Optimal. Leaf size=83 \[ \frac{\sqrt{2} \tan (e+f x) (a \sec (e+f x)+a)^m F_1\left (m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right )}{f (2 m+1) \sqrt{1-\sec (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.056996, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {3779, 3778, 136} \[ \frac{\sqrt{2} \tan (e+f x) (a \sec (e+f x)+a)^m F_1\left (m+\frac{1}{2};\frac{1}{2},1;m+\frac{3}{2};\frac{1}{2} (\sec (e+f x)+1),\sec (e+f x)+1\right )}{f (2 m+1) \sqrt{1-\sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3779
Rule 3778
Rule 136
Rubi steps
\begin{align*} \int (a+a \sec (e+f x))^m \, dx &=\left ((1+\sec (e+f x))^{-m} (a+a \sec (e+f x))^m\right ) \int (1+\sec (e+f x))^m \, dx\\ &=-\frac{\left ((1+\sec (e+f x))^{-\frac{1}{2}-m} (a+a \sec (e+f x))^m \tan (e+f x)\right ) \operatorname{Subst}\left (\int \frac{(1+x)^{-\frac{1}{2}+m}}{\sqrt{1-x} x} \, dx,x,\sec (e+f x)\right )}{f \sqrt{1-\sec (e+f x)}}\\ &=\frac{\sqrt{2} F_1\left (\frac{1}{2}+m;\frac{1}{2},1;\frac{3}{2}+m;\frac{1}{2} (1+\sec (e+f x)),1+\sec (e+f x)\right ) (a+a \sec (e+f x))^m \tan (e+f x)}{f (1+2 m) \sqrt{1-\sec (e+f x)}}\\ \end{align*}
Mathematica [B] time = 6.57736, size = 711, normalized size = 8.57 \[ \frac{30 \sin (e+f x) \cos ^2\left (\frac{1}{2} (e+f x)\right ) \cos (e+f x) (a (\sec (e+f x)+1))^m F_1\left (\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right ) \left (3 F_1\left (\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-2 \tan ^2\left (\frac{1}{2} (e+f x)\right ) \left (F_1\left (\frac{3}{2};m,2;\frac{5}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-m F_1\left (\frac{3}{2};m+1,1;\frac{5}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )\right )}{f \left (45 \cos ^2\left (\frac{1}{2} (e+f x)\right ) (-2 m \cos (e+f x)+\cos (2 (e+f x))+2 m+1) F_1\left (\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right ){}^2+40 \sin ^2\left (\frac{1}{2} (e+f x)\right ) \cos (e+f x) \tan ^2\left (\frac{1}{2} (e+f x)\right ) \left (F_1\left (\frac{3}{2};m,2;\frac{5}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-m F_1\left (\frac{3}{2};m+1,1;\frac{5}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right ){}^2+6 \sin ^2\left (\frac{1}{2} (e+f x)\right ) F_1\left (\frac{1}{2};m,1;\frac{3}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right ) \left (-5 (-2 (m+2) \cos (e+f x)+\cos (2 (e+f x))+2 m+1) F_1\left (\frac{3}{2};m,2;\frac{5}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+5 m (-2 (m+2) \cos (e+f x)+\cos (2 (e+f x))+2 m+1) F_1\left (\frac{3}{2};m+1,1;\frac{5}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-48 \sin ^4\left (\frac{1}{2} (e+f x)\right ) \cot (e+f x) \csc (e+f x) \left (2 F_1\left (\frac{5}{2};m,3;\frac{7}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )-2 m F_1\left (\frac{5}{2};m+1,2;\frac{7}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )+m (m+1) F_1\left (\frac{5}{2};m+2,1;\frac{7}{2};\tan ^2\left (\frac{1}{2} (e+f x)\right ),-\tan ^2\left (\frac{1}{2} (e+f x)\right )\right )\right )\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.201, size = 0, normalized size = 0. \begin{align*} \int \left ( a+a\sec \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sec \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (a \sec \left (f x + e\right ) + a\right )}^{m}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a \sec{\left (e + f x \right )} + a\right )^{m}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (a \sec \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]