Optimal. Leaf size=210 \[ -\frac {2}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}-\frac {1}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 x \sin ^{-1}(a x)}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x \sin ^{-1}(a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \sin ^{-1}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}} \]
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Rubi [A] time = 0.12, antiderivative size = 210, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4655, 4653, 260, 261} \[ -\frac {2}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}-\frac {1}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}+\frac {8 x \sin ^{-1}(a x)}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 x \sin ^{-1}(a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {x \sin ^{-1}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}} \]
Antiderivative was successfully verified.
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Rule 260
Rule 261
Rule 4653
Rule 4655
Rubi steps
\begin {align*} \int \frac {\sin ^{-1}(a x)}{\left (c-a^2 c x^2\right )^{7/2}} \, dx &=\frac {x \sin ^{-1}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 \int \frac {\sin ^{-1}(a x)}{\left (c-a^2 c x^2\right )^{5/2}} \, dx}{5 c}-\frac {\left (a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{\left (1-a^2 x^2\right )^3} \, dx}{5 c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {1}{20 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)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 \int \frac {\sin ^{-1}(a x)}{\left (c-a^2 c x^2\right )^{3/2}} \, dx}{15 c^2}-\frac {\left (4 a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{\left (1-a^2 x^2\right )^2} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {1}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {2}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)}{15 c^3 \sqrt {c-a^2 c x^2}}-\frac {\left (8 a \sqrt {1-a^2 x^2}\right ) \int \frac {x}{1-a^2 x^2} \, dx}{15 c^3 \sqrt {c-a^2 c x^2}}\\ &=-\frac {1}{20 a c^3 \left (1-a^2 x^2\right )^{3/2} \sqrt {c-a^2 c x^2}}-\frac {2}{15 a c^3 \sqrt {1-a^2 x^2} \sqrt {c-a^2 c x^2}}+\frac {x \sin ^{-1}(a x)}{5 c \left (c-a^2 c x^2\right )^{5/2}}+\frac {4 x \sin ^{-1}(a x)}{15 c^2 \left (c-a^2 c x^2\right )^{3/2}}+\frac {8 x \sin ^{-1}(a x)}{15 c^3 \sqrt {c-a^2 c x^2}}+\frac {4 \sqrt {1-a^2 x^2} \log \left (1-a^2 x^2\right )}{15 a c^3 \sqrt {c-a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.22, size = 111, normalized size = 0.53 \[ -\frac {\sqrt {c-a^2 c x^2} \left (\sqrt {1-a^2 x^2} \left (8 a^2 x^2+16 \left (a^2 x^2-1\right )^2 \log \left (a^2 x^2-1\right )-11\right )+4 a x \left (8 a^4 x^4-20 a^2 x^2+15\right ) \sin ^{-1}(a x)\right )}{60 a c^4 \left (a^2 x^2-1\right )^3} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.94, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {-a^{2} c x^{2} + c} \arcsin \left (a x\right )}{a^{8} c^{4} x^{8} - 4 \, a^{6} c^{4} x^{6} + 6 \, a^{4} c^{4} x^{4} - 4 \, a^{2} c^{4} x^{2} + c^{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.81, size = 128, normalized size = 0.61 \[ -\frac {1}{60} \, \sqrt {c} {\left (\frac {16 \, \log \left ({\left | a^{2} x^{2} - 1 \right |}\right )}{a c^{4}} - \frac {24 \, a^{4} x^{4} - 56 \, a^{2} x^{2} + 35}{{\left (a^{2} x^{2} - 1\right )}^{2} a c^{4}}\right )} - \frac {\sqrt {-a^{2} c x^{2} + c} {\left (4 \, {\left (\frac {2 \, a^{4} x^{2}}{c} - \frac {5 \, a^{2}}{c}\right )} x^{2} + \frac {15}{c}\right )} x \arcsin \left (a x\right )}{15 \, {\left (a^{2} c x^{2} - c\right )}^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.33, size = 409, normalized size = 1.95 \[ \frac {16 i \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \arcsin \left (a x \right )}{15 a \,c^{4} \left (a^{2} x^{2}-1\right )}-\frac {\sqrt {-c \left (a^{2} x^{2}-1\right )}\, \left (8 a^{5} x^{5}-20 a^{3} x^{3}+8 i \sqrt {-a^{2} x^{2}+1}\, x^{4} a^{4}+15 a x -16 i \sqrt {-a^{2} x^{2}+1}\, x^{2} a^{2}+8 i \sqrt {-a^{2} x^{2}+1}\right ) \left (64 i x^{8} a^{8}+64 \sqrt {-a^{2} x^{2}+1}\, x^{7} a^{7}-280 i x^{6} a^{6}-248 \sqrt {-a^{2} x^{2}+1}\, a^{5} x^{5}+160 a^{4} x^{4} \arcsin \left (a x \right )+456 i x^{4} a^{4}+340 a^{3} x^{3} \sqrt {-a^{2} x^{2}+1}-380 a^{2} x^{2} \arcsin \left (a x \right )-328 i x^{2} a^{2}-165 a x \sqrt {-a^{2} x^{2}+1}+256 \arcsin \left (a x \right )+88 i\right )}{60 c^{4} \left (40 a^{10} x^{10}-215 x^{8} a^{8}+469 a^{6} x^{6}-517 a^{4} x^{4}+287 a^{2} x^{2}-64\right ) a}-\frac {8 \sqrt {-c \left (a^{2} x^{2}-1\right )}\, \sqrt {-a^{2} x^{2}+1}\, \ln \left (1+\left (i a x +\sqrt {-a^{2} x^{2}+1}\right )^{2}\right )}{15 a \,c^{4} \left (a^{2} x^{2}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.06, size = 149, normalized size = 0.71 \[ -\frac {1}{60} \, a {\left (\frac {3}{{\left (a^{6} c^{\frac {5}{2}} x^{4} - 2 \, a^{4} c^{\frac {5}{2}} x^{2} + a^{2} c^{\frac {5}{2}}\right )} c} - \frac {8}{{\left (a^{4} c^{\frac {3}{2}} x^{2} - a^{2} c^{\frac {3}{2}}\right )} c^{2}} + \frac {16 \, \log \left (x^{2} - \frac {1}{a^{2}}\right )}{a^{2} c^{\frac {7}{2}}}\right )} + \frac {1}{15} \, {\left (\frac {8 \, x}{\sqrt {-a^{2} c x^{2} + c} c^{3}} + \frac {4 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {3}{2}} c^{2}} + \frac {3 \, x}{{\left (-a^{2} c x^{2} + c\right )}^{\frac {5}{2}} c}\right )} \arcsin \left (a x\right ) \]
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 \[ \int \frac {\mathrm {asin}\left (a\,x\right )}{{\left (c-a^2\,c\,x^2\right )}^{7/2}} \,d x \]
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 \[ \int \frac {\operatorname {asin}{\left (a x \right )}}{\left (- c \left (a x - 1\right ) \left (a x + 1\right )\right )^{\frac {7}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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