Optimal. Leaf size=191 \[ \frac{a^3 c^2 \cos ^5(e+f x)}{4 f (c-c \sin (e+f x))^{13/2}}-\frac{5 a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{5 a^3 \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{5/2}}-\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{128 \sqrt{2} c^{9/2} f}-\frac{5 a^3 \cos ^3(e+f x)}{24 f (c-c \sin (e+f x))^{9/2}} \]
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Rubi [A] time = 0.353228, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179, Rules used = {2736, 2680, 2650, 2649, 206} \[ \frac{a^3 c^2 \cos ^5(e+f x)}{4 f (c-c \sin (e+f x))^{13/2}}-\frac{5 a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{5 a^3 \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{5/2}}-\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{128 \sqrt{2} c^{9/2} f}-\frac{5 a^3 \cos ^3(e+f x)}{24 f (c-c \sin (e+f x))^{9/2}} \]
Antiderivative was successfully verified.
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Rule 2736
Rule 2680
Rule 2650
Rule 2649
Rule 206
Rubi steps
\begin{align*} \int \frac{(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^{9/2}} \, dx &=\left (a^3 c^3\right ) \int \frac{\cos ^6(e+f x)}{(c-c \sin (e+f x))^{15/2}} \, dx\\ &=\frac{a^3 c^2 \cos ^5(e+f x)}{4 f (c-c \sin (e+f x))^{13/2}}-\frac{1}{8} \left (5 a^3 c\right ) \int \frac{\cos ^4(e+f x)}{(c-c \sin (e+f x))^{11/2}} \, dx\\ &=\frac{a^3 c^2 \cos ^5(e+f x)}{4 f (c-c \sin (e+f x))^{13/2}}-\frac{5 a^3 \cos ^3(e+f x)}{24 f (c-c \sin (e+f x))^{9/2}}+\frac{\left (5 a^3\right ) \int \frac{\cos ^2(e+f x)}{(c-c \sin (e+f x))^{7/2}} \, dx}{16 c}\\ &=\frac{a^3 c^2 \cos ^5(e+f x)}{4 f (c-c \sin (e+f x))^{13/2}}-\frac{5 a^3 \cos ^3(e+f x)}{24 f (c-c \sin (e+f x))^{9/2}}+\frac{5 a^3 \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{5/2}}-\frac{\left (5 a^3\right ) \int \frac{1}{(c-c \sin (e+f x))^{3/2}} \, dx}{64 c^3}\\ &=\frac{a^3 c^2 \cos ^5(e+f x)}{4 f (c-c \sin (e+f x))^{13/2}}-\frac{5 a^3 \cos ^3(e+f x)}{24 f (c-c \sin (e+f x))^{9/2}}+\frac{5 a^3 \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{5/2}}-\frac{5 a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}-\frac{\left (5 a^3\right ) \int \frac{1}{\sqrt{c-c \sin (e+f x)}} \, dx}{256 c^4}\\ &=\frac{a^3 c^2 \cos ^5(e+f x)}{4 f (c-c \sin (e+f x))^{13/2}}-\frac{5 a^3 \cos ^3(e+f x)}{24 f (c-c \sin (e+f x))^{9/2}}+\frac{5 a^3 \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{5/2}}-\frac{5 a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac{\left (5 a^3\right ) \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 )}{128 c^4 f}\\ &=-\frac{5 a^3 \tanh ^{-1}\left (\frac{\sqrt{c} \cos (e+f x)}{\sqrt{2} \sqrt{c-c \sin (e+f x)}}\right )}{128 \sqrt{2} c^{9/2} f}+\frac{a^3 c^2 \cos ^5(e+f x)}{4 f (c-c \sin (e+f x))^{13/2}}-\frac{5 a^3 \cos ^3(e+f x)}{24 f (c-c \sin (e+f x))^{9/2}}+\frac{5 a^3 \cos (e+f x)}{32 c^2 f (c-c \sin (e+f x))^{5/2}}-\frac{5 a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{3/2}}\\ \end{align*}
Mathematica [C] time = 2.57538, size = 371, normalized size = 1.94 \[ \frac{a^3 (\sin (e+f x)+1)^3 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right ) \left (768 \sin \left (\frac{1}{2} (e+f x)\right )-15 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^7-30 \sin \left (\frac{1}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^6+236 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^5+472 \sin \left (\frac{1}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^4-544 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^3-1088 \sin \left (\frac{1}{2} (e+f x)\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^2+384 \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )+(15+15 i) \sqrt [4]{-1} \tan ^{-1}\left (\left (\frac{1}{2}+\frac{i}{2}\right ) \sqrt [4]{-1} \left (\tan \left (\frac{1}{4} (e+f x)\right )+1\right )\right ) \left (\cos \left (\frac{1}{2} (e+f x)\right )-\sin \left (\frac{1}{2} (e+f x)\right )\right )^8\right )}{384 f (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )\right )^6} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.991, size = 299, normalized size = 1.6 \begin{align*} -{\frac{{a}^{3}}{768\, \left ( -1+\sin \left ( fx+e \right ) \right ) ^{3}\cos \left ( fx+e \right ) f} \left ( -15\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}}{\sqrt{c}}} \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{4}{c}^{5}+30\, \left ( c \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{7/2}{c}^{3/2}+60\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}}{\sqrt{c}}} \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{3}{c}^{5}+292\, \left ( c \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{5/2}{c}^{5/2}-90\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}}{\sqrt{c}}} \right ) \left ( \sin \left ( fx+e \right ) \right ) ^{2}{c}^{5}-440\, \left ( c \left ( 1+\sin \left ( fx+e \right ) \right ) \right ) ^{3/2}{c}^{7/2}+60\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}}{\sqrt{c}}} \right ) \sin \left ( fx+e \right ){c}^{5}+240\,\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }{c}^{9/2}-15\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }\sqrt{2}}{\sqrt{c}}} \right ){c}^{5} \right ) \sqrt{c \left ( 1+\sin \left ( fx+e \right ) \right ) }{c}^{-{\frac{19}{2}}}{\frac{1}{\sqrt{c-c\sin \left ( fx+e \right ) }}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac{9}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.23502, size = 1338, normalized size = 7.01 \begin{align*} \frac{15 \, \sqrt{2}{\left (a^{3} \cos \left (f x + e\right )^{5} + 5 \, a^{3} \cos \left (f x + e\right )^{4} - 8 \, a^{3} \cos \left (f x + e\right )^{3} - 20 \, a^{3} \cos \left (f x + e\right )^{2} + 8 \, a^{3} \cos \left (f x + e\right ) + 16 \, a^{3} -{\left (a^{3} \cos \left (f x + e\right )^{4} - 4 \, a^{3} \cos \left (f x + e\right )^{3} - 12 \, a^{3} \cos \left (f x + e\right )^{2} + 8 \, a^{3} \cos \left (f x + e\right ) + 16 \, a^{3}\right )} \sin \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 \,{\left (15 \, a^{3} \cos \left (f x + e\right )^{4} - 191 \, a^{3} \cos \left (f x + e\right )^{3} - 338 \, a^{3} \cos \left (f x + e\right )^{2} + 252 \, a^{3} \cos \left (f x + e\right ) + 384 \, a^{3} -{\left (15 \, a^{3} \cos \left (f x + e\right )^{3} + 206 \, a^{3} \cos \left (f x + e\right )^{2} - 132 \, a^{3} \cos \left (f x + e\right ) - 384 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt{-c \sin \left (f x + e\right ) + c}}{1536 \,{\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f -{\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{2} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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