Optimal. Leaf size=117 \[ \frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{4 \sqrt {2} a c^{3/2} f}+\frac {3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.19, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2736, 2687, 2650, 2649, 206} \[ \frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{4 \sqrt {2} a c^{3/2} f}+\frac {3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2650
Rule 2687
Rule 2736
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x)) (c-c \sin (e+f x))^{3/2}} \, dx &=\frac {\int \frac {\sec ^2(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx}{a c}\\ &=-\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}}+\frac {3 \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{2 a}\\ &=\frac {3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}}+\frac {3 \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{8 a c}\\ &=\frac {3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}}-\frac {3 \operatorname {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{4 a c f}\\ &=\frac {3 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{4 \sqrt {2} a c^{3/2} f}+\frac {3 \cos (e+f x)}{4 a f (c-c \sin (e+f x))^{3/2}}-\frac {\sec (e+f x)}{a c f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
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Mathematica [C] time = 0.62, size = 125, normalized size = 1.07 \[ -\frac {\sec (e+f x) \left (-3 \sin (e+f x)+(3+3 i) \sqrt [4]{-1} \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (\tan \left (\frac {1}{4} (e+f x)\right )+1\right )\right ) \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+1\right )}{4 a c f \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 207, normalized size = 1.77 \[ \frac {3 \, \sqrt {2} {\left (\cos \left (f x + e\right ) \sin \left (f x + e\right ) - \cos \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, \sqrt {-c \sin \left (f x + e\right ) + c} {\left (3 \, \sin \left (f x + e\right ) - 1\right )}}{16 \, {\left (a c^{2} f \cos \left (f x + e\right ) \sin \left (f x + e\right ) - a c^{2} f \cos \left (f x + e\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: NotImplementedError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.82, size = 134, normalized size = 1.15 \[ -\frac {3 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c -3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c \sqrt {c \left (1+\sin \left (f x +e \right )\right )}-6 c^{\frac {3}{2}} \sin \left (f x +e \right )+2 c^{\frac {3}{2}}}{8 c^{\frac {5}{2}} a \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {1}{{\left (a \sin \left (f x + e\right ) + a\right )} {\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {1}{\left (a+a\,\sin \left (e+f\,x\right )\right )\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \frac {\int \frac {1}{- c \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )} + c \sqrt {- c \sin {\left (e + f x \right )} + c}}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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