Optimal. Leaf size=150 \[ \frac {b^2}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {b x (a+b \text {ArcSin}(c x))}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b x (a+b \text {ArcSin}(c x))}{3 c d^3 \sqrt {1-c^2 x^2}}+\frac {(a+b \text {ArcSin}(c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{6 c^2 d^3} \]
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Rubi [A]
time = 0.10, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4767, 4747,
4745, 266, 267} \begin {gather*} -\frac {b x (a+b \text {ArcSin}(c x))}{3 c d^3 \sqrt {1-c^2 x^2}}-\frac {b x (a+b \text {ArcSin}(c x))}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {(a+b \text {ArcSin}(c x))^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b^2}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {b^2 \log \left (1-c^2 x^2\right )}{6 c^2 d^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 266
Rule 267
Rule 4745
Rule 4747
Rule 4767
Rubi steps
\begin {align*} \int \frac {x \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^3} \, dx &=\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b \int \frac {a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{2 c d^3}\\ &=-\frac {b x \left (a+b \sin ^{-1}(c x)\right )}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b^2 \int \frac {x}{\left (1-c^2 x^2\right )^2} \, dx}{6 d^3}-\frac {b \int \frac {a+b \sin ^{-1}(c x)}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{3 c d^3}\\ &=\frac {b^2}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {b x \left (a+b \sin ^{-1}(c x)\right )}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^3 \sqrt {1-c^2 x^2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}+\frac {b^2 \int \frac {x}{1-c^2 x^2} \, dx}{3 d^3}\\ &=\frac {b^2}{12 c^2 d^3 \left (1-c^2 x^2\right )}-\frac {b x \left (a+b \sin ^{-1}(c x)\right )}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}-\frac {b x \left (a+b \sin ^{-1}(c x)\right )}{3 c d^3 \sqrt {1-c^2 x^2}}+\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^2 d^3 \left (1-c^2 x^2\right )^2}-\frac {b^2 \log \left (1-c^2 x^2\right )}{6 c^2 d^3}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 162, normalized size = 1.08 \begin {gather*} \frac {3 a^2+b^2-b^2 c^2 x^2-6 a b c x \sqrt {1-c^2 x^2}+4 a b c^3 x^3 \sqrt {1-c^2 x^2}+2 b \left (3 a+b c x \sqrt {1-c^2 x^2} \left (-3+2 c^2 x^2\right )\right ) \text {ArcSin}(c x)+3 b^2 \text {ArcSin}(c x)^2-2 b^2 \left (-1+c^2 x^2\right )^2 \log \left (1-c^2 x^2\right )}{12 c^2 d^3 \left (-1+c^2 x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(290\) vs.
\(2(136)=272\).
time = 0.09, size = 291, normalized size = 1.94
method | result | size |
derivativedivides | \(\frac {\frac {a^{2}}{4 d^{3} \left (c^{2} x^{2}-1\right )^{2}}+\frac {b^{2} \arcsin \left (c x \right )^{2}}{4 d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x}{6 d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b^{2}}{12 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c x}{3 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \ln \left (-c^{2} x^{2}+1\right )}{6 d^{3}}-\frac {2 a b \left (-\frac {\arcsin \left (c x \right )}{4 \left (c^{2} x^{2}-1\right )^{2}}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{48 \left (c x -1\right )^{2}}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{12 \left (c x -1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{12 \left (c x +1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{48 \left (c x +1\right )^{2}}\right )}{d^{3}}}{c^{2}}\) | \(291\) |
default | \(\frac {\frac {a^{2}}{4 d^{3} \left (c^{2} x^{2}-1\right )^{2}}+\frac {b^{2} \arcsin \left (c x \right )^{2}}{4 d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b^{2} \arcsin \left (c x \right ) \sqrt {-c^{2} x^{2}+1}\, c x}{6 d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b^{2}}{12 d^{3} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) c x}{3 d^{3} \left (c^{2} x^{2}-1\right )}-\frac {b^{2} \ln \left (-c^{2} x^{2}+1\right )}{6 d^{3}}-\frac {2 a b \left (-\frac {\arcsin \left (c x \right )}{4 \left (c^{2} x^{2}-1\right )^{2}}+\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{48 \left (c x -1\right )^{2}}-\frac {\sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{12 \left (c x -1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{12 \left (c x +1\right )}-\frac {\sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{48 \left (c x +1\right )^{2}}\right )}{d^{3}}}{c^{2}}\) | \(291\) |
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 [A]
time = 3.14, size = 165, normalized size = 1.10 \begin {gather*} -\frac {b^{2} c^{2} x^{2} - 3 \, b^{2} \arcsin \left (c x\right )^{2} - 6 \, a b \arcsin \left (c x\right ) - 3 \, a^{2} - b^{2} + 2 \, {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} - 1\right ) - 2 \, {\left (2 \, a b c^{3} x^{3} - 3 \, a b c x + {\left (2 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{12 \, {\left (c^{6} d^{3} x^{4} - 2 \, c^{4} d^{3} x^{2} + c^{2} d^{3}\right )}} \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*} - \frac {\int \frac {a^{2} x}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x \operatorname {asin}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 395 vs.
\(2 (134) = 268\).
time = 0.48, size = 395, normalized size = 2.63 \begin {gather*} \frac {b^{2} c^{2} x^{4} \arcsin \left (c x\right )^{2}}{4 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac {a b c^{2} x^{4} \arcsin \left (c x\right )}{2 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac {a^{2} c^{2} x^{4}}{4 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac {b^{2} c x^{3} \arcsin \left (c x\right )}{6 \, {\left (c^{2} x^{2} - 1\right )} \sqrt {-c^{2} x^{2} + 1} d^{3}} - \frac {b^{2} x^{2} \arcsin \left (c x\right )^{2}}{2 \, {\left (c^{2} x^{2} - 1\right )} d^{3}} + \frac {a b c x^{3}}{6 \, {\left (c^{2} x^{2} - 1\right )} \sqrt {-c^{2} x^{2} + 1} d^{3}} - \frac {a b x^{2} \arcsin \left (c x\right )}{{\left (c^{2} x^{2} - 1\right )} d^{3}} - \frac {a^{2} x^{2}}{2 \, {\left (c^{2} x^{2} - 1\right )} d^{3}} - \frac {b^{2} x^{2}}{12 \, {\left (c^{2} x^{2} - 1\right )} d^{3}} - \frac {b^{2} x \arcsin \left (c x\right )}{2 \, \sqrt {-c^{2} x^{2} + 1} c d^{3}} + \frac {b^{2} \arcsin \left (c x\right )^{2}}{4 \, c^{2} d^{3}} - \frac {a b x}{2 \, \sqrt {-c^{2} x^{2} + 1} c d^{3}} + \frac {a b \arcsin \left (c x\right )}{2 \, c^{2} d^{3}} - \frac {b^{2} \log \left (2\right )}{3 \, c^{2} d^{3}} - \frac {b^{2} \log \left ({\left | -c^{2} x^{2} + 1 \right |}\right )}{6 \, c^{2} d^{3}} + \frac {a^{2}}{4 \, c^{2} d^{3}} + \frac {b^{2}}{12 \, c^{2} d^{3}} \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 {x\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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