Optimal. Leaf size=61 \[ \frac {b \tan (e+f x) \sqrt {b \tan ^2(e+f x)}}{2 f}+\frac {b \cot (e+f x) \sqrt {b \tan ^2(e+f x)} \log (\cos (e+f x))}{f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.03, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3658, 3473, 3475} \[ \frac {b \tan (e+f x) \sqrt {b \tan ^2(e+f x)}}{2 f}+\frac {b \cot (e+f x) \sqrt {b \tan ^2(e+f x)} \log (\cos (e+f x))}{f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3473
Rule 3475
Rule 3658
Rubi steps
\begin {align*} \int \left (b \tan ^2(e+f x)\right )^{3/2} \, dx &=\left (b \cot (e+f x) \sqrt {b \tan ^2(e+f x)}\right ) \int \tan ^3(e+f x) \, dx\\ &=\frac {b \tan (e+f x) \sqrt {b \tan ^2(e+f x)}}{2 f}-\left (b \cot (e+f x) \sqrt {b \tan ^2(e+f x)}\right ) \int \tan (e+f x) \, dx\\ &=\frac {b \cot (e+f x) \log (\cos (e+f x)) \sqrt {b \tan ^2(e+f x)}}{f}+\frac {b \tan (e+f x) \sqrt {b \tan ^2(e+f x)}}{2 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.11, size = 47, normalized size = 0.77 \[ \frac {\cot ^3(e+f x) \left (b \tan ^2(e+f x)\right )^{3/2} \left (\tan ^2(e+f x)+2 \log (\cos (e+f x))\right )}{2 f} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.40, size = 52, normalized size = 0.85 \[ \frac {{\left (b \tan \left (f x + e\right )^{2} + b \log \left (\frac {1}{\tan \left (f x + e\right )^{2} + 1}\right ) + b\right )} \sqrt {b \tan \left (f x + e\right )^{2}}}{2 \, f \tan \left (f x + e\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.26, size = 48, normalized size = 0.79 \[ -\frac {\left (b \left (\tan ^{2}\left (f x +e \right )\right )\right )^{\frac {3}{2}} \left (-\left (\tan ^{2}\left (f x +e \right )\right )+\ln \left (1+\tan ^{2}\left (f x +e \right )\right )\right )}{2 f \tan \left (f x +e \right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.83, size = 34, normalized size = 0.56 \[ \frac {b^{\frac {3}{2}} \tan \left (f x + e\right )^{2} - b^{\frac {3}{2}} \log \left (\tan \left (f x + e\right )^{2} + 1\right )}{2 \, f} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int {\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^2\right )}^{3/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (b \tan ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________