Optimal. Leaf size=179 \[ -\frac {a^{3/2} (c+7 d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{4 d^{3/2} f (c+d)^{5/2}}-\frac {a^2 (c+7 d) \cos (e+f x)}{4 d f (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac {a^2 (c-d) \cos (e+f x)}{2 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2} \]
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Rubi [A] time = 0.29, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {2762, 21, 2772, 2773, 208} \[ -\frac {a^{3/2} (c+7 d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{4 d^{3/2} f (c+d)^{5/2}}-\frac {a^2 (c+7 d) \cos (e+f x)}{4 d f (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac {a^2 (c-d) \cos (e+f x)}{2 d f (c+d) \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))^2} \]
Antiderivative was successfully verified.
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Rule 21
Rule 208
Rule 2762
Rule 2772
Rule 2773
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^{3/2}}{(c+d \sin (e+f x))^3} \, dx &=\frac {a^2 (c-d) \cos (e+f x)}{2 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {a \int \frac {-\frac {1}{2} a (c+7 d)-\frac {1}{2} a (c+7 d) \sin (e+f x)}{\sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2} \, dx}{2 d (c+d)}\\ &=\frac {a^2 (c-d) \cos (e+f x)}{2 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}+\frac {(a (c+7 d)) \int \frac {\sqrt {a+a \sin (e+f x)}}{(c+d \sin (e+f x))^2} \, dx}{4 d (c+d)}\\ &=\frac {a^2 (c-d) \cos (e+f x)}{2 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {a^2 (c+7 d) \cos (e+f x)}{4 d (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}+\frac {(a (c+7 d)) \int \frac {\sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{8 d (c+d)^2}\\ &=\frac {a^2 (c-d) \cos (e+f x)}{2 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {a^2 (c+7 d) \cos (e+f x)}{4 d (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}-\frac {\left (a^2 (c+7 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 )}{4 d (c+d)^2 f}\\ &=-\frac {a^{3/2} (c+7 d) \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{4 d^{3/2} (c+d)^{5/2} f}+\frac {a^2 (c-d) \cos (e+f x)}{2 d (c+d) f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))^2}-\frac {a^2 (c+7 d) \cos (e+f x)}{4 d (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}\\ \end {align*}
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Mathematica [A] time = 4.06, size = 313, normalized size = 1.75 \[ \frac {(a (\sin (e+f x)+1))^{3/2} \left (\frac {4 \sqrt {d} \sin \left (\frac {1}{2} (e+f x)\right ) \left (-c^2+d (c+7 d) \sin (e+f x)+7 c d+2 d^2\right )}{(c+d)^2 (c+d \sin (e+f x))^2}-\frac {4 \sqrt {d} \cos \left (\frac {1}{2} (e+f x)\right ) \left (-c^2+d (c+7 d) \sin (e+f x)+7 c d+2 d^2\right )}{(c+d)^2 (c+d \sin (e+f x))^2}-\frac {2 (c+7 d) \left (\log \left (-\sec ^2\left (\frac {1}{4} (e+f x)\right ) \left (-\sqrt {d} \sqrt {c+d} \sin \left (\frac {1}{2} (e+f x)\right )+\sqrt {d} \sqrt {c+d} \cos \left (\frac {1}{2} (e+f x)\right )+c+d\right )\right )-\log \left (\sqrt {d} \sqrt {c+d} \left (\tan ^2\left (\frac {1}{4} (e+f x)\right )+2 \tan \left (\frac {1}{4} (e+f x)\right )-1\right )+(c+d) \sec ^2\left (\frac {1}{4} (e+f x)\right )\right )\right )}{(c+d)^{5/2}}\right )}{16 d^{3/2} f \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.66, size = 1558, normalized size = 8.70 \[ \text {result too large to display} \]
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 = 1.50, size = 429, normalized size = 2.40 \[ \frac {\left (-\arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, d}{\sqrt {a \left (c +d \right ) d}}\right ) \left (\sin ^{2}\left (f x +e \right )\right ) a^{2} c \,d^{2}-7 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, d}{\sqrt {a \left (c +d \right ) d}}\right ) \left (\sin ^{2}\left (f x +e \right )\right ) a^{2} d^{3}-2 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, d}{\sqrt {a \left (c +d \right ) d}}\right ) \sin \left (f x +e \right ) a^{2} c^{2} d -14 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, d}{\sqrt {a \left (c +d \right ) d}}\right ) \sin \left (f x +e \right ) a^{2} c \,d^{2}+\left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, c d +7 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, d^{2}-\arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, d}{\sqrt {a \left (c +d \right ) d}}\right ) a^{2} c^{3}-7 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, d}{\sqrt {a \left (c +d \right ) d}}\right ) a^{2} c^{2} d +\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (c +d \right ) d}\, a \,c^{2}-8 \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (c +d \right ) d}\, a c d -9 \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a \left (c +d \right ) d}\, a \,d^{2}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (1+\sin \left (f x +e \right )\right )}{4 \sqrt {a \left (c +d \right ) d}\, \left (c +d \sin \left (f x +e \right )\right )^{2} \left (c +d \right )^{2} d \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{3}}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]
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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