+6 \pi \arcsin (c x) \log \left (-\frac {1}{2} \sqrt [4]{-1} e^{-\frac {1}{2} i \arcsin (c x)} \left (-i+e^{i \arcsin (c x)}\right )\right )-8 \arcsin (c x) \log \left (1+e^{i \arcsin (c x)}\right )+6 \arcsin (c x)^2 \log \left (\frac {1}{2} e^{-\frac {1}{2} i \arcsin (c x)} \left ((1+i)+(1-i) e^{i \arcsin (c x)}\right )\right )-6 \pi \arcsin (c x) \log \left (-\cos \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )-4 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+6 \arcsin (c x)^2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )+4 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-6 \arcsin (c x)^2 \log \left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )-6 \pi \arcsin (c x) \log \left (\sin \left (\frac {1}{4} (\pi +2 \arcsin (c x))\right )\right )+8 i \operatorname {PolyLog}\left (2,-e^{i \arcsin (c x)}\right )+12 i \arcsin (c x) \operatorname {PolyLog}\left (2,-i e^{i \arcsin (c x)}\right )-12 i \arcsin (c x) \operatorname {PolyLog}\left (2,i e^{i \arcsin (c x)}\right )-8 i \operatorname {PolyLog}\left (2,e^{i \arcsin (c x)}\right )-12 \operatorname {PolyLog}\left (3,-i e^{i \arcsin (c x)}\right )+12 \operatorname {PolyLog}\left (3,i e^{i \arcsin (c x)}\right )+\frac {\arcsin (c x)^2}{\left (\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^2}-\frac {4 \arcsin (c x) \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\cos \left (\frac {1}{2} \arcsin (c x)\right )-\sin \left (\frac {1}{2} \arcsin (c x)\right )}-\frac {\arcsin (c x)^2}{\left (\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )\right )^2}+\frac {4 \arcsin (c x) \sin \left (\frac {1}{2} \arcsin (c x)\right )}{\cos \left (\frac {1}{2} \arcsin (c x)\right )+\sin \left (\frac {1}{2} \arcsin (c x)\right )}-2 \arcsin (c x)^2 \tan \left (\frac {1}{2} \arcsin (c x)\right )\right )}{4 d^2} \]
[In]
[Out]
Time = 0.44 (sec) , antiderivative size = 605, normalized size of antiderivative = 1.87
method | result | size |
derivativedivides | \(c \left (\frac {a^{2} \left (-\frac {1}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {3 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\) | \(605\) |
default | \(c \left (\frac {a^{2} \left (-\frac {1}{c x}-\frac {1}{4 \left (c x -1\right )}-\frac {3 \ln \left (c x -1\right )}{4}-\frac {1}{4 \left (c x +1\right )}+\frac {3 \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} \left (-\frac {\left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b \left (-\frac {3 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\right )\) | \(605\) |
parts | \(\frac {a^{2} \left (-\frac {1}{x}-\frac {c}{4 \left (c x -1\right )}-\frac {3 c \ln \left (c x -1\right )}{4}-\frac {c}{4 \left (c x +1\right )}+\frac {3 c \ln \left (c x +1\right )}{4}\right )}{d^{2}}+\frac {b^{2} c \left (-\frac {\left (3 c^{2} x^{2} \arcsin \left (c x \right )-2 c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )\right ) \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}-\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )-3 \operatorname {polylog}\left (3, -i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+\frac {3 \arcsin \left (c x \right )^{2} \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-3 i \arcsin \left (c x \right ) \operatorname {polylog}\left (2, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+3 \operatorname {polylog}\left (3, i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )+2 i \operatorname {dilog}\left (i c x +\sqrt {-c^{2} x^{2}+1}\right )+2 i \operatorname {dilog}\left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 i \arctan \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )-2 \arcsin \left (c x \right ) \ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{d^{2}}+\frac {2 a b c \left (-\frac {3 c^{2} x^{2} \arcsin \left (c x \right )-c x \sqrt {-c^{2} x^{2}+1}-2 \arcsin \left (c x \right )}{2 c x \left (c^{2} x^{2}-1\right )}+\ln \left (i c x +\sqrt {-c^{2} x^{2}+1}-1\right )-\ln \left (1+i c x +\sqrt {-c^{2} x^{2}+1}\right )-\frac {3 \arcsin \left (c x \right ) \ln \left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 \arcsin \left (c x \right ) \ln \left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}+\frac {3 i \operatorname {dilog}\left (1+i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}-\frac {3 i \operatorname {dilog}\left (1-i \left (i c x +\sqrt {-c^{2} x^{2}+1}\right )\right )}{2}\right )}{d^{2}}\) | \(606\) |
[In]
[Out]
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\frac {\int \frac {a^{2}}{c^{4} x^{6} - 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{4} x^{6} - 2 c^{2} x^{4} + x^{2}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{4} x^{6} - 2 c^{2} x^{4} + x^{2}}\, dx}{d^{2}} \]
[In]
[Out]
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{2}} \,d x } \]
[In]
[Out]
\[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\int { \frac {{\left (b \arcsin \left (c x\right ) + a\right )}^{2}}{{\left (c^{2} d x^{2} - d\right )}^{2} x^{2}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(a+b \arcsin (c x))^2}{x^2 \left (d-c^2 d x^2\right )^2} \, dx=\int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x^2\,{\left (d-c^2\,d\,x^2\right )}^2} \,d x \]
[In]
[Out]