Optimal. Leaf size=72 \[ \frac{8 \sqrt{b \tan (e+f x)}}{5 b d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \sqrt{b \tan (e+f x)}}{5 b f (d \sec (e+f x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0973784, antiderivative size = 72, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08, Rules used = {2612, 2605} \[ \frac{8 \sqrt{b \tan (e+f x)}}{5 b d^2 f \sqrt{d \sec (e+f x)}}+\frac{2 \sqrt{b \tan (e+f x)}}{5 b f (d \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2612
Rule 2605
Rubi steps
\begin{align*} \int \frac{1}{(d \sec (e+f x))^{5/2} \sqrt{b \tan (e+f x)}} \, dx &=\frac{2 \sqrt{b \tan (e+f x)}}{5 b f (d \sec (e+f x))^{5/2}}+\frac{4 \int \frac{1}{\sqrt{d \sec (e+f x)} \sqrt{b \tan (e+f x)}} \, dx}{5 d^2}\\ &=\frac{2 \sqrt{b \tan (e+f x)}}{5 b f (d \sec (e+f x))^{5/2}}+\frac{8 \sqrt{b \tan (e+f x)}}{5 b d^2 f \sqrt{d \sec (e+f x)}}\\ \end{align*}
Mathematica [A] time = 1.15039, size = 112, normalized size = 1.56 \[ \frac{\sqrt{\frac{1}{\cos (e+f x)+1}} \cos (2 (e+f x)) \tan (e+f x)+9 \tan \left (\frac{1}{2} (e+f x)\right ) \sqrt{\sec (e+f x)} \sqrt{\sec (e+f x)+1}}{5 d^2 f \sqrt{\frac{1}{\cos (e+f x)+1}} \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.17, size = 60, normalized size = 0.8 \begin{align*}{\frac{2\,\sin \left ( fx+e \right ) \left ( \left ( \cos \left ( fx+e \right ) \right ) ^{2}+4 \right ) }{5\,f \left ( \cos \left ( fx+e \right ) \right ) ^{3}} \left ({\frac{d}{\cos \left ( fx+e \right ) }} \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{{\frac{b\sin \left ( fx+e \right ) }{\cos \left ( fx+e \right ) }}}}}} \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{1}{\left (d \sec \left (f x + e\right )\right )^{\frac{5}{2}} \sqrt{b \tan \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.696, size = 140, normalized size = 1.94 \begin{align*} \frac{2 \,{\left (\cos \left (f x + e\right )^{3} + 4 \, \cos \left (f x + e\right )\right )} \sqrt{\frac{b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt{\frac{d}{\cos \left (f x + e\right )}}}{5 \, b d^{3} f} \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] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (d \sec \left (f x + e\right )\right )^{\frac{5}{2}} \sqrt{b \tan \left (f x + e\right )}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]