Optimal. Leaf size=37 \[ \frac {2 (c \sin (a+b x))^{3/2}}{3 b c d (d \cos (a+b x))^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {2563} \[ \frac {2 (c \sin (a+b x))^{3/2}}{3 b c d (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2563
Rubi steps
\begin {align*} \int \frac {\sqrt {c \sin (a+b x)}}{(d \cos (a+b x))^{5/2}} \, dx &=\frac {2 (c \sin (a+b x))^{3/2}}{3 b c d (d \cos (a+b x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 37, normalized size = 1.00 \[ \frac {2 (c \sin (a+b x))^{3/2}}{3 b c d (d \cos (a+b x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.46, size = 42, normalized size = 1.14 \[ \frac {2 \, \sqrt {d \cos \left (b x + a\right )} \sqrt {c \sin \left (b x + a\right )} \sin \left (b x + a\right )}{3 \, b d^{3} \cos \left (b x + a\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c \sin \left (b x + a\right )}}{\left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.14, size = 38, normalized size = 1.03 \[ \frac {2 \sin \left (b x +a \right ) \cos \left (b x +a \right ) \sqrt {c \sin \left (b x +a \right )}}{3 b \left (d \cos \left (b x +a \right )\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c \sin \left (b x + a\right )}}{\left (d \cos \left (b x + a\right )\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.99, size = 50, normalized size = 1.35 \[ \frac {2\,\sin \left (2\,a+2\,b\,x\right )\,\sqrt {c\,\sin \left (a+b\,x\right )}}{3\,b\,d^2\,\left (\cos \left (2\,a+2\,b\,x\right )+1\right )\,\sqrt {d\,\cos \left (a+b\,x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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