Optimal. Leaf size=97 \[ \frac {\tan (e+f x) (d \sec (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (m+n p+1)} \left (b (c \tan (e+f x))^n\right )^p \, _2F_1\left (\frac {1}{2} (n p+1),\frac {1}{2} (m+n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right )}{f (n p+1)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {3659, 2617} \[ \frac {\tan (e+f x) (d \sec (e+f x))^m \cos ^2(e+f x)^{\frac {1}{2} (m+n p+1)} \left (b (c \tan (e+f x))^n\right )^p \, _2F_1\left (\frac {1}{2} (n p+1),\frac {1}{2} (m+n p+1);\frac {1}{2} (n p+3);\sin ^2(e+f x)\right )}{f (n p+1)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2617
Rule 3659
Rubi steps
\begin {align*} \int (d \sec (e+f x))^m \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 (d \sec (e+f x))^m (c \tan (e+f x))^{n p} \, dx\\ &=\frac {\cos ^2(e+f x)^{\frac {1}{2} (1+m+n p)} \, _2F_1\left (\frac {1}{2} (1+n p),\frac {1}{2} (1+m+n p);\frac {1}{2} (3+n p);\sin ^2(e+f x)\right ) (d \sec (e+f x))^m \tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1+n p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.18, size = 89, normalized size = 0.92 \[ \frac {\cot (e+f x) (d \sec (e+f x))^m \left (-\tan ^2(e+f x)\right )^{\frac {1}{2} (1-n p)} \left (b (c \tan (e+f x))^n\right )^p \, _2F_1\left (\frac {m}{2},\frac {1}{2} (1-n p);\frac {m+2}{2};\sec ^2(e+f x)\right )}{f m} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \sec \left (f x + e\right )\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \sec \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 1.25, size = 0, normalized size = 0.00 \[ \int \left (d \sec \left (f x +e \right )\right )^{m} \left (b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (\left (c \tan \left (f x + e\right )\right )^{n} b\right )^{p} \left (d \sec \left (f x + e\right )\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^m\,{\left (b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \left (c \tan {\left (e + f x \right )}\right )^{n}\right )^{p} \left (d \sec {\left (e + f x \right )}\right )^{m}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________