Optimal. Leaf size=36 \[ -\frac {2 (e \cos (c+d x))^{5/2}}{5 d e (a \sin (c+d x)+a)^{5/2}} \]
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Rubi [A] time = 0.07, antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {2671} \[ -\frac {2 (e \cos (c+d x))^{5/2}}{5 d e (a \sin (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
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Rule 2671
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{3/2}}{(a+a \sin (c+d x))^{5/2}} \, dx &=-\frac {2 (e \cos (c+d x))^{5/2}}{5 d e (a+a \sin (c+d x))^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 49, normalized size = 1.36 \[ -\frac {2 \sqrt {a (\sin (c+d x)+1)} (e \cos (c+d x))^{5/2}}{5 a^3 d e (\sin (c+d x)+1)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.67, size = 70, normalized size = 1.94 \[ -\frac {2 \, \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} {\left (e \sin \left (d x + c\right ) - e\right )}}{5 \, {\left (a^{3} d \cos \left (d x + c\right )^{2} - 2 \, a^{3} d \sin \left (d x + c\right ) - 2 \, a^{3} d\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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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maple [A] time = 0.19, size = 34, normalized size = 0.94 \[ -\frac {2 \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \cos \left (d x +c \right )}{5 d \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.96, size = 131, normalized size = 3.64 \[ -\frac {2 \, {\left (\sqrt {a} e^{\frac {3}{2}} - \frac {\sqrt {a} e^{\frac {3}{2}} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}}{5 \, {\left (a^{3} + \frac {a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.57, size = 102, normalized size = 2.83 \[ -\frac {4\,e\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (\sin \left (c+d\,x\right )+2\,\cos \left (2\,c+2\,d\,x\right )+\sin \left (3\,c+3\,d\,x\right )+2\right )}{5\,a^3\,d\,\left (56\,\sin \left (c+d\,x\right )-28\,\cos \left (2\,c+2\,d\,x\right )+\cos \left (4\,c+4\,d\,x\right )-8\,\sin \left (3\,c+3\,d\,x\right )+35\right )} \]
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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