Optimal. Leaf size=65 \[ -\frac{2 d^5}{11 b (d \tan (a+b x))^{11/2}}-\frac{4 d^3}{7 b (d \tan (a+b x))^{7/2}}-\frac{2 d}{3 b (d \tan (a+b x))^{3/2}} \]
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Rubi [A] time = 0.0489785, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.095, Rules used = {2591, 270} \[ -\frac{2 d^5}{11 b (d \tan (a+b x))^{11/2}}-\frac{4 d^3}{7 b (d \tan (a+b x))^{7/2}}-\frac{2 d}{3 b (d \tan (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2591
Rule 270
Rubi steps
\begin{align*} \int \frac{\csc ^6(a+b x)}{\sqrt{d \tan (a+b x)}} \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{\left (d^2+x^2\right )^2}{x^{13/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=\frac{d \operatorname{Subst}\left (\int \left (\frac{d^4}{x^{13/2}}+\frac{2 d^2}{x^{9/2}}+\frac{1}{x^{5/2}}\right ) \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac{2 d^5}{11 b (d \tan (a+b x))^{11/2}}-\frac{4 d^3}{7 b (d \tan (a+b x))^{7/2}}-\frac{2 d}{3 b (d \tan (a+b x))^{3/2}}\\ \end{align*}
Mathematica [A] time = 0.157082, size = 50, normalized size = 0.77 \[ \frac{2 d (28 \cos (2 (a+b x))-4 \cos (4 (a+b x))-45) \csc ^4(a+b x)}{231 b (d \tan (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.194, size = 60, normalized size = 0.9 \begin{align*} -{\frac{ \left ( 64\, \left ( \cos \left ( bx+a \right ) \right ) ^{4}-176\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}+154 \right ) \cos \left ( bx+a \right ) }{231\,b \left ( \sin \left ( bx+a \right ) \right ) ^{5}}{\frac{1}{\sqrt{{\frac{d\sin \left ( bx+a \right ) }{\cos \left ( bx+a \right ) }}}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.14466, size = 65, normalized size = 1. \begin{align*} -\frac{2 \,{\left (77 \, d^{4} \tan \left (b x + a\right )^{4} + 66 \, d^{4} \tan \left (b x + a\right )^{2} + 21 \, d^{4}\right )} d}{231 \, \left (d \tan \left (b x + a\right )\right )^{\frac{11}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.60822, size = 235, normalized size = 3.62 \begin{align*} \frac{2 \,{\left (32 \, \cos \left (b x + a\right )^{6} - 88 \, \cos \left (b x + a\right )^{4} + 77 \, \cos \left (b x + a\right )^{2}\right )} \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{231 \,{\left (b d \cos \left (b x + a\right )^{6} - 3 \, b d \cos \left (b x + a\right )^{4} + 3 \, b d \cos \left (b x + a\right )^{2} - b d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Giac [A] time = 1.10829, size = 78, normalized size = 1.2 \begin{align*} -\frac{2 \,{\left (77 \, d^{5} \tan \left (b x + a\right )^{4} + 66 \, d^{5} \tan \left (b x + a\right )^{2} + 21 \, d^{5}\right )}}{231 \, \sqrt{d \tan \left (b x + a\right )} b d^{5} \tan \left (b x + a\right )^{5}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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