Optimal. Leaf size=30 \[ \frac{(a+b) \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.032005, antiderivative size = 30, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.048, Rules used = {3191} \[ \frac{(a+b) \tan ^3(e+f x)}{3 f}+\frac{a \tan (e+f x)}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3191
Rubi steps
\begin{align*} \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right ) \, dx &=\frac{\operatorname{Subst}\left (\int \left (a+(a+b) x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{a \tan (e+f x)}{f}+\frac{(a+b) \tan ^3(e+f x)}{3 f}\\ \end{align*}
Mathematica [A] time = 0.0741321, size = 41, normalized size = 1.37 \[ \frac{a \left (\frac{1}{3} \tan ^3(e+f x)+\tan (e+f x)\right )}{f}+\frac{b \tan ^3(e+f x)}{3 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.056, size = 46, normalized size = 1.5 \begin{align*}{\frac{1}{f} \left ( -a \left ( -{\frac{2}{3}}-{\frac{ \left ( \sec \left ( fx+e \right ) \right ) ^{2}}{3}} \right ) \tan \left ( fx+e \right ) +{\frac{b \left ( \sin \left ( fx+e \right ) \right ) ^{3}}{3\, \left ( \cos \left ( fx+e \right ) \right ) ^{3}}} \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 0.979309, size = 36, normalized size = 1.2 \begin{align*} \frac{{\left (a + b\right )} \tan \left (f x + e\right )^{3} + 3 \, a \tan \left (f x + e\right )}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.87285, size = 97, normalized size = 3.23 \begin{align*} \frac{{\left ({\left (2 \, a - b\right )} \cos \left (f x + e\right )^{2} + a + b\right )} \sin \left (f x + e\right )}{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 = 1.16849, size = 51, normalized size = 1.7 \begin{align*} \frac{a \tan \left (f x + e\right )^{3} + b \tan \left (f x + e\right )^{3} + 3 \, a \tan \left (f x + e\right )}{3 \, f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]