Optimal. Leaf size=105 \[ -\frac{2 d \csc ^3(a+b x)}{7 b \sqrt{d \tan (a+b x)}}-\frac{4 d \csc (a+b x)}{7 b \sqrt{d \tan (a+b x)}}+\frac{4 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{d \tan (a+b x)}}{7 b} \]
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Rubi [A] time = 0.142963, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19, Rules used = {2599, 2601, 2573, 2641} \[ -\frac{2 d \csc ^3(a+b x)}{7 b \sqrt{d \tan (a+b x)}}-\frac{4 d \csc (a+b x)}{7 b \sqrt{d \tan (a+b x)}}+\frac{4 \sqrt{\sin (2 a+2 b x)} \csc (a+b x) F\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{d \tan (a+b x)}}{7 b} \]
Antiderivative was successfully verified.
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Rule 2599
Rule 2601
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int \csc ^5(a+b x) \sqrt{d \tan (a+b x)} \, dx &=-\frac{2 d \csc ^3(a+b x)}{7 b \sqrt{d \tan (a+b x)}}+\frac{6}{7} \int \csc ^3(a+b x) \sqrt{d \tan (a+b x)} \, dx\\ &=-\frac{4 d \csc (a+b x)}{7 b \sqrt{d \tan (a+b x)}}-\frac{2 d \csc ^3(a+b x)}{7 b \sqrt{d \tan (a+b x)}}+\frac{4}{7} \int \csc (a+b x) \sqrt{d \tan (a+b x)} \, dx\\ &=-\frac{4 d \csc (a+b x)}{7 b \sqrt{d \tan (a+b x)}}-\frac{2 d \csc ^3(a+b x)}{7 b \sqrt{d \tan (a+b x)}}+\frac{\left (4 \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}{7 \sqrt{\sin (a+b x)}}\\ &=-\frac{4 d \csc (a+b x)}{7 b \sqrt{d \tan (a+b x)}}-\frac{2 d \csc ^3(a+b x)}{7 b \sqrt{d \tan (a+b x)}}+\frac{1}{7} \left (4 \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{4 d \csc (a+b x)}{7 b \sqrt{d \tan (a+b x)}}-\frac{2 d \csc ^3(a+b x)}{7 b \sqrt{d \tan (a+b x)}}+\frac{4 \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)}}{7 b}\\ \end{align*}
Mathematica [C] time = 1.42776, size = 124, normalized size = 1.18 \[ -\frac{2 d \cos (2 (a+b x)) \csc ^3(a+b x) \left ((\cos (2 (a+b x))-2) \sec ^2(a+b x)^{3/2}-4 \sqrt [4]{-1} \tan ^{\frac{7}{2}}(a+b x) F\left (\left .i \sinh ^{-1}\left (\sqrt [4]{-1} \sqrt{\tan (a+b x)}\right )\right |-1\right )\right )}{7 b \left (\tan ^2(a+b x)-1\right ) \sqrt{\sec ^2(a+b x)} \sqrt{d \tan (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.189, size = 558, normalized size = 5.3 \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 \sqrt{d \tan \left (b x + a\right )} \csc \left (b x + a\right )^{5}\,{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 )} \csc \left (b x + a\right )^{5}, 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 \sqrt{d \tan \left (b x + a\right )} \csc \left (b x + a\right )^{5}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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