Optimal. Leaf size=110 \[ \frac{8 d^5 \sqrt{d \csc (a+b x)}}{45 b c \sqrt{c \sec (a+b x)}}+\frac{2 d^3 (d \csc (a+b x))^{5/2}}{45 b c \sqrt{c \sec (a+b x)}}-\frac{2 d (d \csc (a+b x))^{9/2}}{9 b c \sqrt{c \sec (a+b x)}} \]
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Rubi [A] time = 0.153996, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.12, Rules used = {2623, 2625, 2619} \[ \frac{8 d^5 \sqrt{d \csc (a+b x)}}{45 b c \sqrt{c \sec (a+b x)}}+\frac{2 d^3 (d \csc (a+b x))^{5/2}}{45 b c \sqrt{c \sec (a+b x)}}-\frac{2 d (d \csc (a+b x))^{9/2}}{9 b c \sqrt{c \sec (a+b x)}} \]
Antiderivative was successfully verified.
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Rule 2623
Rule 2625
Rule 2619
Rubi steps
\begin{align*} \int \frac{(d \csc (a+b x))^{11/2}}{(c \sec (a+b x))^{3/2}} \, dx &=-\frac{2 d (d \csc (a+b x))^{9/2}}{9 b c \sqrt{c \sec (a+b x)}}-\frac{d^2 \int (d \csc (a+b x))^{7/2} \sqrt{c \sec (a+b x)} \, dx}{9 c^2}\\ &=\frac{2 d^3 (d \csc (a+b x))^{5/2}}{45 b c \sqrt{c \sec (a+b x)}}-\frac{2 d (d \csc (a+b x))^{9/2}}{9 b c \sqrt{c \sec (a+b x)}}-\frac{\left (4 d^4\right ) \int (d \csc (a+b x))^{3/2} \sqrt{c \sec (a+b x)} \, dx}{45 c^2}\\ &=\frac{8 d^5 \sqrt{d \csc (a+b x)}}{45 b c \sqrt{c \sec (a+b x)}}+\frac{2 d^3 (d \csc (a+b x))^{5/2}}{45 b c \sqrt{c \sec (a+b x)}}-\frac{2 d (d \csc (a+b x))^{9/2}}{9 b c \sqrt{c \sec (a+b x)}}\\ \end{align*}
Mathematica [A] time = 0.280943, size = 57, normalized size = 0.52 \[ \frac{2 d^3 (2 \cos (2 (a+b x))-7) \cot ^2(a+b x) (d \csc (a+b x))^{5/2}}{45 b c \sqrt{c \sec (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.158, size = 54, normalized size = 0.5 \begin{align*}{\frac{ \left ( 8\, \left ( \cos \left ( bx+a \right ) \right ) ^{2}-18 \right ) \cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{45\,b} \left ({\frac{d}{\sin \left ( bx+a \right ) }} \right ) ^{{\frac{11}{2}}} \left ({\frac{c}{\cos \left ( bx+a \right ) }} \right ) ^{-{\frac{3}{2}}}} \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 \frac{\left (d \csc \left (b x + a\right )\right )^{\frac{11}{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 [A] time = 2.21246, size = 203, normalized size = 1.85 \begin{align*} \frac{2 \,{\left (4 \, d^{5} \cos \left (b x + a\right )^{5} - 9 \, d^{5} \cos \left (b x + a\right )^{3}\right )} \sqrt{\frac{c}{\cos \left (b x + a\right )}} \sqrt{\frac{d}{\sin \left (b x + a\right )}}}{45 \,{\left (b c^{2} \cos \left (b x + a\right )^{4} - 2 \, b c^{2} \cos \left (b x + a\right )^{2} + b c^{2}\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 \frac{\left (d \csc \left (b x + a\right )\right )^{\frac{11}{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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