Optimal. Leaf size=72 \[ \frac{a^2 \cos ^3(e+f x)}{3 f}-\frac{a (a-2 b) \cos (e+f x)}{f}+\frac{b (2 a-b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.072152, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087, Rules used = {4133, 448} \[ \frac{a^2 \cos ^3(e+f x)}{3 f}-\frac{a (a-2 b) \cos (e+f x)}{f}+\frac{b (2 a-b) \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4133
Rule 448
Rubi steps
\begin{align*} \int \left (a+b \sec ^2(e+f x)\right )^2 \sin ^3(e+f x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right ) \left (b+a x^2\right )^2}{x^4} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{\operatorname{Subst}\left (\int \left (a (a-2 b)+\frac{b^2}{x^4}+\frac{(2 a-b) b}{x^2}-a^2 x^2\right ) \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{a (a-2 b) \cos (e+f x)}{f}+\frac{a^2 \cos ^3(e+f x)}{3 f}+\frac{(2 a-b) b \sec (e+f x)}{f}+\frac{b^2 \sec ^3(e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.464596, size = 83, normalized size = 1.15 \[ \frac{\sec ^3(e+f x) \left (-3 \left (11 a^2-64 a b+16 b^2\right ) \cos (2 (e+f x))+a^2 \cos (6 (e+f x))-26 a^2-6 a (a-4 b) \cos (4 (e+f x))+168 a b-16 b^2\right )}{96 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.053, size = 125, normalized size = 1.7 \begin{align*}{\frac{1}{f} \left ( -{\frac{{a}^{2} \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}+2\,ab \left ({\frac{ \left ( \sin \left ( fx+e \right ) \right ) ^{4}}{\cos \left ( fx+e \right ) }}+ \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) \right ) +{b}^{2} \left ({\frac{ \left ( \sin \left ( fx+e \right ) \right ) ^{4}}{3\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}}}-{\frac{ \left ( \sin \left ( fx+e \right ) \right ) ^{4}}{3\,\cos \left ( fx+e \right ) }}-{\frac{ \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01258, size = 90, normalized size = 1.25 \begin{align*} \frac{a^{2} \cos \left (f x + e\right )^{3} - 3 \,{\left (a^{2} - 2 \, a b\right )} \cos \left (f x + e\right ) + \frac{3 \,{\left (2 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}}{\cos \left (f x + e\right )^{3}}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 0.496206, size = 158, normalized size = 2.19 \begin{align*} \frac{a^{2} \cos \left (f x + e\right )^{6} - 3 \,{\left (a^{2} - 2 \, a b\right )} \cos \left (f x + e\right )^{4} + 3 \,{\left (2 \, a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + b^{2}}{3 \, f \cos \left (f x + e\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.33672, size = 131, normalized size = 1.82 \begin{align*} \frac{6 \, a b \cos \left (f x + e\right )^{2} - 3 \, b^{2} \cos \left (f x + e\right )^{2} + b^{2}}{3 \, f \cos \left (f x + e\right )^{3}} + \frac{a^{2} f^{11} \cos \left (f x + e\right )^{3} - 3 \, a^{2} f^{11} \cos \left (f x + e\right ) + 6 \, a b f^{11} \cos \left (f x + e\right )}{3 \, f^{12}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]