Optimal. Leaf size=78 \[ -\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{a \sin ^{-1}(a x)}-\frac {5 c^2 \text {Si}\left (\sin ^{-1}(a x)\right )}{8 a}-\frac {15 c^2 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{16 a}-\frac {5 c^2 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{16 a} \]
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Rubi [A] time = 0.16, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4659, 4723, 4406, 3299} \[ -\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{a \sin ^{-1}(a x)}-\frac {5 c^2 \text {Si}\left (\sin ^{-1}(a x)\right )}{8 a}-\frac {15 c^2 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{16 a}-\frac {5 c^2 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{16 a} \]
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 )^2}{\sin ^{-1}(a x)^2} \, dx &=-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{a \sin ^{-1}(a x)}-\left (5 a c^2\right ) \int \frac {x \left (1-a^2 x^2\right )^{3/2}}{\sin ^{-1}(a x)} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{a \sin ^{-1}(a x)}-\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {\cos ^4(x) \sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{a \sin ^{-1}(a x)}-\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \left (\frac {\sin (x)}{8 x}+\frac {3 \sin (3 x)}{16 x}+\frac {\sin (5 x)}{16 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{a \sin ^{-1}(a x)}-\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{16 a}-\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{8 a}-\frac {\left (15 c^2\right ) \operatorname {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{16 a}\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{a \sin ^{-1}(a x)}-\frac {5 c^2 \text {Si}\left (\sin ^{-1}(a x)\right )}{8 a}-\frac {15 c^2 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{16 a}-\frac {5 c^2 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{16 a}\\ \end {align*}
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Mathematica [A] time = 0.48, size = 70, normalized size = 0.90 \[ -\frac {c^2 \left (16 \left (1-a^2 x^2\right )^{5/2}+10 \sin ^{-1}(a x) \text {Si}\left (\sin ^{-1}(a x)\right )+15 \sin ^{-1}(a x) \text {Si}\left (3 \sin ^{-1}(a x)\right )+5 \sin ^{-1}(a x) \text {Si}\left (5 \sin ^{-1}(a x)\right )\right )}{16 a \sin ^{-1}(a x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.40, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}{\arcsin \left (a x\right )^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 81, normalized size = 1.04 \[ -\frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} c^{2}}{a \arcsin \left (a x\right )} - \frac {5 \, c^{2} \operatorname {Si}\left (5 \, \arcsin \left (a x\right )\right )}{16 \, a} - \frac {15 \, c^{2} \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{16 \, a} - \frac {5 \, c^{2} \operatorname {Si}\left (\arcsin \left (a x\right )\right )}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.09, size = 83, normalized size = 1.06 \[ -\frac {c^{2} \left (10 \Si \left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+15 \Si \left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+5 \Si \left (5 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+10 \sqrt {-a^{2} x^{2}+1}+5 \cos \left (3 \arcsin \left (a x \right )\right )+\cos \left (5 \arcsin \left (a x \right )\right )\right )}{16 a \arcsin \left (a x \right )} \]
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 \[ \frac {5 \, a \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right ) \int \frac {{\left (a^{3} c^{2} x^{3} - a c^{2} x\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{\arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )}\,{d x} - {\left (a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}\right )} \sqrt {a x + 1} \sqrt {-a x + 1}}{a \arctan \left (a x, \sqrt {a x + 1} \sqrt {-a x + 1}\right )} \]
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 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^2}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \]
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 \[ c^{2} \left (\int \left (- \frac {2 a^{2} x^{2}}{\operatorname {asin}^{2}{\left (a x \right )}}\right )\, dx + \int \frac {a^{4} x^{4}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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