Optimal. Leaf size=93 \[ -\frac{\cot ^2(e+f x) \sqrt{b \sec (e+f x)}}{2 b^3 f}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{4 b^{5/2} f}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{4 b^{5/2} f} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0728972, 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, 212, 206, 203} \[ -\frac{\cot ^2(e+f x) \sqrt{b \sec (e+f x)}}{2 b^3 f}+\frac{3 \tan ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{4 b^{5/2} f}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{4 b^{5/2} f} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2622
Rule 290
Rule 329
Rule 212
Rule 206
Rule 203
Rubi steps
\begin{align*} \int \frac{\csc ^3(e+f x)}{(b \sec (e+f x))^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\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) \sqrt{b \sec (e+f x)}}{2 b^3 f}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{\sqrt{x} \left (-1+\frac{x^2}{b^2}\right )} \, dx,x,b \sec (e+f x)\right )}{4 b^3 f}\\ &=-\frac{\cot ^2(e+f x) \sqrt{b \sec (e+f x)}}{2 b^3 f}-\frac{3 \operatorname{Subst}\left (\int \frac{1}{-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) \sqrt{b \sec (e+f x)}}{2 b^3 f}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{b-x^2} \, dx,x,\sqrt{b \sec (e+f x)}\right )}{4 b^2 f}+\frac{3 \operatorname{Subst}\left (\int \frac{1}{b+x^2} \, dx,x,\sqrt{b \sec (e+f x)}\right )}{4 b^2 f}\\ &=\frac{3 \tan ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{4 b^{5/2} f}+\frac{3 \tanh ^{-1}\left (\frac{\sqrt{b \sec (e+f x)}}{\sqrt{b}}\right )}{4 b^{5/2} f}-\frac{\cot ^2(e+f x) \sqrt{b \sec (e+f x)}}{2 b^3 f}\\ \end{align*}
Mathematica [A] time = 2.34729, size = 98, normalized size = 1.05 \[ \frac{\sqrt{\sec (e+f x)} \left (-3 \log \left (1-\sqrt{\sec (e+f x)}\right )+3 \log \left (\sqrt{\sec (e+f x)}+1\right )-\frac{4 \csc ^2(e+f x)}{\sec ^{\frac{3}{2}}(e+f x)}+6 \tan ^{-1}\left (\sqrt{\sec (e+f x)}\right )\right )}{8 b^2 f \sqrt{b \sec (e+f x)}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.127, size = 437, normalized size = 4.7 \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 = 2.75014, size = 969, normalized size = 10.42 \begin{align*} \left [-\frac{6 \,{\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 ) - 8 \, \sqrt{\frac{b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} + 3 \,{\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 )}{16 \,{\left (b^{3} f \cos \left (f x + e\right )^{2} - b^{3} f\right )}}, -\frac{6 \,{\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 ) - 8 \, \sqrt{\frac{b}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )^{2} - 3 \,{\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 )}{16 \,{\left (b^{3} f \cos \left (f x + e\right )^{2} - b^{3} f\right )}}\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{\csc \left (f x + e\right )^{3}}{\left (b \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]