Optimal. Leaf size=43 \[ \frac{2 b^3}{11 f (b \sec (e+f x))^{11/2}}-\frac{2 b}{7 f (b \sec (e+f x))^{7/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0526156, antiderivative size = 43, 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 = {2622, 14} \[ \frac{2 b^3}{11 f (b \sec (e+f x))^{11/2}}-\frac{2 b}{7 f (b \sec (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2622
Rule 14
Rubi steps
\begin{align*} \int \frac{\sin ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx &=\frac{b^3 \operatorname{Subst}\left (\int \frac{-1+\frac{x^2}{b^2}}{x^{13/2}} \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac{b^3 \operatorname{Subst}\left (\int \left (-\frac{1}{x^{13/2}}+\frac{1}{b^2 x^{9/2}}\right ) \, dx,x,b \sec (e+f x)\right )}{f}\\ &=\frac{2 b^3}{11 f (b \sec (e+f x))^{11/2}}-\frac{2 b}{7 f (b \sec (e+f x))^{7/2}}\\ \end{align*}
Mathematica [A] time = 0.148857, size = 42, normalized size = 0.98 \[ \frac{\cos ^4(e+f x) (7 \cos (2 (e+f x))-15) \sqrt{b \sec (e+f x)}}{77 b^3 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.113, size = 36, normalized size = 0.8 \begin{align*}{\frac{ \left ( 14\, \left ( \cos \left ( fx+e \right ) \right ) ^{2}-22 \right ) \cos \left ( fx+e \right ) }{77\,f} \left ({\frac{b}{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{5}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.01223, size = 50, normalized size = 1.16 \begin{align*} \frac{2 \,{\left (7 \, b^{2} - \frac{11 \, b^{2}}{\cos \left (f x + e\right )^{2}}\right )} b}{77 \, f \left (\frac{b}{\cos \left (f x + e\right )}\right )^{\frac{11}{2}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 2.4716, size = 100, normalized size = 2.33 \begin{align*} \frac{2 \,{\left (7 \, \cos \left (f x + e\right )^{6} - 11 \, \cos \left (f x + e\right )^{4}\right )} \sqrt{\frac{b}{\cos \left (f x + e\right )}}}{77 \, b^{3} f} \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 = 1.56658, size = 68, normalized size = 1.58 \begin{align*} \frac{2 \,{\left (7 \, b^{2} - \frac{11 \, b^{2}}{\cos \left (f x + e\right )^{2}}\right )} \cos \left (f x + e\right )^{5}}{77 \, b^{4} f \sqrt{\frac{b}{\cos \left (f x + e\right )}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]