Optimal. Leaf size=140 \[ -\frac{20 d^3 \csc ^3(a+b x)}{21 b \sqrt{d \tan (a+b x)}}-\frac{40 d^3 \csc (a+b x)}{21 b \sqrt{d \tan (a+b x)}}+\frac{40 d^2 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{d \tan (a+b x)}}{21 b}+\frac{2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b} \]
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Rubi [A] time = 0.184841, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {2593, 2599, 2601, 2573, 2641} \[ -\frac{20 d^3 \csc ^3(a+b x)}{21 b \sqrt{d \tan (a+b x)}}-\frac{40 d^3 \csc (a+b x)}{21 b \sqrt{d \tan (a+b x)}}+\frac{40 d^2 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{d \tan (a+b x)}}{21 b}+\frac{2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 2593
Rule 2599
Rule 2601
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int \csc ^7(a+b x) (d \tan (a+b x))^{5/2} \, dx &=\frac{2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac{1}{3} \left (10 d^2\right ) \int \csc ^5(a+b x) \sqrt{d \tan (a+b x)} \, dx\\ &=-\frac{20 d^3 \csc ^3(a+b x)}{21 b \sqrt{d \tan (a+b x)}}+\frac{2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac{1}{7} \left (20 d^2\right ) \int \csc ^3(a+b x) \sqrt{d \tan (a+b x)} \, dx\\ &=-\frac{40 d^3 \csc (a+b x)}{21 b \sqrt{d \tan (a+b x)}}-\frac{20 d^3 \csc ^3(a+b x)}{21 b \sqrt{d \tan (a+b x)}}+\frac{2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac{1}{21} \left (40 d^2\right ) \int \csc (a+b x) \sqrt{d \tan (a+b x)} \, dx\\ &=-\frac{40 d^3 \csc (a+b x)}{21 b \sqrt{d \tan (a+b x)}}-\frac{20 d^3 \csc ^3(a+b x)}{21 b \sqrt{d \tan (a+b x)}}+\frac{2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac{\left (40 d^2 \sqrt{\cos (a+b x)} \sqrt{d \tan (a+b x)}\right ) \int \frac{1}{\sqrt{\cos (a+b x)} \sqrt{\sin (a+b x)}} \, dx}{21 \sqrt{\sin (a+b x)}}\\ &=-\frac{40 d^3 \csc (a+b x)}{21 b \sqrt{d \tan (a+b x)}}-\frac{20 d^3 \csc ^3(a+b x)}{21 b \sqrt{d \tan (a+b x)}}+\frac{2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}+\frac{1}{21} \left (40 d^2 \csc (a+b x) \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}\right ) \int \frac{1}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=-\frac{40 d^3 \csc (a+b x)}{21 b \sqrt{d \tan (a+b x)}}-\frac{20 d^3 \csc ^3(a+b x)}{21 b \sqrt{d \tan (a+b x)}}+\frac{40 d^2 \csc (a+b x) F\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sqrt{\sin (2 a+2 b x)} \sqrt{d \tan (a+b x)}}{21 b}+\frac{2 d \csc ^5(a+b x) (d \tan (a+b x))^{3/2}}{3 b}\\ \end{align*}
Mathematica [C] time = 1.60996, size = 130, normalized size = 0.93 \[ -\frac{d^2 \csc (a+b x) \sqrt{d \tan (a+b x)} \left ((10 \cos (2 (a+b x))-5 \cos (4 (a+b x))+1) \csc ^3(a+b x) \sec (a+b x) \sqrt{\sec ^2(a+b x)}+80 \sqrt [4]{-1} \sqrt{\tan (a+b x)} F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{\tan (a+b x)}\right )\right |-1\right )\right )}{21 b \sqrt{\sec ^2(a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.187, size = 571, normalized size = 4.1 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan \left (b x + a\right )\right )^{\frac{5}{2}} \csc \left (b x + a\right )^{7}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\sqrt{d \tan \left (b x + a\right )} d^{2} \csc \left (b x + a\right )^{7} \tan \left (b x + a\right )^{2}, x\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 [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (d \tan \left (b x + a\right )\right )^{\frac{5}{2}} \csc \left (b x + a\right )^{7}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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