Optimal. Leaf size=403 \[ \frac {b^2 c^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c (a+b \text {ArcSin}(c x))}{d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x (a+b \text {ArcSin}(c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c^3 x (a+b \text {ArcSin}(c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 (a+b \text {ArcSin}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {(a+b \text {ArcSin}(c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 (a+b \text {ArcSin}(c x))^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {6 c^2 (a+b \text {ArcSin}(c x))^2 \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right )}{d^3}+\frac {b^2 c^2 \log (x)}{d^3}-\frac {7 b^2 c^2 \log \left (1-c^2 x^2\right )}{6 d^3}+\frac {3 i b c^2 (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )}{d^3}-\frac {3 i b c^2 (a+b \text {ArcSin}(c x)) \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )}{d^3}-\frac {3 b^2 c^2 \text {PolyLog}\left (3,-e^{2 i \text {ArcSin}(c x)}\right )}{2 d^3}+\frac {3 b^2 c^2 \text {PolyLog}\left (3,e^{2 i \text {ArcSin}(c x)}\right )}{2 d^3} \]
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Rubi [A]
time = 0.55, antiderivative size = 403, normalized size of antiderivative = 1.00, number of steps
used = 23, number of rules used = 19, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.704, Rules used = {4789, 4793,
4769, 4504, 4268, 2611, 2320, 6724, 4745, 266, 4747, 267, 277, 198, 197, 4779, 12, 1265, 907}
\begin {gather*} \frac {3 i b c^2 \text {Li}_2\left (-e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d^3}-\frac {3 i b c^2 \text {Li}_2\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))}{d^3}+\frac {3 c^2 (a+b \text {ArcSin}(c x))^2}{2 d^3 \left (1-c^2 x^2\right )}+\frac {3 c^2 (a+b \text {ArcSin}(c x))^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b c (a+b \text {ArcSin}(c x))}{d^3 x \left (1-c^2 x^2\right )^{3/2}}-\frac {(a+b \text {ArcSin}(c x))^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}-\frac {6 c^2 \tanh ^{-1}\left (e^{2 i \text {ArcSin}(c x)}\right ) (a+b \text {ArcSin}(c x))^2}{d^3}-\frac {4 b c^3 x (a+b \text {ArcSin}(c x))}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {5 b c^3 x (a+b \text {ArcSin}(c x))}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {3 b^2 c^2 \text {Li}_3\left (-e^{2 i \text {ArcSin}(c x)}\right )}{2 d^3}+\frac {3 b^2 c^2 \text {Li}_3\left (e^{2 i \text {ArcSin}(c x)}\right )}{2 d^3}+\frac {b^2 c^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {7 b^2 c^2 \log \left (1-c^2 x^2\right )}{6 d^3}+\frac {b^2 c^2 \log (x)}{d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 197
Rule 198
Rule 266
Rule 267
Rule 277
Rule 907
Rule 1265
Rule 2320
Rule 2611
Rule 4268
Rule 4504
Rule 4745
Rule 4747
Rule 4769
Rule 4779
Rule 4789
Rule 4793
Rule 6724
Rubi steps
\begin {align*} \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x^3 \left (d-c^2 d x^2\right )^3} \, dx &=-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\left (3 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^3} \, dx+\frac {(b c) \int \frac {a+b \sin ^{-1}(c x)}{x^2 \left (1-c^2 x^2\right )^{5/2}} \, dx}{d^3}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {4 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {8 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}-\frac {\left (b^2 c^2\right ) \int \frac {-3+12 c^2 x^2-8 c^4 x^4}{3 x \left (1-c^2 x^2\right )^2} \, dx}{d^3}-\frac {\left (3 b c^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{2 d^3}+\frac {\left (3 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )^2} \, dx}{d}\\ &=-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {8 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {\left (b^2 c^2\right ) \int \frac {-3+12 c^2 x^2-8 c^4 x^4}{x \left (1-c^2 x^2\right )^2} \, dx}{3 d^3}-\frac {\left (b c^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^3}-\frac {\left (3 b c^3\right ) \int \frac {a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{d^3}+\frac {\left (b^2 c^4\right ) \int \frac {x}{\left (1-c^2 x^2\right )^2} \, dx}{2 d^3}+\frac {\left (3 c^2\right ) \int \frac {\left (a+b \sin ^{-1}(c x)\right )^2}{x \left (d-c^2 d x^2\right )} \, dx}{d^2}\\ &=\frac {b^2 c^2}{4 d^3 \left (1-c^2 x^2\right )}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}+\frac {\left (3 c^2\right ) \text {Subst}\left (\int (a+b x)^2 \csc (x) \sec (x) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}-\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \frac {-3+12 c^2 x-8 c^4 x^2}{x \left (1-c^2 x\right )^2} \, dx,x,x^2\right )}{6 d^3}+\frac {\left (b^2 c^4\right ) \int \frac {x}{1-c^2 x^2} \, dx}{d^3}+\frac {\left (3 b^2 c^4\right ) \int \frac {x}{1-c^2 x^2} \, dx}{d^3}\\ &=\frac {b^2 c^2}{4 d^3 \left (1-c^2 x^2\right )}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {2 b^2 c^2 \log \left (1-c^2 x^2\right )}{d^3}+\frac {\left (6 c^2\right ) \text {Subst}\left (\int (a+b x)^2 \csc (2 x) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}-\frac {\left (b^2 c^2\right ) \text {Subst}\left (\int \left (-\frac {3}{x}+\frac {c^2}{\left (-1+c^2 x\right )^2}-\frac {5 c^2}{-1+c^2 x}\right ) \, dx,x,x^2\right )}{6 d^3}\\ &=\frac {b^2 c^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {6 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}+\frac {b^2 c^2 \log (x)}{d^3}-\frac {7 b^2 c^2 \log \left (1-c^2 x^2\right )}{6 d^3}-\frac {\left (6 b c^2\right ) \text {Subst}\left (\int (a+b x) \log \left (1-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}+\frac {\left (6 b c^2\right ) \text {Subst}\left (\int (a+b x) \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}\\ &=\frac {b^2 c^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {6 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}+\frac {b^2 c^2 \log (x)}{d^3}-\frac {7 b^2 c^2 \log \left (1-c^2 x^2\right )}{6 d^3}+\frac {3 i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac {3 i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac {\left (3 i b^2 c^2\right ) \text {Subst}\left (\int \text {Li}_2\left (-e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}+\frac {\left (3 i b^2 c^2\right ) \text {Subst}\left (\int \text {Li}_2\left (e^{2 i x}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{d^3}\\ &=\frac {b^2 c^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {6 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}+\frac {b^2 c^2 \log (x)}{d^3}-\frac {7 b^2 c^2 \log \left (1-c^2 x^2\right )}{6 d^3}+\frac {3 i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac {3 i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac {\left (3 b^2 c^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(-x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}+\frac {\left (3 b^2 c^2\right ) \text {Subst}\left (\int \frac {\text {Li}_2(x)}{x} \, dx,x,e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}\\ &=\frac {b^2 c^2}{12 d^3 \left (1-c^2 x^2\right )}-\frac {b c \left (a+b \sin ^{-1}(c x)\right )}{d^3 x \left (1-c^2 x^2\right )^{3/2}}+\frac {5 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{6 d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {4 b c^3 x \left (a+b \sin ^{-1}(c x)\right )}{3 d^3 \sqrt {1-c^2 