Optimal. Leaf size=95 \[ -\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \text {ArcSin}(a x)}-\frac {35 c^3 \text {Si}(\text {ArcSin}(a x))}{64 a}-\frac {63 c^3 \text {Si}(3 \text {ArcSin}(a x))}{64 a}-\frac {35 c^3 \text {Si}(5 \text {ArcSin}(a x))}{64 a}-\frac {7 c^3 \text {Si}(7 \text {ArcSin}(a x))}{64 a} \]
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Rubi [A]
time = 0.12, 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 = {4751, 4809,
4491, 3380} \begin {gather*} -\frac {c^3 \left (1-a^2 x^2\right )^{7/2}}{a \text {ArcSin}(a x)}-\frac {35 c^3 \text {Si}(\text {ArcSin}(a x))}{64 a}-\frac {63 c^3 \text {Si}(3 \text {ArcSin}(a x))}{64 a}-\frac {35 c^3 \text {Si}(5 \text {ArcSin}(a x))}{64 a}-\frac {7 c^3 \text {Si}(7 \text {ArcSin}(a x))}{64 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 4491
Rule 4751
Rule 4809
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 ) \text {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 ) \text {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 ) \text {Subst}\left (\int \frac {\sin (7 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}-\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}-\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{64 a}-\frac {\left (63 c^3\right ) \text {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 = 0.29, size = 83, normalized size = 0.87 \begin {gather*} -\frac {c^3 \left (64 \left (1-a^2 x^2\right )^{7/2}+35 \text {ArcSin}(a x) \text {Si}(\text {ArcSin}(a x))+63 \text {ArcSin}(a x) \text {Si}(3 \text {ArcSin}(a x))+35 \text {ArcSin}(a x) \text {Si}(5 \text {ArcSin}(a x))+7 \text {ArcSin}(a x) \text {Si}(7 \text {ArcSin}(a x))\right )}{64 a \text {ArcSin}(a x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.24, size = 105, normalized size = 1.11
method | result | size |
derivativedivides | \(-\frac {c^{3} \left (35 \sinIntegral \left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+63 \sinIntegral \left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+35 \sinIntegral \left (5 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+7 \sinIntegral \left (7 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+35 \sqrt {-a^{2} x^{2}+1}+21 \cos \left (3 \arcsin \left (a x \right )\right )+7 \cos \left (5 \arcsin \left (a x \right )\right )+\cos \left (7 \arcsin \left (a x \right )\right )\right )}{64 a \arcsin \left (a x \right )}\) | \(105\) |
default | \(-\frac {c^{3} \left (35 \sinIntegral \left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+63 \sinIntegral \left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+35 \sinIntegral \left (5 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+7 \sinIntegral \left (7 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+35 \sqrt {-a^{2} x^{2}+1}+21 \cos \left (3 \arcsin \left (a x \right )\right )+7 \cos \left (5 \arcsin \left (a x \right )\right )+\cos \left (7 \arcsin \left (a x \right )\right )\right )}{64 a \arcsin \left (a x \right )}\) | \(105\) |
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*} - 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 ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.48, size = 95, normalized size = 1.00 \begin {gather*} \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} \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 = -0.01 \begin {gather*} \int \frac {{\left (c-a^2\,c\,x^2\right )}^3}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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