Optimal. Leaf size=57 \[ -\frac {3 a^2 x}{c}+\frac {3 a^2 \cos (e+f x)}{c f}+\frac {2 a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^2} \]
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Rubi [A]
time = 0.10, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {2815, 2759,
2761, 8} \begin {gather*} \frac {3 a^2 \cos (e+f x)}{c f}+\frac {2 a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^2}-\frac {3 a^2 x}{c} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2759
Rule 2761
Rule 2815
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^2}{c-c \sin (e+f x)} \, dx &=\left (a^2 c^2\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^3} \, dx\\ &=\frac {2 a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^2}-\left (3 a^2\right ) \int \frac {\cos ^2(e+f x)}{c-c \sin (e+f x)} \, dx\\ &=\frac {3 a^2 \cos (e+f x)}{c f}+\frac {2 a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^2}-\frac {\left (3 a^2\right ) \int 1 \, dx}{c}\\ &=-\frac {3 a^2 x}{c}+\frac {3 a^2 \cos (e+f x)}{c f}+\frac {2 a^2 c \cos ^3(e+f x)}{f (c-c \sin (e+f x))^2}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(130\) vs. \(2(57)=114\).
time = 0.27, size = 130, normalized size = 2.28 \begin {gather*} \frac {a^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right ) (3 (e+f x)-\cos (e+f x))+(-8-3 e-3 f x+\cos (e+f x)) \sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^2}{c f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 (-1+\sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 55, normalized size = 0.96
method | result | size |
derivativedivides | \(\frac {2 a^{2} \left (\frac {1}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}-3 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {4}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}\right )}{f c}\) | \(55\) |
default | \(\frac {2 a^{2} \left (\frac {1}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}-3 \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {4}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1}\right )}{f c}\) | \(55\) |
risch | \(-\frac {3 a^{2} x}{c}+\frac {a^{2} {\mathrm e}^{i \left (f x +e \right )}}{2 c f}+\frac {a^{2} {\mathrm e}^{-i \left (f x +e \right )}}{2 c f}+\frac {8 a^{2}}{f c \left ({\mathrm e}^{i \left (f x +e \right )}-i\right )}\) | \(76\) |
norman | \(\frac {\frac {3 a^{2} x}{c}-\frac {8 a^{2}}{c f}-\frac {3 a^{2} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c}+\frac {6 a^{2} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {6 a^{2} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}+\frac {3 a^{2} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {3 a^{2} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c}-\frac {2 a^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {2 a^{2} \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {6 a^{2} \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}-\frac {14 a^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )-1\right )}\) | \(237\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 236 vs.
\(2 (61) = 122\).
time = 0.53, size = 236, normalized size = 4.14 \begin {gather*} -\frac {2 \, {\left (a^{2} {\left (\frac {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} - \frac {\sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - 2}{c - \frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {c \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {c \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c}\right )} + 2 \, a^{2} {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{c} - \frac {1}{c - \frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} - \frac {a^{2}}{c - \frac {c \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.32, size = 111, normalized size = 1.95 \begin {gather*} -\frac {3 \, a^{2} f x - a^{2} \cos \left (f x + e\right )^{2} - 4 \, a^{2} + {\left (3 \, a^{2} f x - 5 \, a^{2}\right )} \cos \left (f x + e\right ) - {\left (3 \, a^{2} f x - a^{2} \cos \left (f x + e\right ) + 4 \, a^{2}\right )} \sin \left (f x + e\right )}{c f \cos \left (f x + e\right ) - c f \sin \left (f x + e\right ) + c f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 454 vs.
\(2 (51) = 102\).
time = 1.17, size = 454, normalized size = 7.96 \begin {gather*} \begin {cases} - \frac {3 a^{2} f x \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{c f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f} + \frac {3 a^{2} f x \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{c f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f} - \frac {3 a^{2} f x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{c f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f} + \frac {3 a^{2} f x}{c f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f} - \frac {8 a^{2} \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{c f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f} + \frac {2 a^{2} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{c f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f} - \frac {10 a^{2}}{c f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + c f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - c f} & \text {for}\: f \neq 0 \\\frac {x \left (a \sin {\left (e \right )} + a\right )^{2}}{- c \sin {\left (e \right )} + c} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.45, size = 103, normalized size = 1.81 \begin {gather*} -\frac {\frac {3 \, {\left (f x + e\right )} a^{2}}{c} + \frac {2 \, {\left (4 \, a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 5 \, a^{2}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 1\right )} c}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.91, size = 118, normalized size = 2.07 \begin {gather*} \frac {3\,\sqrt {2}\,a^2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (e+f\,x\right )-\frac {\sqrt {2}\,a^2\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (6\,e+6\,f\,x-16\right )}{2}}{c\,f\,\left (\sqrt {2}\,\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )-\sqrt {2}\,\sin \left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )}-\frac {3\,a^2\,x}{c}+\frac {2\,a^2\,{\cos \left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2}{c\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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