Optimal. Leaf size=95 \[ \frac{4 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \tan (e+f x)}}{5 d^2 f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{2 (b \tan (e+f x))^{3/2}}{5 b f (d \sec (e+f x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.113922, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2612, 2616, 2640, 2639} \[ \frac{4 E\left (\left .\frac{1}{2} \left (e+f x-\frac{\pi }{2}\right )\right |2\right ) \sqrt{b \tan (e+f x)}}{5 d^2 f \sqrt{\sin (e+f x)} \sqrt{d \sec (e+f x)}}+\frac{2 (b \tan (e+f x))^{3/2}}{5 b f (d \sec (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2612
Rule 2616
Rule 2640
Rule 2639
Rubi steps
\begin{align*} \int \frac{\sqrt{b \tan (e+f x)}}{(d \sec (e+f x))^{5/2}} \, dx &=\frac{2 (b \tan (e+f x))^{3/2}}{5 b f (d \sec (e+f x))^{5/2}}+\frac{2 \int \frac{\sqrt{b \tan (e+f x)}}{\sqrt{d \sec (e+f x)}} \, dx}{5 d^2}\\ &=\frac{2 (b \tan (e+f x))^{3/2}}{5 b f (d \sec (e+f x))^{5/2}}+\frac{\left (2 \sqrt{b \tan (e+f x)}\right ) \int \sqrt{b \sin (e+f x)} \, dx}{5 d^2 \sqrt{d \sec (e+f x)} \sqrt{b \sin (e+f x)}}\\ &=\frac{2 (b \tan (e+f x))^{3/2}}{5 b f (d \sec (e+f x))^{5/2}}+\frac{\left (2 \sqrt{b \tan (e+f x)}\right ) \int \sqrt{\sin (e+f x)} \, dx}{5 d^2 \sqrt{d \sec (e+f x)} \sqrt{\sin (e+f x)}}\\ &=\frac{4 E\left (\left .\frac{1}{2} \left (e-\frac{\pi }{2}+f x\right )\right |2\right ) \sqrt{b \tan (e+f x)}}{5 d^2 f \sqrt{d \sec (e+f x)} \sqrt{\sin (e+f x)}}+\frac{2 (b \tan (e+f x))^{3/2}}{5 b f (d \sec (e+f x))^{5/2}}\\ \end{align*}
Mathematica [C] time = 0.670819, size = 79, normalized size = 0.83 \[ -\frac{b \left (4 \sqrt [4]{-\tan ^2(e+f x)} \, _2F_1\left (-\frac{1}{4},\frac{1}{4};\frac{3}{4};\sec ^2(e+f x)\right )+\cos (2 (e+f x))-1\right )}{5 d^2 f \sqrt{b \tan (e+f x)} \sqrt{d \sec (e+f x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 0.242, size = 573, normalized size = 6. \begin{align*} \text{result too large to display} \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{\sqrt{b \tan \left (f x + e\right )}}{\left (d \sec \left (f x + e\right )\right )^{\frac{5}{2}}}\,{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{\sqrt{d \sec \left (f x + e\right )} \sqrt{b \tan \left (f x + e\right )}}{d^{3} \sec \left (f x + e\right )^{3}}, x\right ) \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{\sqrt{b \tan \left (f x + e\right )}}{\left (d \sec \left (f x + e\right )\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]