Optimal. Leaf size=236 \[ \frac {3 e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{4 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac {3 e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{4 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}+\frac {3 e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{4 d} \]
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Rubi [A] time = 0.36, antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2678, 2685, 2677, 2775, 203, 2833, 63, 215} \[ \frac {3 e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \tan ^{-1}\left (\frac {\sqrt {e} \sin (c+d x)}{\sqrt {\cos (c+d x)+1} \sqrt {e \cos (c+d x)}}\right )}{4 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac {3 e^{3/2} \sqrt {\cos (c+d x)+1} \sqrt {a \sin (c+d x)+a} \sinh ^{-1}\left (\frac {\sqrt {e \cos (c+d x)}}{\sqrt {e}}\right )}{4 d (\sin (c+d x)+\cos (c+d x)+1)}-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a \sin (c+d x)+a}}+\frac {3 e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}}{4 d} \]
Antiderivative was successfully verified.
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Rule 63
Rule 203
Rule 215
Rule 2677
Rule 2678
Rule 2685
Rule 2775
Rule 2833
Rubi steps
\begin {align*} \int (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)} \, dx &=-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a+a \sin (c+d x)}}+\frac {1}{4} (3 a) \int \frac {(e \cos (c+d x))^{3/2}}{\sqrt {a+a \sin (c+d x)}} \, dx\\ &=-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a+a \sin (c+d x)}}+\frac {3 e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d}+\frac {1}{8} \left (3 e^2\right ) \int \frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {e \cos (c+d x)}} \, dx\\ &=-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a+a \sin (c+d x)}}+\frac {3 e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d}+\frac {\left (3 a 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}{8 (a+a \cos (c+d x)+a \sin (c+d x))}+\frac {\left (3 a 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}{8 (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a+a \sin (c+d x)}}+\frac {3 e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {\left (3 a 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 )}{8 d (a+a \cos (c+d x)+a \sin (c+d x))}-\frac {\left (3 a 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 )}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a+a \sin (c+d x)}}+\frac {3 e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d}+\frac {3 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)}}{4 d (1+\cos (c+d x)+\sin (c+d x))}-\frac {\left (3 a 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 )}{4 d (a+a \cos (c+d x)+a \sin (c+d x))}\\ &=-\frac {a (e \cos (c+d x))^{5/2}}{2 d e \sqrt {a+a \sin (c+d x)}}+\frac {3 e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}{4 d}-\frac {3 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)}}{4 d (1+\cos (c+d x)+\sin (c+d x))}+\frac {3 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)}}{4 d (1+\cos (c+d x)+\sin (c+d x))}\\ \end {align*}
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Mathematica [C] time = 0.95, size = 269, normalized size = 1.14 \[ -\frac {i e e^{-i (c+d x)} \sqrt {a (\sin (c+d x)+1)} \sqrt {e \cos (c+d x)} \left (-3 d x e^{2 i (c+d x)}-2 e^{i (c+d x)} \sqrt {1+e^{2 i (c+d x)}}+2 i e^{2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}}+e^{3 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}}-i \sqrt {1+e^{2 i (c+d x)}}-3 i e^{2 i (c+d x)} \log \left (1+\sqrt {1+e^{2 i (c+d x)}}\right )+3 e^{2 i (c+d x)} \sinh ^{-1}\left (e^{i (c+d x)}\right )\right )}{4 d \left (e^{i (c+d x)}+i\right ) \sqrt {1+e^{2 i (c+d x)}}} \]
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] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.35, size = 241, normalized size = 1.02 \[ -\frac {\left (3 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctan \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sqrt {2}}{2}\right ) \sin \left (d x +c \right )-3 \sqrt {2}\, \sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (d x +c \right )}{1+\cos \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {2}}{2 \cos \left (d x +c \right )}\right ) \sin \left (d x +c \right )-4 \left (\cos ^{3}\left (d x +c \right )\right )-4 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-2 \left (\cos ^{2}\left (d x +c \right )\right )+6 \cos \left (d x +c \right ) \sin \left (d x +c \right )+6 \cos \left (d x +c \right )\right ) \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {a \left (1+\sin \left (d x +c \right )\right )}}{8 d \left (-1+\cos \left (d x +c \right )-\sin \left (d x +c \right )\right ) \cos \left (d x +c \right )^{2}} \]
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 \[ \int \left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (d x + c\right ) + a}\,{d x} \]
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 \[ \int {\left (e\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\sqrt {a+a\,\sin \left (c+d\,x\right )} \,d x \]
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 \[ \int \sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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