Optimal. Leaf size=100 \[ \frac{2}{b d^3 \sqrt{d \cos (a+b x)}}+\frac{\tan ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b d^{7/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b d^{7/2}}+\frac{2}{5 b d (d \cos (a+b x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0778657, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.316, Rules used = {2565, 325, 329, 298, 203, 206} \[ \frac{2}{b d^3 \sqrt{d \cos (a+b x)}}+\frac{\tan ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b d^{7/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b d^{7/2}}+\frac{2}{5 b d (d \cos (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 2565
Rule 325
Rule 329
Rule 298
Rule 203
Rule 206
Rubi steps
\begin{align*} \int \frac{\csc (a+b x)}{(d \cos (a+b x))^{7/2}} \, dx &=-\frac{\operatorname{Subst}\left (\int \frac{1}{x^{7/2} \left (1-\frac{x^2}{d^2}\right )} \, dx,x,d \cos (a+b x)\right )}{b d}\\ &=\frac{2}{5 b d (d \cos (a+b x))^{5/2}}-\frac{\operatorname{Subst}\left (\int \frac{1}{x^{3/2} \left (1-\frac{x^2}{d^2}\right )} \, dx,x,d \cos (a+b x)\right )}{b d^3}\\ &=\frac{2}{5 b d (d \cos (a+b x))^{5/2}}+\frac{2}{b d^3 \sqrt{d \cos (a+b x)}}-\frac{\operatorname{Subst}\left (\int \frac{\sqrt{x}}{1-\frac{x^2}{d^2}} \, dx,x,d \cos (a+b x)\right )}{b d^5}\\ &=\frac{2}{5 b d (d \cos (a+b x))^{5/2}}+\frac{2}{b d^3 \sqrt{d \cos (a+b x)}}-\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{1-\frac{x^4}{d^2}} \, dx,x,\sqrt{d \cos (a+b x)}\right )}{b d^5}\\ &=\frac{2}{5 b d (d \cos (a+b x))^{5/2}}+\frac{2}{b d^3 \sqrt{d \cos (a+b x)}}-\frac{\operatorname{Subst}\left (\int \frac{1}{d-x^2} \, dx,x,\sqrt{d \cos (a+b x)}\right )}{b d^3}+\frac{\operatorname{Subst}\left (\int \frac{1}{d+x^2} \, dx,x,\sqrt{d \cos (a+b x)}\right )}{b d^3}\\ &=\frac{\tan ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b d^{7/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d \cos (a+b x)}}{\sqrt{d}}\right )}{b d^{7/2}}+\frac{2}{5 b d (d \cos (a+b x))^{5/2}}+\frac{2}{b d^3 \sqrt{d \cos (a+b x)}}\\ \end{align*}
Mathematica [C] time = 0.0682588, size = 38, normalized size = 0.38 \[ \frac{2 \, _2F_1\left (-\frac{5}{4},1;-\frac{1}{4};\cos ^2(a+b x)\right )}{5 b d (d \cos (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.22, size = 882, normalized size = 8.8 \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 = 2.70545, size = 942, normalized size = 9.42 \begin{align*} \left [\frac{10 \, \sqrt{-d} \arctan \left (\frac{\sqrt{d \cos \left (b x + a\right )} \sqrt{-d}{\left (\cos \left (b x + a\right ) + 1\right )}}{2 \, d \cos \left (b x + a\right )}\right ) \cos \left (b x + a\right )^{3} - 5 \, \sqrt{-d} \cos \left (b x + a\right )^{3} \log \left (\frac{d \cos \left (b x + a\right )^{2} - 4 \, \sqrt{d \cos \left (b x + a\right )} \sqrt{-d}{\left (\cos \left (b x + a\right ) - 1\right )} - 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} + 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, \sqrt{d \cos \left (b x + a\right )}{\left (5 \, \cos \left (b x + a\right )^{2} + 1\right )}}{20 \, b d^{4} \cos \left (b x + a\right )^{3}}, \frac{10 \, \sqrt{d} \arctan \left (\frac{\sqrt{d \cos \left (b x + a\right )}{\left (\cos \left (b x + a\right ) - 1\right )}}{2 \, \sqrt{d} \cos \left (b x + a\right )}\right ) \cos \left (b x + a\right )^{3} + 5 \, \sqrt{d} \cos \left (b x + a\right )^{3} \log \left (\frac{d \cos \left (b x + a\right )^{2} - 4 \, \sqrt{d \cos \left (b x + a\right )} \sqrt{d}{\left (\cos \left (b x + a\right ) + 1\right )} + 6 \, d \cos \left (b x + a\right ) + d}{\cos \left (b x + a\right )^{2} - 2 \, \cos \left (b x + a\right ) + 1}\right ) + 8 \, \sqrt{d \cos \left (b x + a\right )}{\left (5 \, \cos \left (b x + a\right )^{2} + 1\right )}}{20 \, b d^{4} \cos \left (b x + a\right )^{3}}\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 [A] time = 1.13774, size = 127, normalized size = 1.27 \begin{align*} \frac{d{\left (\frac{5 \, \arctan \left (\frac{\sqrt{d \cos \left (b x + a\right )}}{\sqrt{-d}}\right )}{\sqrt{-d} d^{4}} + \frac{5 \, \arctan \left (\frac{\sqrt{d \cos \left (b x + a\right )}}{\sqrt{d}}\right )}{d^{\frac{9}{2}}} + \frac{2 \,{\left (5 \, d^{2} \cos \left (b x + a\right )^{2} + d^{2}\right )}}{\sqrt{d \cos \left (b x + a\right )} d^{6} \cos \left (b x + a\right )^{2}}\right )}}{5 \, b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]