Optimal. Leaf size=44 \[ -\frac{(a-b) \cos (e+f x)}{f}+\frac{a \cos ^3(e+f x)}{3 f}+\frac{b \sec (e+f x)}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0389122, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {4133, 448} \[ -\frac{(a-b) \cos (e+f x)}{f}+\frac{a \cos ^3(e+f x)}{3 f}+\frac{b \sec (e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4133
Rule 448
Rubi steps
\begin{align*} \int \left (a+b \sec ^2(e+f x)\right ) \sin ^3(e+f x) \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{\left (1-x^2\right ) \left (b+a x^2\right )}{x^2} \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{\operatorname{Subst}\left (\int \left (a \left (1-\frac{b}{a}\right )+\frac{b}{x^2}-a x^2\right ) \, dx,x,\cos (e+f x)\right )}{f}\\ &=-\frac{(a-b) \cos (e+f x)}{f}+\frac{a \cos ^3(e+f x)}{3 f}+\frac{b \sec (e+f x)}{f}\\ \end{align*}
Mathematica [A] time = 0.0324777, size = 53, normalized size = 1.2 \[ -\frac{3 a \cos (e+f x)}{4 f}+\frac{a \cos (3 (e+f x))}{12 f}+\frac{b \cos (e+f x)}{f}+\frac{b \sec (e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.041, size = 62, normalized size = 1.4 \begin{align*}{\frac{1}{f} \left ( -{\frac{a \left ( 2+ \left ( \sin \left ( fx+e \right ) \right ) ^{2} \right ) \cos \left ( fx+e \right ) }{3}}+b \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 ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.975774, size = 54, normalized size = 1.23 \begin{align*} \frac{a \cos \left (f x + e\right )^{3} - 3 \,{\left (a - b\right )} \cos \left (f x + e\right ) + \frac{3 \, b}{\cos \left (f x + e\right )}}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.01928, size = 100, normalized size = 2.27 \begin{align*} \frac{a \cos \left (f x + e\right )^{4} - 3 \,{\left (a - b\right )} \cos \left (f x + e\right )^{2} + 3 \, b}{3 \, f \cos \left (f x + e\right )} \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 = 1.27419, size = 82, normalized size = 1.86 \begin{align*} \frac{b}{f \cos \left (f x + e\right )} + \frac{a f^{5} \cos \left (f x + e\right )^{3} - 3 \, a f^{5} \cos \left (f x + e\right ) + 3 \, b f^{5} \cos \left (f x + e\right )}{3 \, f^{6}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]