Optimal. Leaf size=577 \[ \frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x (a+b \text {ArcSin}(c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {ArcSin}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {(a+b \text {ArcSin}(c x))^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 i b \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x))^2 \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {7 i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {7 i b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (3,-e^{i \text {ArcSin}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \text {PolyLog}\left (3,e^{i \text {ArcSin}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}} \]
[Out]
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Rubi [A]
time = 0.52, antiderivative size = 577, normalized size of antiderivative = 1.00, number of steps
used = 24, number of rules used = 12, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.414, Rules used = {4793, 4803,
4268, 2611, 2320, 6724, 4749, 4266, 2317, 2438, 4747, 267} \begin {gather*} \frac {14 i b \sqrt {1-c^2 x^2} \text {ArcTan}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} \text {Li}_2\left (-e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} \text {Li}_2\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x (a+b \text {ArcSin}(c x))}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {ArcSin}(c x))^2}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \tanh ^{-1}\left (e^{i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {(a+b \text {ArcSin}(c x))^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}-\frac {7 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {7 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \text {ArcSin}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (-e^{i \text {ArcSin}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (e^{i \text {ArcSin}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 2317
Rule 2320
Rule 2438
Rule 2611
Rule 4266
Rule 4268
Rule 4747
Rule 4749
Rule 4793
Rule 4803
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^{5/2}} \, dx &=\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^{3/2}} \, dx}{d}-\frac {\left (2 b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=-\frac {b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt {d-c^2 d x^2}} \, dx}{d^2}-\frac {\left (b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b c \sqrt {1-c^2 x^2}\right ) \int \frac {a+b \sin ^{-1}(c x)}{1-c^2 x^2} \, dx}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 c^2 \sqrt {1-c^2 x^2}\right ) \int \frac {x}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \sqrt {1-c^2 x^2}} \, dx}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int (a+b x)^2 \csc (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1-i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+i e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1-i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+i x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {7 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {7 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 i b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {7 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {7 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {\left (2 b^2 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ &=\frac {b^2}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {b c x \left (a+b \sin ^{-1}(c x)\right )}{3 d^2 \sqrt {1-c^2 x^2} \sqrt {d-c^2 d x^2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{3 d \left (d-c^2 d x^2\right )^{3/2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{d^2 \sqrt {d-c^2 d x^2}}+\frac {14 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \tan ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {7 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (-i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}+\frac {7 i b^2 \sqrt {1-c^2 x^2} \text {Li}_2\left (i e^{i \sin ^{-1}(c x)}\right )}{3 d^2 \sqrt {d-c^2 d x^2}}-\frac {2 i b \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}-\frac {2 b^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (-e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}+\frac {2 b^2 \sqrt {1-c^2 x^2} \text {Li}_3\left (e^{i \sin ^{-1}(c x)}\right )}{d^2 \sqrt {d-c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 7.58, size = 935, normalized size = 1.62 \begin {gather*} \sqrt {-d \left (-1+c^2 x^2\right )} \left (\frac {a^2}{3 d^3 \left (-1+c^2 x^2\right )^2}-\frac {a^2}{d^3 \left (-1+c^2 x^2\right )}\right )+\frac {a^2 \log (c x)}{d^{5/2}}-\frac {a^2 \log \left (d+\sqrt {d} \sqrt {-d \left (-1+c^2 x^2\right )}\right )}{d^{5/2}}+\frac {b^2 \left (1-c^2 x^2\right )^{3/2} \left (4-\frac {(-2+\text {ArcSin}(c x)) \text {ArcSin}(c x)}{-1+c x}+14 \text {ArcSin}(c x)^2+12 \text {ArcSin}(c x)^2 \left (\log \left (1-e^{i \text {ArcSin}(c x)}\right )-\log \left (1+e^{i \text {ArcSin}(c x)}\right )\right )-28 \left (\text {ArcSin}(c x) \left (\log \left (1-i e^{i \text {ArcSin}(c x)}\right )-\log \left (1+i e^{i \text {ArcSin}(c x)}\right )\right )+i \left (\text {PolyLog}\left (2,-i e^{i \text {ArcSin}(c x)}\right )-\text {PolyLog}\left (2,i e^{i \text {ArcSin}(c x)}\right )\right )\right )+24 i \text {ArcSin}(c x) \left (\text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-\text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )\right )+24 \left (-\text {PolyLog}\left (3,-e^{i \text {ArcSin}(c x)}\right )+\text {PolyLog}\left (3,e^{i \text {ArcSin}(c x)}\right )\right )+\frac {2 \text {ArcSin}(c x)^2 \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^3}+\frac {2 \left (2+7 \text {ArcSin}(c x)^2\right ) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}-\frac {2 \text {ArcSin}(c x)^2 \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^3}+\frac {\text {ArcSin}(c x) (2+\text {ArcSin}(c x))}{\left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )^2}-\frac {2 \left (2+7 \text {ArcSin}(c x)^2\right ) \sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}{\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )}\right )}{12 d \left (d \left (1-c^2 x^2\right )\right )^{3/2}}+\frac {a b \left (20 \text {ArcSin}(c x)+12 \text {ArcSin}(c x) \cos (2 \text {ArcSin}(c x))+18 \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1-e^{i \text {ArcSin}(c x)}\right )+6 \text {ArcSin}(c x) \cos (3 \text {ArcSin}(c x)) \log \left (1-e^{i \text {ArcSin}(c x)}\right )-18 \sqrt {1-c^2 x^2} \text {ArcSin}(c x) \log \left (1+e^{i \text {ArcSin}(c x)}\right )-6 \text {ArcSin}(c x) \cos (3 \text {ArcSin}(c x)) \log \left (1+e^{i \text {ArcSin}(c x)}\right )+21 \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+7 \cos (3 \text {ArcSin}(c x)) \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )-\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-21 \sqrt {1-c^2 x^2} \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )-7 \cos (3 \text {ArcSin}(c x)) \log \left (\cos \left (\frac {1}{2} \text {ArcSin}(c x)\right )+\sin \left (\frac {1}{2} \text {ArcSin}(c x)\right )\right )+24 i \left (1-c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,-e^{i \text {ArcSin}(c x)}\right )-24 i \left (1-c^2 x^2\right )^{3/2} \text {PolyLog}\left (2,e^{i \text {ArcSin}(c x)}\right )-2 \sin (2 \text {ArcSin}(c x))\right )}{12 d \left (d \left (1-c^2 x^2\right )\right )^{3/2}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice
the leaf count of optimal. 1372 vs. \(2 (580 ) = 1160\).
time = 0.33, size = 1373, normalized size = 2.38
method | result | size |
default | \(\text {Expression too large to display}\) | \(1373\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{x\,{\left (d-c^2\,d\,x^2\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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