Optimal. Leaf size=43 \[ -\frac{2 d^3}{7 b (d \tan (a+b x))^{7/2}}-\frac{2 d}{3 b (d \tan (a+b x))^{3/2}} \]
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Rubi [A] time = 0.0427622, antiderivative size = 43, 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, 14} \[ -\frac{2 d^3}{7 b (d \tan (a+b x))^{7/2}}-\frac{2 d}{3 b (d \tan (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2591
Rule 14
Rubi steps
\begin{align*} \int \frac{\csc ^4(a+b x)}{\sqrt{d \tan (a+b x)}} \, dx &=\frac{d \operatorname{Subst}\left (\int \frac{d^2+x^2}{x^{9/2}} \, dx,x,d \tan (a+b x)\right )}{b}\\ &=\frac{d \operatorname{Subst}\left (\int \left (\frac{d^2}{x^{9/2}}+\frac{1}{x^{5/2}}\right ) \, dx,x,d \tan (a+b x)\right )}{b}\\ &=-\frac{2 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 = 0.127127, size = 40, normalized size = 0.93 \[ \frac{2 d (2 \cos (2 (a+b x))-5) \csc ^2(a+b x)}{21 b (d \tan (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.173, size = 50, normalized size = 1.2 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-14 \right ) \cos \left ( bx+a \right ) }{21\,b \left ( \sin \left ( bx+a \right ) \right ) ^{3}}{\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 = 1.0649, size = 47, normalized size = 1.09 \begin{align*} -\frac{2 \,{\left (7 \, d^{2} \tan \left (b x + a\right )^{2} + 3 \, d^{2}\right )} d}{21 \, \left (d \tan \left (b x + a\right )\right )^{\frac{7}{2}} b} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.31248, size = 173, normalized size = 4.02 \begin{align*} \frac{2 \,{\left (4 \, \cos \left (b x + a\right )^{4} - 7 \, \cos \left (b x + a\right )^{2}\right )} \sqrt{\frac{d \sin \left (b x + a\right )}{\cos \left (b x + a\right )}}}{21 \,{\left (b d \cos \left (b x + a\right )^{4} - 2 \, 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 = 1.13789, size = 61, normalized size = 1.42 \begin{align*} -\frac{2 \,{\left (7 \, d^{3} \tan \left (b x + a\right )^{2} + 3 \, d^{3}\right )}}{21 \, \sqrt{d \tan \left (b x + a\right )} b d^{3} \tan \left (b x + a\right )^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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