Optimal. Leaf size=93 \[ -\frac{\cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{2 b^3 f}-\frac{\tan ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{4 b^{3/2} f}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{4 b^{3/2} f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0738407, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286, Rules used = {2622, 290, 329, 298, 203, 206} \[ -\frac{\cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{2 b^3 f}-\frac{\tan ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{4 b^{3/2} f}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{4 b^{3/2} f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2622
Rule 290
Rule 329
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{\csc ^3(e+f x)}{(b \sec (e+f x))^{3/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\sqrt{x}}{\left (-1+\frac{x^2}{b^2}\right )^2} \, dx,x,b \sec (e+f x)\right )}{b^3 f}\\ &=-\frac{\cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{2 b^3 f}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{x}}{-1+\frac{x^2}{b^2}} \, dx,x,b \sec (e+f x)\right )}{4 b^3 f}\\ &=-\frac{\cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{2 b^3 f}-\frac{\operatorname{Subst}\left (\int \frac{x^2}{-1+\frac{x^4}{b^2}} \, dx,x,\sqrt{b \sec (e+f x)}\right )}{2 b^3 f}\\ &=-\frac{\cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{2 b^3 f}+\frac{\operatorname{Subst}\left (\int \frac{1}{b-x^2} \, dx,x,\sqrt{b \sec (e+f x)}\right )}{4 b f}-\frac{\operatorname{Subst}\left (\int \frac{1}{b+x^2} \, dx,x,\sqrt{b \sec (e+f x)}\right )}{4 b f}\\ &=-\frac{\tan ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{4 b^{3/2} f}+\frac{\tanh ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{4 b^{3/2} f}-\frac{\cot ^2(e+f x) (b \sec (e+f x))^{3/2}}{2 b^3 f}\\ \end{align*}
Mathematica [A] time = 0.466018, size = 98, normalized size = 1.05 \[ \frac{-4 \csc ^2(e+f x)+\sqrt{\sec (e+f x)} \left (\log \left (\sqrt{\sec (e+f x)}+1\right )-\log \left (1-\sqrt{\sec (e+f x)}\right )\right )-2 \sqrt{\sec (e+f x)} \tan ^{-1}\left (\sqrt{\sec (e+f x)}\right )}{8 b f \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.125, size = 426, normalized size = 4.6 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 3.0209, size = 957, normalized size = 10.29 \begin{align*} \left [-\frac{2 \,{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} \sqrt{\frac{b}{\cos \left (f x + e\right )}}{\left (\cos \left (f x + e\right ) + 1\right )}}{2 \, b}\right ) +{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt{-b} \log \left (\frac{b \cos \left (f x + e\right )^{2} - 4 \,{\left (\cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt{-b} \sqrt{\frac{b}{\cos \left (f x + e\right )}} - 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) - 8 \, \sqrt{\frac{b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{16 \,{\left (b^{2} f \cos \left (f x + e\right )^{2} - b^{2} f\right )}}, \frac{2 \,{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt{b} \arctan \left (\frac{\sqrt{\frac{b}{\cos \left (f x + e\right )}}{\left (\cos \left (f x + e\right ) - 1\right )}}{2 \, \sqrt{b}}\right ) +{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sqrt{b} \log \left (\frac{b \cos \left (f x + e\right )^{2} + 4 \,{\left (\cos \left (f x + e\right )^{2} + \cos \left (f x + e\right )\right )} \sqrt{b} \sqrt{\frac{b}{\cos \left (f x + e\right )}} + 6 \, b \cos \left (f x + e\right ) + b}{\cos \left (f x + e\right )^{2} - 2 \, \cos \left (f x + e\right ) + 1}\right ) + 8 \, \sqrt{\frac{b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{16 \,{\left (b^{2} f \cos \left (f x + e\right )^{2} - b^{2} f\right )}}\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{\csc ^{3}{\left (e + f x \right )}}{\left (b \sec{\left (e + f x \right )}\right )^{\frac{3}{2}}}\, 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{\csc \left (f x + e\right )^{3}}{\left (b \sec \left (f x + e\right )\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]