Optimal. Leaf size=390 \[ \frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}+\frac {x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {\text {ArcSin}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {4 \text {ArcSin}(a x)}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \text {ArcSin}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \text {ArcSin}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \text {ArcSin}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {16 \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \log \left (1+e^{2 i \text {ArcSin}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}} \]
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Rubi [A]
time = 0.24, antiderivative size = 390, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 10, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.454, Rules used = {4747, 4745,
4765, 3800, 2221, 2317, 2438, 4767, 197, 198} \begin {gather*} -\frac {8 i \sqrt {1-a^2 x^2} \text {Li}_2\left (-e^{2 i \text {ArcSin}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 x \text {ArcSin}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \text {ArcSin}(a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {4 \text {ArcSin}(a x)}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}-\frac {\text {ArcSin}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}+\frac {16 \sqrt {1-a^2 x^2} \text {ArcSin}(a x) \log \left (1+e^{2 i \text {ArcSin}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x \text {ArcSin}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \text {ArcSin}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}+\frac {x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 197
Rule 198
Rule 2221
Rule 2317
Rule 2438
Rule 3800
Rule 4745
Rule 4747
Rule 4765
Rule 4767
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=\frac {x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 \int \frac {\sin ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{5 c}-\frac {\left (2 a \sqrt {1-a^2 x^2}\right ) \int \frac {x \sin ^{-1}(a x)}{\left (1-a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 \int \frac {\sin ^{-1}(a x)^2}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{15 c^2}+\frac {\sqrt {1-a^2 x^2} \int \frac {1}{\left (1-a^2 x^2\right )^{5/2}} \, dx}{10 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (8 a \sqrt {1-a^2 x^2}\right ) \int \frac {x \sin ^{-1}(a x)}{\left (1-a^2 x^2\right )^2} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {4 \sin ^{-1}(a x)}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {\sqrt {1-a^2 x^2} \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (4 \sqrt {1-a^2 x^2}\right ) \int \frac {1}{\left (1-a^2 x^2\right )^{3/2}} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (16 a \sqrt {1-a^2 x^2}\right ) \int \frac {x \sin ^{-1}(a x)}{1-a^2 x^2} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}+\frac {x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {4 \sin ^{-1}(a x)}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (16 \sqrt {1-a^2 x^2}\right ) \text {Subst}\left (\int x \tan (x) \, dx,x,\sin ^{-1}(a x)\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}+\frac {x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {4 \sin ^{-1}(a x)}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (32 i \sqrt {1-a^2 x^2}\right ) \text {Subst}\left (\int \frac {e^{2 i x} x}{1+e^{2 i x}} \, dx,x,\sin ^{-1}(a x)\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}+\frac {x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {4 \sin ^{-1}(a x)}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {16 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (16 \sqrt {1-a^2 x^2}\right ) \text {Subst}\left (\int \log \left (1+e^{2 i x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}+\frac {x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {4 \sin ^{-1}(a x)}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {16 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {\left (8 i \sqrt {1-a^2 x^2}\right ) \text {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 i \sin ^{-1}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}\\ &=\frac {x}{3 c^3 \sqrt {c-a^2 c x^2}}+\frac {x}{30 c^3 \left (1-a^2 x^2\right ) \sqrt {c-a^2 c x^2}}-\frac {\sin ^{-1}(a x)}{10 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {4 \sin ^{-1}(a x)}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)^2}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)^2}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)^2}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \sin ^{-1}(a x)^2}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {16 \sqrt {1-a^2 x^2} \sin ^{-1}(a x) \log \left (1+e^{2 i \sin ^{-1}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}-\frac {8 i \sqrt {1-a^2 x^2} \text {Li}_2\left (-e^{2 i \sin ^{-1}(a x)}\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.60, size = 234, normalized size = 0.60 \begin {gather*} \frac {\sqrt {1-a^2 x^2} \left (\frac {a^3 x^3}{\left (1-a^2 x^2\right )^{3/2}}+\frac {11 a x}{\sqrt {1-a^2 x^2}}-16 i \text {ArcSin}(a x)^2+\frac {16 a x \text {ArcSin}(a x)^2}{\sqrt {1-a^2 x^2}}+\frac {8 \text {ArcSin}(a x) \left (-1+\frac {a x \text {ArcSin}(a x)}{\sqrt {1-a^2 x^2}}\right )}{1-a^2 x^2}+\frac {3 \text {ArcSin}(a x) \left (-1+\frac {2 a x \text {ArcSin}(a x)}{\sqrt {1-a^2 x^2}}\right )}{\left (1-a^2 x^2\right )^2}+32 \text {ArcSin}(a x) \log \left (1+e^{2 i \text {ArcSin}(a x)}\right )-16 i \text {PolyLog}\left (2,-e^{2 i \text {ArcSin}(a x)}\right )\right )}{30 a c^3 \sqrt {c \left (1-a^2 x^2\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 556, normalized size = 1.43
method | result | size |
default | \(-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-20 a^{3} x^{3}+8 i \sqrt {-a^{2} x^{2}+1}\, a^{4} x^{4}+15 a x -16 i \sqrt {-a^{2} x^{2}+1}\, a^{2} x^{2}+8 i \sqrt {-a^{2} x^{2}+1}\right ) \left (62 i \sqrt {-a^{2} x^{2}+1}\, a x +64 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}+126 i \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+32 a^{8} x^{8}+456 i \arcsin \left (a x \right ) a^{4} x^{4}-248 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}-328 i \arcsin \left (a x \right ) a^{2} x^{2}-142 a^{6} x^{6}+80 a^{4} x^{4} \arcsin \left (a x \right )^{2}-32 i \sqrt {-a^{2} x^{2}+1}\, a^{7} x^{7}+340 \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}+88 i \arcsin \left (a x \right )+265 a^{4} x^{4}-190 \arcsin \left (a x \right )^{2} a^{2} x^{2}+64 i \arcsin \left (a x \right ) a^{8} x^{8}-165 a x \arcsin \left (a x \right ) \sqrt {-a^{2} x^{2}+1}-156 i \sqrt {-a^{2} x^{2}+1}\, a^{3} x^{3}-235 a^{2} x^{2}+128 \arcsin \left (a x \right )^{2}-280 i \arcsin \left (a x \right ) a^{6} x^{6}+80\right )}{30 c^{4} \left (40 a^{10} x^{10}-215 a^{8} x^{8}+469 a^{6} x^{6}-517 a^{4} x^{4}+287 a^{2} x^{2}-64\right ) a}+\frac {8 i \sqrt {-a^{2} x^{2}+1}\, \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (2 i \arcsin \left (a x \right ) \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )+2 \arcsin \left (a x \right )^{2}+\polylog \left (2, -\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )\right )}{15 a \,c^{4} \left (a^{2} x^{2}-1\right )}\) | \(556\) |
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*} \int \frac {\operatorname {asin}^{2}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \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 {{\mathrm {asin}\left (a\,x\right )}^2}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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