Optimal. Leaf size=111 \[ \frac {a^{3/2} (c-3 d) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{d^{3/2} f}-\frac {a^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.21, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.138, Rules used = {2763, 21, 2775, 205} \[ \frac {a^{3/2} (c-3 d) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a} \sqrt {c+d \sin (e+f x)}}\right )}{d^{3/2} f}-\frac {a^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 205
Rule 2763
Rule 2775
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{3/2}}{\sqrt {c+d \sin (e+f x)}} \, dx &=-\frac {a^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+a \sin (e+f x)}}+\frac {\int \frac {-\frac {1}{2} a^2 (c-3 d)-\frac {1}{2} a^2 (c-3 d) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}} \, dx}{d}\\ &=-\frac {a^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+a \sin (e+f x)}}-\frac {(a (c-3 d)) \int \frac {\sqrt {a+a \sin (e+f x)}}{\sqrt {c+d \sin (e+f x)}} \, dx}{2 d}\\ &=-\frac {a^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+a \sin (e+f x)}}+\frac {\left (a^2 (c-3 d)\right ) \operatorname {Subst}\left (\int \frac {1}{a+d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{d f}\\ &=\frac {a^{3/2} (c-3 d) \tan ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)} \sqrt {c+d \sin (e+f x)}}\right )}{d^{3/2} f}-\frac {a^2 \cos (e+f x) \sqrt {c+d \sin (e+f x)}}{d f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 0.58, size = 301, normalized size = 2.71 \[ \frac {(a (\sin (e+f x)+1))^{3/2} \left (2 \sqrt {d} \sin \left (\frac {1}{2} (e+f x)\right ) \sqrt {c+d \sin (e+f x)}-2 (c-3 d) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \sin \left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{\sqrt {c+d \sin (e+f x)}}\right )-2 \sqrt {d} \cos \left (\frac {1}{2} (e+f x)\right ) \sqrt {c+d \sin (e+f x)}+c \log \left (\sqrt {c+d \sin (e+f x)}+\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )-3 d \log \left (\sqrt {c+d \sin (e+f x)}+\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e+2 f x-\pi )\right )\right )-(c-3 d) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {d} \cos \left (\frac {1}{4} (2 e+2 f x-\pi )\right )}{\sqrt {c+d \sin (e+f x)}}\right )\right )}{2 d^{3/2} f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 989, normalized size = 8.91 \[ \text {result too large to display} \]
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 [F] time = 0.49, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}{\sqrt {c +d \sin \left (f x +e \right )}}\, dx \]
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 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{\sqrt {d \sin \left (f x + e\right ) + c}}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{\sqrt {c+d\,\sin \left (e+f\,x\right )}} \,d x \]
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 \[ \int \frac {\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}{\sqrt {c + d \sin {\left (e + f x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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