Optimal. Leaf size=76 \[ \frac {4 \sqrt {a \sin (c+d x)+a}}{3 a d e \sqrt {e \cos (c+d x)}}-\frac {2}{3 d e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}} \]
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Rubi [A] time = 0.13, antiderivative size = 76, 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} \[ \frac {4 \sqrt {a \sin (c+d x)+a}}{3 a d e \sqrt {e \cos (c+d x)}}-\frac {2}{3 d e \sqrt {a \sin (c+d x)+a} \sqrt {e \cos (c+d x)}} \]
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} \sqrt {a+a \sin (c+d x)}} \, dx &=-\frac {2}{3 d e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}+\frac {2 \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{3/2}} \, dx}{3 a}\\ &=-\frac {2}{3 d e \sqrt {e \cos (c+d x)} \sqrt {a+a \sin (c+d x)}}+\frac {4 \sqrt {a+a \sin (c+d x)}}{3 a d e \sqrt {e \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 46, normalized size = 0.61 \[ \frac {2 (2 \sin (c+d x)+1)}{3 d e \sqrt {a (\sin (c+d x)+1)} \sqrt {e \cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.74, size = 67, normalized size = 0.88 \[ \frac {2 \, \sqrt {e \cos \left (d x + c\right )} \sqrt {a \sin \left (d x + c\right ) + a} {\left (2 \, \sin \left (d x + c\right ) + 1\right )}}{3 \, {\left (a d e^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a d e^{2} \cos \left (d x + c\right )\right )}} \]
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 \[ \int \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {3}{2}} \sqrt {a \sin \left (d x + c\right ) + a}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.18, size = 44, normalized size = 0.58 \[ \frac {2 \left (2 \sin \left (d x +c \right )+1\right ) \cos \left (d x +c \right )}{3 d \left (e \cos \left (d x +c \right )\right )^{\frac {3}{2}} \sqrt {a \left (1+\sin \left (d x +c \right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.03, size = 210, normalized size = 2.76 \[ \frac {2 \, {\left (\sqrt {a} \sqrt {e} + \frac {4 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {4 \, \sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {\sqrt {a} \sqrt {e} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{2}}{3 \, {\left (a e^{2} + \frac {2 \, a e^{2} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {a e^{2} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.01, size = 77, normalized size = 1.01 \[ \frac {4\,\sqrt {a\,\left (\sin \left (c+d\,x\right )+1\right )}\,\left (3\,\sin \left (c+d\,x\right )-\cos \left (2\,c+2\,d\,x\right )+2\right )}{3\,a\,d\,e\,\sqrt {e\,\cos \left (c+d\,x\right )}\,\left (4\,\sin \left (c+d\,x\right )-\cos \left (2\,c+2\,d\,x\right )+3\right )} \]
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 \[ \int \frac {1}{\sqrt {a \left (\sin {\left (c + d x \right )} + 1\right )} \left (e \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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