Optimal. Leaf size=131 \[ \frac{2^{m+\frac{1}{2}} \left (\frac{1}{\sec (c+d x)+1}\right )^{m+\frac{1}{2}} (e \tan (c+d x))^{m+1} F_1\left (\frac{m+1}{2};m-\frac{1}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d e (m+1) \sqrt{a \sec (c+d x)+a}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0884013, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {3889} \[ \frac{2^{m+\frac{1}{2}} \left (\frac{1}{\sec (c+d x)+1}\right )^{m+\frac{1}{2}} (e \tan (c+d x))^{m+1} F_1\left (\frac{m+1}{2};m-\frac{1}{2},1;\frac{m+3}{2};-\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a},\frac{a-a \sec (c+d x)}{\sec (c+d x) a+a}\right )}{d e (m+1) \sqrt{a \sec (c+d x)+a}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3889
Rubi steps
\begin{align*} \int \frac{(e \tan (c+d x))^m}{\sqrt{a+a \sec (c+d x)}} \, dx &=\frac{2^{\frac{1}{2}+m} F_1\left (\frac{1+m}{2};-\frac{1}{2}+m,1;\frac{3+m}{2};-\frac{a-a \sec (c+d x)}{a+a \sec (c+d x)},\frac{a-a \sec (c+d x)}{a+a \sec (c+d x)}\right ) \left (\frac{1}{1+\sec (c+d x)}\right )^{\frac{1}{2}+m} (e \tan (c+d x))^{1+m}}{d e (1+m) \sqrt{a+a \sec (c+d x)}}\\ \end{align*}
Mathematica [F] time = 2.21428, size = 0, normalized size = 0. \[ \int \frac{(e \tan (c+d x))^m}{\sqrt{a+a \sec (c+d x)}} \, dx \]
Verification is Not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.283, size = 0, normalized size = 0. \begin{align*} \int{ \left ( e\tan \left ( dx+c \right ) \right ) ^{m}{\frac{1}{\sqrt{a+a\sec \left ( dx+c \right ) }}}}\, 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 \frac{\left (e \tan \left (d x + c\right )\right )^{m}}{\sqrt{a \sec \left (d x + c\right ) + a}}\,{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 (\frac{\left (e \tan \left (d x + c\right )\right )^{m}}{\sqrt{a \sec \left (d x + c\right ) + a}}, 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 \frac{\left (e \tan{\left (c + d x \right )}\right )^{m}}{\sqrt{a \left (\sec{\left (c + d x \right )} + 1\right )}}\, 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 \frac{\left (e \tan \left (d x + c\right )\right )^{m}}{\sqrt{a \sec \left (d x + c\right ) + a}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]