Optimal. Leaf size=172 \[ -\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^3}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b x^3 \left (a+b \sin ^{-1}(c x)\right )}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b x \left (a+b \sin ^{-1}(c x)\right )}{2 c^3 d^3 \sqrt {1-c^2 x^2}}+\frac {b^2}{12 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {b^2 \log \left (1-c^2 x^2\right )}{3 c^4 d^3} \]
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Rubi [A] time = 0.33, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4681, 4703, 4641, 260, 266, 43} \[ \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {b x^3 \left (a+b \sin ^{-1}(c x)\right )}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {b x \left (a+b \sin ^{-1}(c x)\right )}{2 c^3 d^3 \sqrt {1-c^2 x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^3}+\frac {b^2}{12 c^4 d^3 \left (1-c^2 x^2\right )}+\frac {b^2 \log \left (1-c^2 x^2\right )}{3 c^4 d^3} \]
Antiderivative was successfully verified.
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Rule 43
Rule 260
Rule 266
Rule 4641
Rule 4681
Rule 4703
Rubi steps
\begin {align*} \int \frac {x^3 \left (a+b \sin ^{-1}(c x)\right )^2}{\left (d-c^2 d x^2\right )^3} \, dx &=\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}-\frac {(b c) \int \frac {x^4 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{5/2}} \, dx}{2 d^3}\\ &=-\frac {b x^3 \left (a+b \sin ^{-1}(c x)\right )}{6 c d^3 \left (1-c^2 x^2\right )^{3/2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b^2 \int \frac {x^3}{\left (1-c^2 x^2\right )^2} \, dx}{6 d^3}+\frac {b \int \frac {x^2 \left (a+b \sin ^{-1}(c x)\right )}{\left (1-c^2 x^2\right )^{3/2}} \, dx}{2 c d^3}\\ &=-\frac {b x^3 \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 )}{2 c^3 d^3 \sqrt {1-c^2 x^2}}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b^2 \operatorname {Subst}\left (\int \frac {x}{\left (1-c^2 x\right )^2} \, dx,x,x^2\right )}{12 d^3}-\frac {b \int \frac {a+b \sin ^{-1}(c x)}{\sqrt {1-c^2 x^2}} \, dx}{2 c^3 d^3}-\frac {b^2 \int \frac {x}{1-c^2 x^2} \, dx}{2 c^2 d^3}\\ &=-\frac {b x^3 \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 )}{2 c^3 d^3 \sqrt {1-c^2 x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^3}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b^2 \log \left (1-c^2 x^2\right )}{4 c^4 d^3}+\frac {b^2 \operatorname {Subst}\left (\int \left (\frac {1}{c^2 \left (-1+c^2 x\right )^2}+\frac {1}{c^2 \left (-1+c^2 x\right )}\right ) \, dx,x,x^2\right )}{12 d^3}\\ &=\frac {b^2}{12 c^4 d^3 \left (1-c^2 x^2\right )}-\frac {b x^3 \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 )}{2 c^3 d^3 \sqrt {1-c^2 x^2}}-\frac {\left (a+b \sin ^{-1}(c x)\right )^2}{4 c^4 d^3}+\frac {x^4 \left (a+b \sin ^{-1}(c x)\right )^2}{4 d^3 \left (1-c^2 x^2\right )^2}+\frac {b^2 \log \left (1-c^2 x^2\right )}{3 c^4 d^3}\\ \end {align*}
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Mathematica [A] time = 0.19, size = 192, normalized size = 1.12 \[ \frac {6 a^2 c^2 x^2-3 a^2+6 a b c x \sqrt {1-c^2 x^2}+2 b \sin ^{-1}(c x) \left (a \left (6 c^2 x^2-3\right )+b c x \sqrt {1-c^2 x^2} \left (3-4 c^2 x^2\right )\right )-8 a b c^3 x^3 \sqrt {1-c^2 x^2}-b^2 c^2 x^2+4 b^2 \left (c^2 x^2-1\right )^2 \log \left (1-c^2 x^2\right )+3 b^2 \left (2 c^2 x^2-1\right ) \sin ^{-1}(c x)^2+b^2}{12 c^4 d^3 \left (c^2 x^2-1\right )^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.80, size = 198, normalized size = 1.15 \[ \frac {{\left (6 \, a^{2} - b^{2}\right )} c^{2} x^{2} + 3 \, {\left (2 \, b^{2} c^{2} x^{2} - b^{2}\right )} \arcsin \left (c x\right )^{2} - 3 \, a^{2} + b^{2} + 6 \, {\left (2 \, a b c^{2} x^{2} - a b\right )} \arcsin \left (c x\right ) + 4 \, {\left (b^{2} c^{4} x^{4} - 2 \, b^{2} c^{2} x^{2} + b^{2}\right )} \log \left (c^{2} x^{2} - 1\right ) - 2 \, {\left (4 \, a b c^{3} x^{3} - 3 \, a b c x + {\left (4 \, b^{2} c^{3} x^{3} - 3 \, b^{2} c x\right )} \arcsin \left (c x\right )\right )} \sqrt {-c^{2} x^{2} + 1}}{12 \, {\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 1.03, size = 318, normalized