Optimal. Leaf size=36 \[ \frac{a \left (a+b \cos ^2(x)\right )^4}{8 b^2}-\frac{\left (a+b \cos ^2(x)\right )^5}{10 b^2} \]
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Rubi [A] time = 0.0872477, antiderivative size = 36, normalized size of antiderivative = 1., 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 = {4335, 266, 43} \[ \frac{a \left (a+b \cos ^2(x)\right )^4}{8 b^2}-\frac{\left (a+b \cos ^2(x)\right )^5}{10 b^2} \]
Antiderivative was successfully verified.
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Rule 4335
Rule 266
Rule 43
Rubi steps
\begin{align*} \int \cos ^3(x) \left (a+b \cos ^2(x)\right )^3 \sin (x) \, dx &=-\operatorname{Subst}\left (\int x^3 \left (a+b x^2\right )^3 \, dx,x,\cos (x)\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int x (a+b x)^3 \, dx,x,\cos ^2(x)\right )\right )\\ &=-\left (\frac{1}{2} \operatorname{Subst}\left (\int \left (-\frac{a (a+b x)^3}{b}+\frac{(a+b x)^4}{b}\right ) \, dx,x,\cos ^2(x)\right )\right )\\ &=\frac{a \left (a+b \cos ^2(x)\right )^4}{8 b^2}-\frac{\left (a+b \cos ^2(x)\right )^5}{10 b^2}\\ \end{align*}
Mathematica [B] time = 0.301158, size = 137, normalized size = 3.81 \[ \frac{1}{32} \left (-12 a^2 b \cos ^4(x)-4 a^2 b \cos (3 x) \cos ^3(x)-4 a^3 \cos (2 x)-a^3 \cos (4 x)-8 a b^2 \cos ^6(x)-\frac{1}{32} a b^2 (48 \cos (2 x)+36 \cos (4 x)+16 \cos (6 x)+3 \cos (8 x))-2 b^3 \cos ^8(x)-\frac{1}{320} b^3 (140 \cos (2 x)+100 \cos (4 x)+50 \cos (6 x)+15 \cos (8 x)+2 \cos (10 x))\right ) \]
Antiderivative was successfully verified.
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Maple [A] time = 0.009, size = 40, normalized size = 1.1 \begin{align*} -{\frac{{b}^{3} \left ( \cos \left ( x \right ) \right ) ^{10}}{10}}-{\frac{3\,a{b}^{2} \left ( \cos \left ( x \right ) \right ) ^{8}}{8}}-{\frac{{a}^{2}b \left ( \cos \left ( x \right ) \right ) ^{6}}{2}}-{\frac{ \left ( \cos \left ( x \right ) \right ) ^{4}{a}^{3}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] time = 0.980933, size = 139, normalized size = 3.86 \begin{align*} \frac{1}{10} \, b^{3} \sin \left (x\right )^{10} - \frac{1}{8} \,{\left (3 \, a b^{2} + 4 \, b^{3}\right )} \sin \left (x\right )^{8} + \frac{1}{2} \,{\left (a^{2} b + 3 \, a b^{2} + 2 \, b^{3}\right )} \sin \left (x\right )^{6} - \frac{1}{4} \,{\left (a^{3} + 6 \, a^{2} b + 9 \, a b^{2} + 4 \, b^{3}\right )} \sin \left (x\right )^{4} + \frac{1}{2} \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sin \left (x\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.21531, size = 111, normalized size = 3.08 \begin{align*} -\frac{1}{10} \, b^{3} \cos \left (x\right )^{10} - \frac{3}{8} \, a b^{2} \cos \left (x\right )^{8} - \frac{1}{2} \, a^{2} b \cos \left (x\right )^{6} - \frac{1}{4} \, a^{3} \cos \left (x\right )^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] time = 16.4708, size = 97, normalized size = 2.69 \begin{align*} \frac{a^{3} \sin ^{4}{\left (x \right )}}{4} + \frac{a^{3} \sin ^{2}{\left (x \right )} \cos ^{2}{\left (x \right )}}{2} + \frac{a^{2} b \sin ^{6}{\left (x \right )}}{2} + \frac{3 a^{2} b \sin ^{4}{\left (x \right )} \cos ^{2}{\left (x \right )}}{2} + \frac{3 a^{2} b \sin ^{2}{\left (x \right )} \cos ^{4}{\left (x \right )}}{2} - \frac{3 a b^{2} \cos ^{8}{\left (x \right )}}{8} - \frac{b^{3} \cos ^{10}{\left (x \right )}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.08834, size = 53, normalized size = 1.47 \begin{align*} -\frac{1}{10} \, b^{3} \cos \left (x\right )^{10} - \frac{3}{8} \, a b^{2} \cos \left (x\right )^{8} - \frac{1}{2} \, a^{2} b \cos \left (x\right )^{6} - \frac{1}{4} \, a^{3} \cos \left (x\right )^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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