Optimal. Leaf size=36 \[ \frac {\left (a+b \sin ^2(x)\right )^5}{10 b^2}-\frac {a \left (a+b \sin ^2(x)\right )^4}{8 b^2} \]
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Rubi [A] time = 0.08, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {3198, 266, 43} \[ \frac {\left (a+b \sin ^2(x)\right )^5}{10 b^2}-\frac {a \left (a+b \sin ^2(x)\right )^4}{8 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 3198
Rubi steps
\begin {align*} \int \cos (x) \sin ^3(x) \left (a+b \sin ^2(x)\right )^3 \, dx &=\operatorname {Subst}\left (\int x^3 \left (a+b x^2\right )^3 \, dx,x,\sin (x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int x (a+b x)^3 \, dx,x,\sin ^2(x)\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a (a+b x)^3}{b}+\frac {(a+b x)^4}{b}\right ) \, dx,x,\sin ^2(x)\right )\\ &=-\frac {a \left (a+b \sin ^2(x)\right )^4}{8 b^2}+\frac {\left (a+b \sin ^2(x)\right )^5}{10 b^2}\\ \end {align*}
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Mathematica [B] time = 0.35, size = 128, normalized size = 3.56 \[ \frac {-20 \left (64 a^3+24 a b^2+7 b^3\right ) \cos (2 x)+20 \left (16 a^3+18 a b^2+5 b^3\right ) \cos (4 x)+b \left (3840 a^2 \sin ^4(x)-1280 a^2 \sin (3 x) \sin ^3(x)+2560 a b \sin ^6(x)-10 b (16 a+5 b) \cos (6 x)+15 b (2 a+b) \cos (8 x)+640 b^2 \sin ^8(x)-2 b^2 \cos (10 x)\right )}{10240} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.19, size = 103, normalized size = 2.86 \[ -\frac {1}{10} \, b^{3} \cos \relax (x)^{10} + \frac {1}{8} \, {\left (3 \, a b^{2} + 4 \, b^{3}\right )} \cos \relax (x)^{8} - \frac {1}{2} \, {\left (a^{2} b + 3 \, a b^{2} + 2 \, b^{3}\right )} \cos \relax (x)^{6} + \frac {1}{4} \, {\left (a^{3} + 6 \, a^{2} b + 9 \, a b^{2} + 4 \, b^{3}\right )} \cos \relax (x)^{4} - \frac {1}{2} \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \relax (x)^{2} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.12, size = 39, normalized size = 1.08 \[ \frac {1}{10} \, b^{3} \sin \relax (x)^{10} + \frac {3}{8} \, a b^{2} \sin \relax (x)^{8} + \frac {1}{2} \, a^{2} b \sin \relax (x)^{6} + \frac {1}{4} \, a^{3} \sin \relax (x)^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 40, normalized size = 1.11 \[ \frac {b^{3} \left (\sin ^{10}\relax (x )\right )}{10}+\frac {3 a \,b^{2} \left (\sin ^{8}\relax (x )\right )}{8}+\frac {a^{2} b \left (\sin ^{6}\relax (x )\right )}{2}+\frac {a^{3} \left (\sin ^{4}\relax (x )\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.33, size = 39, normalized size = 1.08 \[ \frac {1}{10} \, b^{3} \sin \relax (x)^{10} + \frac {3}{8} \, a b^{2} \sin \relax (x)^{8} + \frac {1}{2} \, a^{2} b \sin \relax (x)^{6} + \frac {1}{4} \, a^{3} \sin \relax (x)^{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.07, size = 73, normalized size = 2.03 \[ \frac {b^2\,{\cos \relax (x)}^8\,\left (3\,a+4\,b\right )}{8}-\frac {b^3\,{\cos \relax (x)}^{10}}{10}-\frac {{\cos \relax (x)}^2\,{\left (a+b\right )}^3}{2}-\frac {b\,{\cos \relax (x)}^6\,\left (a^2+3\,a\,b+2\,b^2\right )}{2}+\frac {{\cos \relax (x)}^4\,{\left (a+b\right )}^2\,\left (a+4\,b\right )}{4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 11.38, size = 44, normalized size = 1.22 \[ \frac {a^{3} \sin ^{4}{\relax (x )}}{4} + \frac {a^{2} b \sin ^{6}{\relax (x )}}{2} + \frac {3 a b^{2} \sin ^{8}{\relax (x )}}{8} + \frac {b^{3} \sin ^{10}{\relax (x )}}{10} \]
Verification of antiderivative is not currently implemented for this CAS.
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