Optimal. Leaf size=102 \[ -\frac {4 e^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 a d \sqrt {e \sin (c+d x)}}+\frac {2 e \sqrt {e \sin (c+d x)}}{a d}-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 a d} \]
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Rubi [A] time = 0.22, antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3872, 2839, 2564, 30, 2569, 2642, 2641} \[ -\frac {4 e^2 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{3 a d \sqrt {e \sin (c+d x)}}+\frac {2 e \sqrt {e \sin (c+d x)}}{a d}-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2564
Rule 2569
Rule 2641
Rule 2642
Rule 2839
Rule 3872
Rubi steps
\begin {align*} \int \frac {(e \sin (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx &=-\int \frac {\cos (c+d x) (e \sin (c+d x))^{3/2}}{-a-a \cos (c+d x)} \, dx\\ &=\frac {e^2 \int \frac {\cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx}{a}-\frac {e^2 \int \frac {\cos ^2(c+d x)}{\sqrt {e \sin (c+d x)}} \, dx}{a}\\ &=-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 a d}+\frac {e \operatorname {Subst}\left (\int \frac {1}{\sqrt {x}} \, dx,x,e \sin (c+d x)\right )}{a d}-\frac {\left (2 e^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{3 a}\\ &=\frac {2 e \sqrt {e \sin (c+d x)}}{a d}-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 a d}-\frac {\left (2 e^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a \sqrt {e \sin (c+d x)}}\\ &=-\frac {4 e^2 F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a d \sqrt {e \sin (c+d x)}}+\frac {2 e \sqrt {e \sin (c+d x)}}{a d}-\frac {2 e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 a d}\\ \end {align*}
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Mathematica [A] time = 20.65, size = 69, normalized size = 0.68 \[ -\frac {2 (e \sin (c+d x))^{3/2} \left (\sqrt {\sin (c+d x)} (\cos (c+d x)-3)-2 F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )\right )}{3 a d \sin ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.67, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \sin \left (d x + c\right )} e \sin \left (d x + c\right )}{a \sec \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}}{a \sec \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 3.28, size = 112, normalized size = 1.10 \[ \frac {2 e^{2} \left (\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-\left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 \cos \left (d x +c \right ) \sin \left (d x +c \right )\right )}{3 a \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d} \]
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 \[ \int \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}}{a \sec \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\cos \left (c+d\,x\right )\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}}{a\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]
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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