Optimal. Leaf size=65 \[ \frac{\tan ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+3)}+\frac{\tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.111186, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3659, 2607, 14} \[ \frac{\tan ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+3)}+\frac{\tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3659
Rule 2607
Rule 14
Rubi steps
\begin{align*} \int \sec ^4(e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx &=\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int \sec ^4(e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\frac{\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \operatorname{Subst}\left (\int (c x)^{n p} \left (1+x^2\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \operatorname{Subst}\left (\int \left ((c x)^{n p}+\frac{(c x)^{2+n p}}{c^2}\right ) \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1+n p)}+\frac{\tan ^3(e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (3+n p)}\\ \end{align*}
Mathematica [A] time = 2.09433, size = 87, normalized size = 1.34 \[ \frac{\cot (e+f x) \left (2 \left (-\tan ^2(e+f x)\right )^{\frac{1}{2} (1-n p)}+\tan ^2(e+f x) \left ((n p+1) \sec ^2(e+f x)+2\right )\right ) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1) (n p+3)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.514, size = 0, normalized size = 0. \begin{align*} \int \left ( \sec \left ( fx+e \right ) \right ) ^{4} \left ( b \left ( c\tan \left ( fx+e \right ) \right ) ^{n} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.11682, size = 96, normalized size = 1.48 \begin{align*} \frac{\frac{b^{p}{\left (c^{n}\right )}^{p}{\left (\tan \left (f x + e\right )^{n}\right )}^{p} \tan \left (f x + e\right )^{3}}{n p + 3} + \frac{b^{p}{\left (c^{n}\right )}^{p}{\left (\tan \left (f x + e\right )^{n}\right )}^{p} \tan \left (f x + e\right )}{n p + 1}}{f} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.41611, size = 189, normalized size = 2.91 \begin{align*} \frac{{\left (n p + 2 \, \cos \left (f x + e\right )^{2} + 1\right )} e^{\left (n p \log \left (\frac{c \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right ) + p \log \left (b\right )\right )} \sin \left (f x + e\right )}{{\left (f n^{2} p^{2} + 4 \, f n p + 3 \, f\right )} \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 [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]