Optimal. Leaf size=119 \[ \frac {a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {a^{3/2} (c+3 d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{d^{3/2} f (c+d)^{3/2}} \]
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Rubi [A] time = 0.19, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2762, 21, 2773, 208} \[ \frac {a^2 (c-d) \cos (e+f x)}{d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}-\frac {a^{3/2} (c+3 d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{d^{3/2} f (c+d)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 21
Rule 208
Rule 2762
Rule 2773
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^2} \, dx &=\frac {a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}-\frac {a \int \frac {-\frac {1}{2} a (c+3 d)-\frac {1}{2} a (c+3 d) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \, dx}{d (c+d)}\\ &=\frac {a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}+\frac {(a (c+3 d)) \int \frac {\sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{2 d (c+d)}\\ &=\frac {a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}-\frac {\left (a^2 (c+3 d)\right ) \operatorname {Subst}\left (\int \frac {1}{a c+a d-d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{d (c+d) f}\\ &=-\frac {a^{3/2} (c+3 d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{d^{3/2} (c+d)^{3/2} f}+\frac {a^2 (c-d) \cos (e+f x)}{d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}\\ \end {align*}
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Mathematica [B] time = 2.35, size = 268, normalized size = 2.25 \[ -\frac {(a (\sin (e+f x)+1))^{3/2} \left (2 \sqrt {d} (c-d) \sqrt {c+d} \sin \left (\frac {1}{2} (e+f x)\right )-2 \sqrt {d} (c-d) \sqrt {c+d} \cos \left (\frac {1}{2} (e+f x)\right )+(c+3 d) (c+d \sin (e+f x)) \left (\log \left (-\sec ^2\left (\frac {1}{4} (e+f x)\right ) \left (-\sqrt {d} \sqrt {c+d} \sin \left (\frac {1}{2} (e+f x)\right )+\sqrt {d} \sqrt {c+d} \cos \left (\frac {1}{2} (e+f x)\right )+c+d\right )\right )-\log \left (\sqrt {d} \sqrt {c+d} \left (\tan ^2\left (\frac {1}{4} (e+f x)\right )+2 \tan \left (\frac {1}{4} (e+f x)\right )-1\right )+(c+d) \sec ^2\left (\frac {1}{4} (e+f x)\right )\right )\right )\right )}{2 d^{3/2} f (c+d)^{3/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3 (c+d \sin (e+f x))} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.60, size = 970, normalized size = 8.15 \[ \text {result too large to display} \]
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 [B] time = 1.42, size = 233, normalized size = 1.96 \[ \frac {a \left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (-\sin \left (f x +e \right ) \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a d \left (c +3 d \right )-\arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a \,c^{2}-3 \arctanh \left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a c d +\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, c -\sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, d \right )}{d \left (c +d \right ) \left (c +d \sin \left (f x +e \right )\right ) \sqrt {a \left (c +d \right ) d}\, \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{2}}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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