Optimal. Leaf size=190 \[ \frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{256 \sqrt {2} c^{9/2} f}+\frac {3 a^2 \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {a^2 \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {a^2 c \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{11/2}}-\frac {a^2 \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{7/2}} \]
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Rubi [A] time = 0.30, antiderivative size = 190, normalized size of antiderivative = 1.00, 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} \[ \frac {3 a^2 \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {a^2 \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{256 \sqrt {2} c^{9/2} f}+\frac {a^2 c \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{11/2}}-\frac {a^2 \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{7/2}} \]
Antiderivative was successfully verified.
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Rule 206
Rule 2649
Rule 2650
Rule 2680
Rule 2736
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^2}{(c-c \sin (e+f x))^{9/2}} \, dx &=\left (a^2 c^2\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{13/2}} \, dx\\ &=\frac {a^2 c \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{11/2}}-\frac {1}{8} \left (3 a^2\right ) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{9/2}} \, dx\\ &=\frac {a^2 c \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{11/2}}-\frac {a^2 \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{7/2}}+\frac {a^2 \int \frac {1}{(c-c \sin (e+f x))^{5/2}} \, dx}{16 c^2}\\ &=\frac {a^2 c \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{11/2}}-\frac {a^2 \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{7/2}}+\frac {a^2 \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {\left (3 a^2\right ) \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{128 c^3}\\ &=\frac {a^2 c \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{11/2}}-\frac {a^2 \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{7/2}}+\frac {a^2 \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {3 a^2 \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}+\frac {\left (3 a^2\right ) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{512 c^4}\\ &=\frac {a^2 c \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{11/2}}-\frac {a^2 \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{7/2}}+\frac {a^2 \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {3 a^2 \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}-\frac {\left (3 a^2\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 )}{256 c^4 f}\\ &=\frac {3 a^2 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{256 \sqrt {2} c^{9/2} f}+\frac {a^2 c \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{11/2}}-\frac {a^2 \cos (e+f x)}{8 c f (c-c \sin (e+f x))^{7/2}}+\frac {a^2 \cos (e+f x)}{64 c^2 f (c-c \sin (e+f x))^{5/2}}+\frac {3 a^2 \cos (e+f x)}{256 c^3 f (c-c \sin (e+f x))^{3/2}}\\ \end {align*}
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Mathematica [C] time = 1.46, size = 371, normalized size = 1.95 \[ \frac {a^2 (\sin (e+f x)+1)^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (256 \sin \left (\frac {1}{2} (e+f x)\right )+3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7+6 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6+4 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+8 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4-96 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-192 \sin \left (\frac {1}{2} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+128 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )+(-3-3 i) \sqrt [4]{-1} \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (\tan \left (\frac {1}{4} (e+f x)\right )+1\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8\right )}{256 f (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.47, size = 523, normalized size = 2.75 \[ \frac {3 \, \sqrt {2} {\left (a^{2} \cos \left (f x + e\right )^{5} + 5 \, a^{2} \cos \left (f x + e\right )^{4} - 8 \, a^{2} \cos \left (f x + e\right )^{3} - 20 \, a^{2} \cos \left (f x + e\right )^{2} + 8 \, a^{2} \cos \left (f x + e\right ) + 16 \, a^{2} - {\left (a^{2} \cos \left (f x + e\right )^{4} - 4 \, a^{2} \cos \left (f x + e\right )^{3} - 12 \, a^{2} \cos \left (f x + e\right )^{2} + 8 \, a^{2} \cos \left (f x + e\right ) + 16 \, a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (3 \, a^{2} \cos \left (f x + e\right )^{4} + 13 \, a^{2} \cos \left (f x + e\right )^{3} + 86 \, a^{2} \cos \left (f x + e\right )^{2} - 52 \, a^{2} \cos \left (f x + e\right ) - 128 \, a^{2} - {\left (3 \, a^{2} \cos \left (f x + e\right )^{3} - 10 \, a^{2} \cos \left (f x + e\right )^{2} + 76 \, a^{2} \cos \left (f x + e\right ) + 128 \, a^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{1024 \, {\left (c^{5} f \cos \left (f x + e\right )^{5} + 5 \, c^{5} f \cos \left (f x + e\right )^{4} - 8 \, c^{5} f \cos \left (f x + e\right )^{3} - 20 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f - {\left (c^{5} f \cos \left (f x + e\right )^{4} - 4 \, c^{5} f \cos \left (f x + e\right )^{3} - 12 \, c^{5} f \cos \left (f x + e\right )^{2} + 8 \, c^{5} f \cos \left (f x + e\right ) + 16 \, c^{5} f\right )} \sin \left (f x + e\right )\right )}} \]
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 = 1.30, size = 299, normalized size = 1.57 \[ \frac {a^{2} \left (6 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}} c^{\frac {5}{2}}-3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \left (\sin ^{4}\left (f x +e \right )\right ) c^{6}-44 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} c^{\frac {7}{2}}+12 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \left (\sin ^{3}\left (f x +e \right )\right ) c^{6}-88 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} c^{\frac {9}{2}}-18 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \left (\sin ^{2}\left (f x +e \right )\right ) c^{6}+48 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, c^{\frac {11}{2}}+12 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c^{6}-3 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{6}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{512 c^{\frac {21}{2}} \left (\sin \left (f x +e \right )-1\right )^{3} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{2}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^2}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}} \,d x \]
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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