Optimal. Leaf size=74 \[ \frac {4 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{3/2}}-\frac {2 \sqrt {a \sin (c+d x)+a}}{d e (e \cos (c+d x))^{3/2}} \]
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Rubi [A] time = 0.15, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ \frac {4 (a \sin (c+d x)+a)^{3/2}}{3 a d e (e \cos (c+d x))^{3/2}}-\frac {2 \sqrt {a \sin (c+d x)+a}}{d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2671
Rule 2672
Rubi steps
\begin {align*} \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx &=-\frac {2 \sqrt {a+a \sin (c+d x)}}{d e (e \cos (c+d x))^{3/2}}+\frac {2 \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx}{a}\\ &=-\frac {2 \sqrt {a+a \sin (c+d x)}}{d e (e \cos (c+d x))^{3/2}}+\frac {4 (a+a \sin (c+d x))^{3/2}}{3 a d e (e \cos (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 46, normalized size = 0.62 \[ \frac {2 (2 \sin (c+d x)-1) \sqrt {a (\sin (c+d x)+1)}}{3 d e (e \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.89, size = 48, normalized size = 0.65 \[ \frac {2 \, \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} {\left (2 \, \sin \left (d x + c\right ) - 1\right )}}{3 \, d e^{3} \cos \left (d x + c\right )^{2}} \]
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 = 0.21, size = 44, normalized size = 0.59 \[ \frac {2 \left (2 \sin \left (d x +c \right )-1\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \cos \left (d x +c \right )}{3 d \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.97, size = 206, normalized size = 2.78 \[ -\frac {2 \, {\left (\sqrt {a} \sqrt {e} - \frac {4 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {4 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {\sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{3 \, {\left (e^{3} + \frac {2 \, e^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {e^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.67, size = 61, normalized size = 0.82 \[ -\frac {4\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (\cos \left (c+d\,x\right )-\sin \left (2\,c+2\,d\,x\right )\right )}{3\,d\,e^2\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )\,\sqrt {e\,\cos \left (c+d\,x\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}{\left (e \cos {\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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