Optimal. Leaf size=50 \[ \frac {5 c^2 \text {Ci}\left (\sin ^{-1}(a x)\right )}{8 a}+\frac {5 c^2 \text {Ci}\left (3 \sin ^{-1}(a x)\right )}{16 a}+\frac {c^2 \text {Ci}\left (5 \sin ^{-1}(a x)\right )}{16 a} \]
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Rubi [A] time = 0.09, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {4661, 3312, 3302} \[ \frac {5 c^2 \text {CosIntegral}\left (\sin ^{-1}(a x)\right )}{8 a}+\frac {5 c^2 \text {CosIntegral}\left (3 \sin ^{-1}(a x)\right )}{16 a}+\frac {c^2 \text {CosIntegral}\left (5 \sin ^{-1}(a x)\right )}{16 a} \]
Antiderivative was successfully verified.
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Rule 3302
Rule 3312
Rule 4661
Rubi steps
\begin {align*} \int \frac {\left (c-a^2 c x^2\right )^2}{\sin ^{-1}(a x)} \, dx &=\frac {c^2 \operatorname {Subst}\left (\int \frac {\cos ^5(x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {c^2 \operatorname {Subst}\left (\int \left (\frac {5 \cos (x)}{8 x}+\frac {5 \cos (3 x)}{16 x}+\frac {\cos (5 x)}{16 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=\frac {c^2 \operatorname {Subst}\left (\int \frac {\cos (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{16 a}+\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{16 a}+\frac {\left (5 c^2\right ) \operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{8 a}\\ &=\frac {5 c^2 \text {Ci}\left (\sin ^{-1}(a x)\right )}{8 a}+\frac {5 c^2 \text {Ci}\left (3 \sin ^{-1}(a x)\right )}{16 a}+\frac {c^2 \text {Ci}\left (5 \sin ^{-1}(a x)\right )}{16 a}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 34, normalized size = 0.68 \[ \frac {c^2 \left (10 \text {Ci}\left (\sin ^{-1}(a x)\right )+5 \text {Ci}\left (3 \sin ^{-1}(a x)\right )+\text {Ci}\left (5 \sin ^{-1}(a x)\right )\right )}{16 a} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {a^{4} c^{2} x^{4} - 2 \, a^{2} c^{2} x^{2} + c^{2}}{\arcsin \left (a x\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.35, size = 44, normalized size = 0.88 \[ \frac {c^{2} \operatorname {Ci}\left (5 \, \arcsin \left (a x\right )\right )}{16 \, a} + \frac {5 \, c^{2} \operatorname {Ci}\left (3 \, \arcsin \left (a x\right )\right )}{16 \, a} + \frac {5 \, c^{2} \operatorname {Ci}\left (\arcsin \left (a x\right )\right )}{8 \, a} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 33, normalized size = 0.66 \[ \frac {c^{2} \left (10 \Ci \left (\arcsin \left (a x \right )\right )+5 \Ci \left (3 \arcsin \left (a x \right )\right )+\Ci \left (5 \arcsin \left (a x \right )\right )\right )}{16 a} \]
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 \[ \int \frac {{\left (a^{2} c x^{2} - c\right )}^{2}}{\arcsin \left (a x\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {{\left (c-a^2\,c\,x^2\right )}^2}{\mathrm {asin}\left (a\,x\right )} \,d x \]
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 \[ c^{2} \left (\int \left (- \frac {2 a^{2} x^{2}}{\operatorname {asin}{\left (a x \right )}}\right )\, dx + \int \frac {a^{4} x^{4}}{\operatorname {asin}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {asin}{\left (a x \right )}}\, dx\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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