Optimal. Leaf size=33 \[ -\frac{2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{5/2}} \]
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Rubi [A] time = 0.0533232, antiderivative size = 33, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {2619} \[ -\frac{2 c d (d \csc (a+b x))^{5/2}}{5 b (c \sec (a+b x))^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2619
Rubi steps
\begin{align*} \int \frac{(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}}{5 b (c \sec (a+b x))^{5/2}}\\ \end{align*}
Mathematica [A] time = 0.133327, size = 45, normalized size = 1.36 \[ -\frac{2 d^3 \cot ^2(a+b x) \sqrt{d \csc (a+b x)}}{5 b c \sqrt{c \sec (a+b x)}} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.142, size = 42, normalized size = 1.3 \begin{align*} -{\frac{2\,\cos \left ( bx+a \right ) \sin \left ( bx+a \right ) }{5\,b} \left ({\frac{d}{\sin \left ( bx+a \right ) }} \right ) ^{{\frac{7}{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{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 [B] time = 1.81826, size = 131, normalized size = 3.97 \begin{align*} \frac{2 \, d^{3} \sqrt{\frac{c}{\cos \left (b x + a\right )}} \sqrt{\frac{d}{\sin \left (b x + a\right )}} \cos \left (b x + a\right )^{3}}{5 \,{\left (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{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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