Optimal. Leaf size=166 \[ -\frac{24 c^2 d^4 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{5 b \sqrt{\sin (2 a+2 b x)} \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}+\frac{24 c d^5 \sqrt{c \sec (a+b x)}}{5 b (d \csc (a+b x))^{3/2}}-\frac{12 c d^3 \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}{5 b}-\frac{2 c d \sqrt{c \sec (a+b x)} (d \csc (a+b x))^{5/2}}{5 b} \]
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Rubi [A] time = 0.267906, antiderivative size = 166, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {2625, 2626, 2630, 2572, 2639} \[ -\frac{24 c^2 d^4 E\left (\left .a+b x-\frac{\pi }{4}\right |2\right )}{5 b \sqrt{\sin (2 a+2 b x)} \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}+\frac{24 c d^5 \sqrt{c \sec (a+b x)}}{5 b (d \csc (a+b x))^{3/2}}-\frac{12 c d^3 \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}{5 b}-\frac{2 c d \sqrt{c \sec (a+b x)} (d \csc (a+b x))^{5/2}}{5 b} \]
Antiderivative was successfully verified.
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Rule 2625
Rule 2626
Rule 2630
Rule 2572
Rule 2639
Rubi steps
\begin{align*} \int (d \csc (a+b x))^{7/2} (c \sec (a+b x))^{3/2} \, dx &=-\frac{2 c d (d \csc (a+b x))^{5/2} \sqrt{c \sec (a+b x)}}{5 b}+\frac{1}{5} \left (6 d^2\right ) \int (d \csc (a+b x))^{3/2} (c \sec (a+b x))^{3/2} \, dx\\ &=-\frac{12 c d^3 \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)}}{5 b}-\frac{2 c d (d \csc (a+b x))^{5/2} \sqrt{c \sec (a+b x)}}{5 b}+\frac{1}{5} \left (12 d^4\right ) \int \frac{(c \sec (a+b x))^{3/2}}{\sqrt{d \csc (a+b x)}} \, dx\\ &=\frac{24 c d^5 \sqrt{c \sec (a+b x)}}{5 b (d \csc (a+b x))^{3/2}}-\frac{12 c d^3 \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)}}{5 b}-\frac{2 c d (d \csc (a+b x))^{5/2} \sqrt{c \sec (a+b x)}}{5 b}-\frac{1}{5} \left (24 c^2 d^4\right ) \int \frac{1}{\sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)}} \, dx\\ &=\frac{24 c d^5 \sqrt{c \sec (a+b x)}}{5 b (d \csc (a+b x))^{3/2}}-\frac{12 c d^3 \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)}}{5 b}-\frac{2 c d (d \csc (a+b x))^{5/2} \sqrt{c \sec (a+b x)}}{5 b}-\frac{\left (24 c^2 d^4\right ) \int \sqrt{c \cos (a+b x)} \sqrt{d \sin (a+b x)} \, dx}{5 \sqrt{c \cos (a+b x)} \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)} \sqrt{d \sin (a+b x)}}\\ &=\frac{24 c d^5 \sqrt{c \sec (a+b x)}}{5 b (d \csc (a+b x))^{3/2}}-\frac{12 c d^3 \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)}}{5 b}-\frac{2 c d (d \csc (a+b x))^{5/2} \sqrt{c \sec (a+b x)}}{5 b}-\frac{\left (24 c^2 d^4\right ) \int \sqrt{\sin (2 a+2 b x)} \, dx}{5 \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)} \sqrt{\sin (2 a+2 b x)}}\\ &=\frac{24 c d^5 \sqrt{c \sec (a+b x)}}{5 b (d \csc (a+b x))^{3/2}}-\frac{12 c d^3 \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)}}{5 b}-\frac{2 c d (d \csc (a+b x))^{5/2} \sqrt{c \sec (a+b x)}}{5 b}-\frac{24 c^2 d^4 E\left (\left .a-\frac{\pi }{4}+b x\right |2\right )}{5 b \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)} \sqrt{\sin (2 a+2 b x)}}\\ \end{align*}
Mathematica [C] time = 1.11952, size = 114, normalized size = 0.69 \[ -\frac{2 c d^3 \tan ^2(a+b x) \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)} \left (12 \cos ^2(a+b x) \sqrt [4]{-\cot ^2(a+b x)} \text{Hypergeometric2F1}\left (-\frac{1}{2},\frac{1}{4},\frac{1}{2},\csc ^2(a+b x)\right )+\cot ^2(a+b x) \left (6 \cos (2 (a+b x))+\csc ^2(a+b x)\right )\right )}{5 b} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.207, size = 996, normalized size = 6. \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 \csc \left (b x + a\right )\right )^{\frac{7}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac{3}{2}}\,{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 \csc \left (b x + a\right )} \sqrt{c \sec \left (b x + a\right )} c d^{3} \csc \left (b x + a\right )^{3} \sec \left (b x + a\right ), 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 \csc \left (b x + a\right )\right )^{\frac{7}{2}} \left (c \sec \left (b x + a\right )\right )^{\frac{3}{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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