Optimal. Leaf size=128 \[ \frac{4 d^4 \sqrt{\sin (2 a+2 b x)} \text{EllipticF}\left (a+b x-\frac{\pi }{4},2\right ) \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}{7 b}-\frac{4 c d^3 (d \csc (a+b x))^{3/2}}{7 b \sqrt{c \sec (a+b x)}}-\frac{2 c d (d \csc (a+b x))^{7/2}}{7 b \sqrt{c \sec (a+b x)}} \]
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Rubi [A] time = 0.203931, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {2625, 2630, 2573, 2641} \[ -\frac{4 c d^3 (d \csc (a+b x))^{3/2}}{7 b \sqrt{c \sec (a+b x)}}+\frac{4 d^4 \sqrt{\sin (2 a+2 b x)} F\left (\left .a+b x-\frac{\pi }{4}\right |2\right ) \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)}}{7 b}-\frac{2 c d (d \csc (a+b x))^{7/2}}{7 b \sqrt{c \sec (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2625
Rule 2630
Rule 2573
Rule 2641
Rubi steps
\begin{align*} \int (d \csc (a+b x))^{9/2} \sqrt{c \sec (a+b x)} \, dx &=-\frac{2 c d (d \csc (a+b x))^{7/2}}{7 b \sqrt{c \sec (a+b x)}}+\frac{1}{7} \left (6 d^2\right ) \int (d \csc (a+b x))^{5/2} \sqrt{c \sec (a+b x)} \, dx\\ &=-\frac{4 c d^3 (d \csc (a+b x))^{3/2}}{7 b \sqrt{c \sec (a+b x)}}-\frac{2 c d (d \csc (a+b x))^{7/2}}{7 b \sqrt{c \sec (a+b x)}}+\frac{1}{7} \left (4 d^4\right ) \int \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)} \, dx\\ &=-\frac{4 c d^3 (d \csc (a+b x))^{3/2}}{7 b \sqrt{c \sec (a+b x)}}-\frac{2 c d (d \csc (a+b x))^{7/2}}{7 b \sqrt{c \sec (a+b x)}}+\frac{1}{7} \left (4 d^4 \sqrt{c \cos (a+b x)} \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)} \sqrt{d \sin (a+b x)}\right ) \int \frac{1}{\sqrt{c \cos (a+b x)} \sqrt{d \sin (a+b x)}} \, dx\\ &=-\frac{4 c d^3 (d \csc (a+b x))^{3/2}}{7 b \sqrt{c \sec (a+b x)}}-\frac{2 c d (d \csc (a+b x))^{7/2}}{7 b \sqrt{c \sec (a+b x)}}+\frac{1}{7} \left (4 d^4 \sqrt{d \csc (a+b x)} \sqrt{c \sec (a+b x)} \sqrt{\sin (2 a+2 b x)}\right ) \int \frac{1}{\sqrt{\sin (2 a+2 b x)}} \, dx\\ &=-\frac{4 c d^3 (d \csc (a+b x))^{3/2}}{7 b \sqrt{c \sec (a+b x)}}-\frac{2 c d (d \csc (a+b x))^{7/2}}{7 b \sqrt{c \sec (a+b x)}}+\frac{4 d^4 \sqrt{d \csc (a+b x)} F\left (\left .a-\frac{\pi }{4}+b x\right |2\right ) \sqrt{c \sec (a+b x)} \sqrt{\sin (2 a+2 b x)}}{7 b}\\ \end{align*}
Mathematica [C] time = 1.48662, size = 122, normalized size = 0.95 \[ \frac{2 d^4 \cos (2 (a+b x)) \cot (a+b x) \sqrt{c \sec (a+b x)} \sqrt{d \csc (a+b x)} \left ((\cos (2 (a+b x))-2) \csc ^4(a+b x)-2 \left (-\cot ^2(a+b x)\right )^{3/4} \sec ^2(a+b x) \text{Hypergeometric2F1}\left (\frac{1}{2},\frac{3}{4},\frac{3}{2},\csc ^2(a+b x)\right )\right )}{7 b \left (\csc ^2(a+b x)-2\right )} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.253, size = 542, normalized size = 4.2 \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{9}{2}} \sqrt{c \sec \left (b x + a\right )}\,{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 )} d^{4} \csc \left (b x + a\right )^{4}, 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{9}{2}} \sqrt{c \sec \left (b x + a\right )}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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