Optimal. Leaf size=191 \[ \frac {7 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{16 \sqrt {2} a^3 c^{3/2} f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}-\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{30 a^3 c^2 f}+\frac {7 \cos (e+f x)}{16 a^3 f (c-c \sin (e+f x))^{3/2}}-\frac {7 \sec (e+f x)}{12 a^3 c f \sqrt {c-c \sin (e+f x)}} \]
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Rubi [A] time = 0.33, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {2736, 2675, 2687, 2650, 2649, 206} \[ -\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}-\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{30 a^3 c^2 f}+\frac {7 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{16 \sqrt {2} a^3 c^{3/2} f}+\frac {7 \cos (e+f x)}{16 a^3 f (c-c \sin (e+f x))^{3/2}}-\frac {7 \sec (e+f x)}{12 a^3 c f \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2650
Rule 2675
Rule 2687
Rule 2736
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x))^3 (c-c \sin (e+f x))^{3/2}} \, dx &=\frac {\int \sec ^6(e+f x) (c-c \sin (e+f x))^{3/2} \, dx}{a^3 c^3}\\ &=-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}+\frac {7 \int \sec ^4(e+f x) \sqrt {c-c \sin (e+f x)} \, dx}{10 a^3 c^2}\\ &=-\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{30 a^3 c^2 f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}+\frac {7 \int \frac {\sec ^2(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx}{12 a^3 c}\\ &=-\frac {7 \sec (e+f x)}{12 a^3 c f \sqrt {c-c \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{30 a^3 c^2 f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}+\frac {7 \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{8 a^3}\\ &=\frac {7 \cos (e+f x)}{16 a^3 f (c-c \sin (e+f x))^{3/2}}-\frac {7 \sec (e+f x)}{12 a^3 c f \sqrt {c-c \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{30 a^3 c^2 f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}+\frac {7 \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{32 a^3 c}\\ &=\frac {7 \cos (e+f x)}{16 a^3 f (c-c \sin (e+f x))^{3/2}}-\frac {7 \sec (e+f x)}{12 a^3 c f \sqrt {c-c \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{30 a^3 c^2 f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}-\frac {7 \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 )}{16 a^3 c f}\\ &=\frac {7 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{16 \sqrt {2} a^3 c^{3/2} f}+\frac {7 \cos (e+f x)}{16 a^3 f (c-c \sin (e+f x))^{3/2}}-\frac {7 \sec (e+f x)}{12 a^3 c f \sqrt {c-c \sin (e+f x)}}-\frac {7 \sec ^3(e+f x) \sqrt {c-c \sin (e+f x)}}{30 a^3 c^2 f}-\frac {\sec ^5(e+f x) (c-c \sin (e+f x))^{3/2}}{5 a^3 c^3 f}\\ \end {align*}
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Mathematica [C] time = 1.19, size = 174, normalized size = 0.91 \[ \frac {\left (\frac {1}{1920}+\frac {i}{1920}\right ) \cos (e+f x) \left ((1-i) (-231 \sin (e+f x)+105 \sin (3 (e+f x))+350 \cos (2 (e+f x))+206)+840 \sqrt [4]{-1} \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (\tan \left (\frac {1}{4} (e+f x)\right )+1\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5\right )}{a^3 c f (\sin (e+f x)-1) (\sin (e+f x)+1)^3 \sqrt {c-c \sin (e+f x)}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.47, size = 238, normalized size = 1.25 \[ \frac {105 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) + \cos \left (f x + e\right )^{3}\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (175 \, \cos \left (f x + e\right )^{2} + 21 \, {\left (5 \, \cos \left (f x + e\right )^{2} - 4\right )} \sin \left (f x + e\right ) - 36\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{960 \, {\left (a^{3} c^{2} f \cos \left (f x + e\right )^{3} \sin \left (f x + e\right ) + a^{3} c^{2} f \cos \left (f x + e\right )^{3}\right )}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.25, size = 170, normalized size = 0.89 \[ -\frac {105 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c -105 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c +42 c^{\frac {7}{2}} \sin \left (f x +e \right )-350 c^{\frac {7}{2}} \left (\sin ^{2}\left (f x +e \right )\right )-210 c^{\frac {7}{2}} \left (\sin ^{3}\left (f x +e \right )\right )+278 c^{\frac {7}{2}}}{480 c^{\frac {9}{2}} a^{3} \left (1+\sin \left (f x +e \right )\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-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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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-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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