size = 1.85 \[ \frac {b^{2} x^{4} \arcsin \left (c x\right )^{2}}{4 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac {a b x^{4} \arcsin \left (c x\right )}{2 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac {a^{2} x^{4}}{4 \, {\left (c^{2} x^{2} - 1\right )}^{2} d^{3}} + \frac {b^{2} x^{3} \arcsin \left (c x\right )}{6 \, {\left (c^{2} x^{2} - 1\right )} \sqrt {-c^{2} x^{2} + 1} c d^{3}} + \frac {a b x^{3}}{6 \, {\left (c^{2} x^{2} - 1\right )} \sqrt {-c^{2} x^{2} + 1} c d^{3}} - \frac {b^{2} x^{2}}{12 \, {\left (c^{2} x^{2} - 1\right )} c^{2} d^{3}} + \frac {b^{2} x \arcsin \left (c x\right )}{2 \, \sqrt {-c^{2} x^{2} + 1} c^{3} d^{3}} - \frac {b^{2} \arcsin \left (c x\right )^{2}}{4 \, c^{4} d^{3}} + \frac {a b x}{2 \, \sqrt {-c^{2} x^{2} + 1} c^{3} d^{3}} - \frac {a b \arcsin \left (c x\right )}{2 \, c^{4} d^{3}} + \frac {2 \, b^{2} \log \relax (2)}{3 \, c^{4} d^{3}} + \frac {b^{2} \log \left ({\left | -c^{2} x^{2} + 1 \right |}\right )}{3 \, c^{4} d^{3}} - \frac {a^{2}}{4 \, c^{4} d^{3}} + \frac {b^{2}}{12 \, c^{4} d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.66, size = 472, normalized size = 2.74 \[ \frac {a^{2}}{16 c^{4} d^{3} \left (c x +1\right )^{2}}-\frac {3 a^{2}}{16 c^{4} d^{3} \left (c x +1\right )}+\frac {a^{2}}{16 c^{4} d^{3} \left (c x -1\right )^{2}}+\frac {3 a^{2}}{16 c^{4} d^{3} \left (c x -1\right )}+\frac {b^{2} \arcsin \left (c x \right )^{2}}{4 c^{4} d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b^{2} \arcsin \left (c x \right ) x \sqrt {-c^{2} x^{2}+1}}{6 c^{3} d^{3} \left (c^{2} x^{2}-1\right )^{2}}-\frac {b^{2}}{12 c^{4} d^{3} \left (c^{2} x^{2}-1\right )}-\frac {2 b^{2} \sqrt {-c^{2} x^{2}+1}\, \arcsin \left (c x \right ) x}{3 c^{3} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {b^{2} \ln \left (-c^{2} x^{2}+1\right )}{3 c^{4} d^{3}}+\frac {b^{2} \arcsin \left (c x \right )^{2}}{2 c^{4} d^{3} \left (c^{2} x^{2}-1\right )}+\frac {a b \arcsin \left (c x \right )}{8 c^{4} d^{3} \left (c x +1\right )^{2}}-\frac {3 a b \arcsin \left (c x \right )}{8 c^{4} d^{3} \left (c x +1\right )}+\frac {a b \arcsin \left (c x \right )}{8 c^{4} d^{3} \left (c x -1\right )^{2}}+\frac {3 a b \arcsin \left (c x \right )}{8 c^{4} d^{3} \left (c x -1\right )}-\frac {a b \sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{3 c^{4} d^{3} \left (c x +1\right )}-\frac {a b \sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{24 c^{4} d^{3} \left (c x -1\right )^{2}}-\frac {a b \sqrt {-\left (c x -1\right )^{2}-2 c x +2}}{3 c^{4} d^{3} \left (c x -1\right )}+\frac {a b \sqrt {-\left (c x +1\right )^{2}+2 c x +2}}{24 c^{4} d^{3} \left (c x +1\right )^{2}} \]
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 \[ \frac {{\left (2 \, c^{2} x^{2} - 1\right )} a^{2}}{4 \, {\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}} + \frac {{\left (2 \, b^{2} c^{2} x^{2} - b^{2}\right )} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )^{2} - 2 \, {\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )} \int \frac {4 \, a b c^{3} x^{3} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right ) - {\left (2 \, b^{2} c^{2} x^{2} - b^{2}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} \arctan \left (c x, \sqrt {c x + 1} \sqrt {-c x + 1}\right )}{c^{9} d^{3} x^{6} - 3 \, c^{7} d^{3} x^{4} + 3 \, c^{5} d^{3} x^{2} - c^{3} d^{3}}\,{d x}}{4 \, {\left (c^{8} d^{3} x^{4} - 2 \, c^{6} d^{3} x^{2} + c^{4} d^{3}\right )}} \]
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 \[ \int \frac {x^3\,{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2}{{\left (d-c^2\,d\,x^2\right )}^3} \,d x \]
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 \[ - \frac {\int \frac {a^{2} x^{3}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {b^{2} x^{3} \operatorname {asin}^{2}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx + \int \frac {2 a b x^{3} \operatorname {asin}{\left (c x \right )}}{c^{6} x^{6} - 3 c^{4} x^{4} + 3 c^{2} x^{2} - 1}\, dx}{d^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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