Optimal. Leaf size=95 \[ -\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \sin ^{-1}(a x)}-\frac {35 c^3 \text {Si}\left (\sin ^{-1}(a x)\right )}{64 a}-\frac {63 c^3 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{64 a}-\frac {35 c^3 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{64 a}-\frac {7 c^3 \text {Si}\left (7 \sin ^{-1}(a x)\right )}{64 a} \]
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Rubi [A] time = 0.17, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4659, 4723, 4406, 3299} \[ -\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \sin ^{-1}(a x)}-\frac {35 c^3 \text {Si}\left (\sin ^{-1}(a x)\right )}{64 a}-\frac {63 c^3 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{64 a}-\frac {35 c^3 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{64 a}-\frac {7 c^3 \text {Si}\left (7 \sin ^{-1}(a x)\right )}{64 a} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 4406
Rule 4659
Rule 4723
Rubi steps
\begin {align*} \int \frac {\left (c-a^2 c x^2\right )^3}{\sin ^{-1}(a x)^2} \, dx &=-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \sin ^{-1}(a x)}-\left (7 a c^3\right ) \int \frac {x \left (1-a^2 x^2\right )^{5/2}}{\sin ^{-1}(a x)} \, dx\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \sin ^{-1}(a x)}-\frac {\left (7 c^3\right ) \operatorname {Subst}\left (\int \frac {\cos ^6(x) \sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \sin ^{-1}(a x)}-\frac {\left (7 c^3\right ) \operatorname {Subst}\left (\int \left (\frac {5 \sin (x)}{64 x}+\frac {9 \sin (3 x)}{64 x}+\frac {5 \sin (5 x)}{64 x}+\frac {\sin (7 x)}{64 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \sin ^{-1}(a x)}-\frac {\left (7 c^3\right ) \operatorname {Subst}\left (\int \frac {\sin (7 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}-\frac {\left (35 c^3\right ) \operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}-\frac {\left (35 c^3\right ) \operatorname {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}-\frac {\left (63 c^3\right ) \operatorname {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}\\ &=-\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \sin ^{-1}(a x)}-\frac {35 c^3 \text {Si}\left (\sin ^{-1}(a x)\right )}{64 a}-\frac {63 c^3 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{64 a}-\frac {35 c^3 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{64 a}-\frac {7 c^3 \text {Si}\left (7 \sin ^{-1}(a x)\right )}{64 a}\\ \end {align*}
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Mathematica [A] time = 0.60, size = 83, normalized size = 0.87 \[ -\frac {c^3 \left (64 \left (1-a^2 x^2\right )^{7/2}+35 \sin ^{-1}(a x) \text {Si}\left (\sin ^{-1}(a x)\right )+63 \sin ^{-1}(a x) \text {Si}\left (3 \sin ^{-1}(a x)\right )+35 \sin ^{-1}(a x) \text {Si}\left (5 \sin ^{-1}(a x)\right )+7 \sin ^{-1}(a x) \text {Si}\left (7 \sin ^{-1}(a x)\right )\right )}{64 a \sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.55, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}}{\arcsin \left (a x\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.47, size = 95, normalized size = 1.00 \[ \frac {{\left (a^{2} x^{2} - 1\right )}^{3} \sqrt {-a^{2} x^{2} + 1} c^{3}}{a \arcsin \left (a x\right )} - \frac {7 \, c^{3} \operatorname {Si}\left (7 \, \arcsin \left (a x\right )\right )}{64 \, a} - \frac {35 \, c^{3} \operatorname {Si}\left (5 \, \arcsin \left (a x\right )\right )}{64 \, a} - \frac {63 \, c^{3} \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{64 \, a} - \frac {35 \, c^{3} \operatorname {Si}\left (\arcsin \left (a x\right )\right )}{64 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 105, normalized size = 1.11 \[ -\frac {c^{3} \left (35 \Si \left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+63 \Si \left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+35 \Si \left (5 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+7 \Si \left (7 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+35 \sqrt {-a^{2} x^{2}+1}+\cos \left (7 \arcsin \left (a x \right )\right )+21 \cos \left (3 \arcsin \left (a x \right )\right )+7 \cos \left (5 \arcsin \left (a x \right )\right )\right )}{64 a \arcsin \left (a x \right )} \]
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 \[ -\frac {7 \, a \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right ) \int \frac {{\left (a^{5} c^{3} x^{5} - 2 \, a^{3} c^{3} x^{3} + a c^{3} x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{\arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}\,{d x} - {\left (a^{6} c^{3} x^{6} - 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} - c^{3}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{a \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^3}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]
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 \[ - c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {3 a^{4} x^{4}}{\operatorname {asin}^{2}{\left (a x \right )}}\right )\, dx + \int \frac {a^{6} x^{6}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx + \int \left (- \frac {1}{\operatorname {asin}^{2}{\left (a x \right )}}\right )\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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