Optimal. Leaf size=34 \[ \frac {2 (b \tan (e+f x))^{3/2}}{3 b f (d \sec (e+f x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2605} \[ \frac {2 (b \tan (e+f x))^{3/2}}{3 b f (d \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2605
Rubi steps
\begin {align*} \int \frac {\sqrt {b \tan (e+f x)}}{(d \sec (e+f x))^{3/2}} \, dx &=\frac {2 (b \tan (e+f x))^{3/2}}{3 b f (d \sec (e+f x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.12, size = 34, normalized size = 1.00 \[ \frac {2 (b \tan (e+f x))^{3/2}}{3 b f (d \sec (e+f x))^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.55, size = 50, normalized size = 1.47 \[ \frac {2 \, \sqrt {\frac {b \sin \left (f x + e\right )}{\cos \left (f x + e\right )}} \sqrt {\frac {d}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{3 \, d^{2} f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \tan \left (f x + e\right )}}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.67, size = 50, normalized size = 1.47 \[ \frac {2 \sin \left (f x +e \right ) \sqrt {\frac {b \sin \left (f x +e \right )}{\cos \left (f x +e \right )}}}{3 f \cos \left (f x +e \right ) \left (\frac {d}{\cos \left (f x +e \right )}\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {b \tan \left (f x + e\right )}}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 3.48, size = 55, normalized size = 1.62 \[ \frac {\sin \left (2\,e+2\,f\,x\right )\,\sqrt {\frac {d}{\cos \left (e+f\,x\right )}}\,\sqrt {\frac {b\,\sin \left (2\,e+2\,f\,x\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}}{3\,d^2\,f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [A] time = 26.00, size = 53, normalized size = 1.56 \[ \begin {cases} \frac {2 \sqrt {b} \tan ^{\frac {3}{2}}{\left (e + f x \right )}}{3 d^{\frac {3}{2}} f \sec ^{\frac {3}{2}}{\left (e + f x \right )}} & \text {for}\: f \neq 0 \\\frac {x \sqrt {b \tan {\relax (e )}}}{\left (d \sec {\relax (e )}\right )^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________