Optimal. Leaf size=200 \[ \frac {e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{a d (\sin (c+d x)+\cos (c+d x)+1)}-\frac {e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{a d (\sin (c+d x)+\cos (c+d x)+1)}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d} \]
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Rubi [A] time = 0.28, antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.259, Rules used = {2685, 2677, 2775, 203, 2833, 63, 215} \[ \frac {e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{a d (\sin (c+d x)+\cos (c+d x)+1)}-\frac {e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{a d (\sin (c+d x)+\cos (c+d x)+1)}+\frac {e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{a d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 215
Rule 2677
Rule 2685
Rule 2775
Rule 2833
Rubi steps
\begin {align*} \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {a+a \sin (c+d x)}} \, dx &=\frac {e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{a d}+\frac {e^2 \int \frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx}{2 a}\\ &=\frac {e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{a d}+\frac {\left (e^2 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sqrt {1+\cos (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx}{2 (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (e^2 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \int \frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}} \, dx}{2 (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac {e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{a d}-\frac {\left (e^2 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {e x} \sqrt {1+x}} \, dx,x,\cos (c+d x)\right )}{2 d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (e^2 \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{1+e x^2} \, dx,x,-\frac {\sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right )}{d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac {e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{a d}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (e \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1+\frac {x^2}{e}}} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=\frac {e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{a d}-\frac {e^{3/2} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {e^{3/2} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {e \cos (c+d x)} \sqrt {1+\cos (c+d x)}}\right ) \sqrt {1+\cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{d (a+a \cos (c+d x)+a \sin (c+d x))}\\ \end {align*}
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Mathematica [C] time = 0.11, size = 77, normalized size = 0.38 \[ -\frac {2\ 2^{3/4} (e \cos (c+d x))^{5/2} \, _2F_1\left (\frac {1}{4},\frac {5}{4};\frac {9}{4};\frac {1}{2} (1-\sin (c+d x))\right )}{5 d e (\sin (c+d x)+1)^{3/4} \sqrt {a (\sin (c+d x)+1)}} \]
Antiderivative was successfully verified.
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fricas [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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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 = 0.24, size = 212, normalized size = 1.06 \[ \frac {\left (\sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \sin \left (d x +c \right )-\sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sin \left (d x +c \right )+2 \cos \left (d x +c \right ) \sin \left (d x +c \right )-2 \left (\cos ^{2}\left (d x +c \right )\right )+2 \cos \left (d x +c \right )\right ) \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}}}{2 d \left (-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )\right ) \sqrt {a \left (1+\sin \left (d x +c \right )\right )}\, \cos \left (d x +c \right )} \]
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 \[ \int \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}}}{\sqrt {a \sin \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}}{\sqrt {a+a\,\sin \left (c+d\,x\right )}} \,d x \]
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 \[ \int \frac {\left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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