x^2}}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 x^2 \left (1-c^2 x^2\right )^2}+\frac {3 c^2 \left (a+b \sin ^{-1}(c x)\right )^2}{2 d^3 \left (1-c^2 x^2\right )}-\frac {6 c^2 \left (a+b \sin ^{-1}(c x)\right )^2 \tanh ^{-1}\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}+\frac {b^2 c^2 \log (x)}{d^3}-\frac {7 b^2 c^2 \log \left (1-c^2 x^2\right )}{6 d^3}+\frac {3 i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (-e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac {3 i b c^2 \left (a+b \sin ^{-1}(c x)\right ) \text {Li}_2\left (e^{2 i \sin ^{-1}(c x)}\right )}{d^3}-\frac {3 b^2 c^2 \text {Li}_3\left (-e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}+\frac {3 b^2 c^2 \text {Li}_3\left (e^{2 i \sin ^{-1}(c x)}\right )}{2 d^3}\\ \end {align*}
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Mathematica [A]
time = 4.40, size = 569, normalized size = 1.41 \begin {gather*} -\frac {\frac {6 a^2}{x^2}-\frac {3 a^2 c^2}{\left (-1+c^2 x^2\right )^2}+\frac {12 a^2 c^2}{-1+c^2 x^2}-36 a^2 c^2 \log (x)+18 a^2 c^2 \log \left (1-c^2 x^2\right )+2 a b c^2 \left (\frac {c x}{\left (1-c^2 x^2\right )^{3/2}}+\frac {14 c x}{\sqrt {1-c^2 x^2}}+\frac {6 \sqrt {1-c^2 x^2}}{c x}+\frac {6 \text {ArcSin}(c x)}{c^2 x^2}-\frac {3 \text {ArcSin}(c x)}{\left (-1+c^2 x^2\right )^2}+\frac {12 \text {ArcSin}(c x)}{-1+c^2 x^2}-36 \text {ArcSin}(c x) \log \left (1-e^{2 i \text {ArcSin}(c x)}\right )+36 \text {ArcSin}(c x) \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )-18 i \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )+18 i \text {PolyLog}\left (2,e^{2 i \text {ArcSin}(c x)}\right )\right )+12 b^2 c^2 \left (-3 i \text {ArcSin}(c x) \text {PolyLog}\left (2,e^{-2 i \text {ArcSin}(c x)}\right )-3 i \text {ArcSin}(c x) \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(c x)}\right )+\frac {1}{24} \left (3 i \pi ^3+\frac {2}{-1+c^2 x^2}+\frac {4 c x \text {ArcSin}(c x)}{\left (1-c^2 x^2\right )^{3/2}}+\frac {56 c x \text {ArcSin}(c x)}{\sqrt {1-c^2 x^2}}+\frac {24 \sqrt {1-c^2 x^2} \text {ArcSin}(c x)}{c x}+\frac {12 \text {ArcSin}(c x)^2}{c^2 x^2}-\frac {6 \text {ArcSin}(c x)^2}{\left (-1+c^2 x^2\right )^2}+\frac {24 \text {ArcSin}(c x)^2}{-1+c^2 x^2}-48 i \text {ArcSin}(c x)^3-72 \text {ArcSin}(c x)^2 \log \left (1-e^{-2 i \text {ArcSin}(c x)}\right )+72 \text {ArcSin}(c x)^2 \log \left (1+e^{2 i \text {ArcSin}(c x)}\right )-24 \log (c x)+28 \log \left (1-c^2 x^2\right )-36 \text {PolyLog}\left (3,e^{-2 i \text {ArcSin}(c x)}\right )+36 \text {PolyLog}\left (3,-e^{2 i \text {ArcSin}(c x)}\right )\right )\right )}{12 d^3} \end {gather*}
Antiderivative was successfully verified.
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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. 1466 vs. \(2 (427 ) = 854\).
time = 0.44, size = 1467, normalized size = 3.64
method | result | size |
derivativedivides | \(\text {Expression too large to display}\) | \(1467\) |
default | \(\text {Expression too large to display}\) | \(1467\) |
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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} - \frac {\int \frac {a^{2}}{c^{6} x^{9} - 3 c^{4} x^{7} + 3 c^{2} x^{5} - x^{3}}\, dx + \int \frac {b^{2} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{9} - 3 c^{4} x^{7} + 3 c^{2} x^{5} - x^{3}}\, dx + \int \frac {2 a b \operatorname {asin}{\left (c x \right )}}{c^{6} x^{9} - 3 c^{4} x^{7} + 3 c^{2} x^{5} - x^{3}}\, dx}{d^{3}} \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^3\,{\left (d-c^2\,d\,x^2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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