Optimal. Leaf size=36 \[ \frac{\left (a+b \sin ^2(x)\right )^5}{10 b^2}-\frac{a \left (a+b \sin ^2(x)\right )^4}{8 b^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.076912, 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 = {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.
[In]
[Out]
Rule 3198
Rule 266
Rule 43
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*}
Mathematica [B] time = 0.42499, 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.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.011, size = 40, normalized size = 1.1 \begin{align*}{\frac{{b}^{3} \left ( \sin \left ( x \right ) \right ) ^{10}}{10}}+{\frac{3\,a{b}^{2} \left ( \sin \left ( x \right ) \right ) ^{8}}{8}}+{\frac{{a}^{2}b \left ( \sin \left ( x \right ) \right ) ^{6}}{2}}+{\frac{ \left ( \sin \left ( x \right ) \right ) ^{4}{a}^{3}}{4}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.982705, size = 53, normalized size = 1.47 \begin{align*} \frac{1}{10} \, b^{3} \sin \left (x\right )^{10} + \frac{3}{8} \, a b^{2} \sin \left (x\right )^{8} + \frac{1}{2} \, a^{2} b \sin \left (x\right )^{6} + \frac{1}{4} \, a^{3} \sin \left (x\right )^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 2.22598, size = 258, normalized size = 7.17 \begin{align*} -\frac{1}{10} \, b^{3} \cos \left (x\right )^{10} + \frac{1}{8} \,{\left (3 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (x\right )^{8} - \frac{1}{2} \,{\left (a^{2} b + 3 \, a b^{2} + 2 \, b^{3}\right )} \cos \left (x\right )^{6} + \frac{1}{4} \,{\left (a^{3} + 6 \, a^{2} b + 9 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (x\right )^{4} - \frac{1}{2} \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cos \left (x\right )^{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [B] time = 16.575, size = 102, normalized size = 2.83 \begin{align*} \frac{a^{3} \sin ^{4}{\left (x \right )}}{4} + \frac{a^{2} b \sin ^{6}{\left (x \right )}}{2} + \frac{3 a b^{2} \sin ^{8}{\left (x \right )}}{8} - \frac{b^{3} \sin ^{8}{\left (x \right )} \cos ^{2}{\left (x \right )}}{2} - b^{3} \sin ^{6}{\left (x \right )} \cos ^{4}{\left (x \right )} - b^{3} \sin ^{4}{\left (x \right )} \cos ^{6}{\left (x \right )} - \frac{b^{3} \sin ^{2}{\left (x \right )} \cos ^{8}{\left (x \right )}}{2} - \frac{b^{3} \cos ^{10}{\left (x \right )}}{10} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.13995, size = 53, normalized size = 1.47 \begin{align*} \frac{1}{10} \, b^{3} \sin \left (x\right )^{10} + \frac{3}{8} \, a b^{2} \sin \left (x\right )^{8} + \frac{1}{2} \, a^{2} b \sin \left (x\right )^{6} + \frac{1}{4} \, a^{3} \sin \left (x\right )^{4} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]