Optimal. Leaf size=115 \[ \frac {16 \sqrt {a \sin (c+d x)+a}}{21 a^2 d e \sqrt {e \cos (c+d x)}}-\frac {8}{21 a d e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}-\frac {2}{7 d e (a \sin (c+d x)+a)^{3/2} \sqrt {e \cos (c+d x)}} \]
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Rubi [A] time = 0.21, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2672, 2671} \[ \frac {16 \sqrt {a \sin (c+d x)+a}}{21 a^2 d e \sqrt {e \cos (c+d x)}}-\frac {8}{21 a d e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}-\frac {2}{7 d e (a \sin (c+d x)+a)^{3/2} \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2671
Rule 2672
Rubi steps
\begin {align*} \int \frac {1}{(e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}} \, dx &=-\frac {2}{7 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}+\frac {4 \int \frac {1}{(e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}} \, dx}{7 a}\\ &=-\frac {2}{7 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac {8}{21 a d e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}+\frac {8 \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx}{21 a^2}\\ &=-\frac {2}{7 d e \sqrt {e \cos (c+d x)} (a+a \sin (c+d x))^{3/2}}-\frac {8}{21 a d e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}+\frac {16 \sqrt {a+a \sin (c+d x)}}{21 a^2 d e \sqrt {e \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 56, normalized size = 0.49 \[ \frac {16 \sin ^2(c+d x)+24 \sin (c+d x)+2}{21 d e (a (\sin (c+d x)+1))^{3/2} \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.04, size = 99, normalized size = 0.86 \[ \frac {2 \, \sqrt {e \cos \left (d x + c\right )} {\left (8 \, \cos \left (d x + c\right )^{2} - 12 \, \sin \left (d x + c\right ) - 9\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{21 \, {\left (a^{2} d e^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} d e^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) - 2 \, a^{2} d e^{2} \cos \left (d x + c\right )\right )}} \]
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 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} {\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.17, size = 54, normalized size = 0.47 \[ \frac {2 \left (-8 \left (\cos ^{2}\left (d x +c \right )\right )+12 \sin \left (d x +c \right )+9\right ) \cos \left (d x +c \right )}{21 d \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.89, size = 294, normalized size = 2.56 \[ \frac {2 \, {\left (\sqrt {a} \sqrt {e} + \frac {24 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {33 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {33 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} - \frac {24 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} - \frac {\sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{3}}{21 \, {\left (a^{2} e^{2} + \frac {3 \, a^{2} e^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, a^{2} e^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {a^{2} e^{2} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {9}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.82, size = 119, normalized size = 1.03 \[ \frac {8\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (70\,\sin \left (c+d\,x\right )-41\,\cos \left (2\,c+2\,d\,x\right )+2\,\cos \left (4\,c+4\,d\,x\right )-14\,\sin \left (3\,c+3\,d\,x\right )+41\right )}{21\,a^2\,d\,e\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (56\,\sin \left (c+d\,x\right )-28\,\cos \left (2\,c+2\,d\,x\right )+\cos \left (4\,c+4\,d\,x\right )-8\,\sin \left (3\,c+3\,d\,x\right )+35\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {3}{2}} \left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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