Optimal. Leaf size=62 \[ \frac {(2 c-d) d x}{a}-\frac {d^2 \cos (e+f x)}{a f}-\frac {(c-d)^2 \cos (e+f x)}{a f (1+\sin (e+f x))} \]
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Rubi [A]
time = 0.10, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.120, Rules used = {2825, 2814,
2727} \begin {gather*} -\frac {(c-d)^2 \cos (e+f x)}{a f (\sin (e+f x)+1)}+\frac {d x (2 c-d)}{a}-\frac {d^2 \cos (e+f x)}{a f} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2814
Rule 2825
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^2}{a+a \sin (e+f x)} \, dx &=-\frac {d^2 \cos (e+f x)}{a f}+\frac {\int \frac {a c^2+a (2 c-d) d \sin (e+f x)}{a+a \sin (e+f x)} \, dx}{a}\\ &=\frac {(2 c-d) d x}{a}-\frac {d^2 \cos (e+f x)}{a f}+(c-d)^2 \int \frac {1}{a+a \sin (e+f x)} \, dx\\ &=\frac {(2 c-d) d x}{a}-\frac {d^2 \cos (e+f x)}{a f}-\frac {(c-d)^2 \cos (e+f x)}{f (a+a \sin (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 122, normalized size = 1.97 \begin {gather*} -\frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (d \cos \left (\frac {1}{2} (e+f x)\right ) (-((2 c-d) (e+f x))+d \cos (e+f x))+\left (-2 c^2-2 c d (-2+e+f x)+d^2 (-2+e+f x)+d^2 \cos (e+f x)\right ) \sin \left (\frac {1}{2} (e+f x)\right )\right )}{a f (1+\sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.31, size = 75, normalized size = 1.21
method | result | size |
derivativedivides | \(\frac {2 d \left (-\frac {d}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (2 c -d \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )-\frac {2 \left (c^{2}-2 c d +d^{2}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a f}\) | \(75\) |
default | \(\frac {2 d \left (-\frac {d}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (2 c -d \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )-\frac {2 \left (c^{2}-2 c d +d^{2}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a f}\) | \(75\) |
risch | \(\frac {2 d x c}{a}-\frac {d^{2} x}{a}-\frac {d^{2} {\mathrm e}^{i \left (f x +e \right )}}{2 a f}-\frac {d^{2} {\mathrm e}^{-i \left (f x +e \right )}}{2 a f}-\frac {2 c^{2}}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}+\frac {4 c d}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}-\frac {2 d^{2}}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}\) | \(133\) |
norman | \(\frac {\frac {-2 c^{2}+4 c d -4 d^{2}}{a f}+\frac {\left (2 c -d \right ) d x}{a}-\frac {2 d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {\left (-2 c^{2}+4 c d -2 d^{2}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {\left (2 c -d \right ) d x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a}+\frac {\left (2 c -d \right ) d x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {\left (2 c -d \right ) d x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}-\frac {2 d^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {2 \left (-2 c^{2}+4 c d -3 d^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {2 \left (2 c -d \right ) d x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {2 \left (2 c -d \right ) d x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}\) | \(295\) |
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. 227 vs.
\(2 (65) = 130\).
time = 0.51, size = 227, normalized size = 3.66 \begin {gather*} -\frac {2 \, {\left (d^{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}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a \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 )}{a}\right )} - 2 \, c d {\left (\frac {\arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a} + \frac {1}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}\right )} + \frac {c^{2}}{a + \frac {a \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 148 vs.
\(2 (65) = 130\).
time = 0.35, size = 148, normalized size = 2.39 \begin {gather*} -\frac {d^{2} \cos \left (f x + e\right )^{2} - {\left (2 \, c d - d^{2}\right )} f x + c^{2} - 2 \, c d + d^{2} - {\left ({\left (2 \, c d - d^{2}\right )} f x - c^{2} + 2 \, c d - 2 \, d^{2}\right )} \cos \left (f x + e\right ) - {\left ({\left (2 \, c d - d^{2}\right )} f x - d^{2} \cos \left (f x + e\right ) + c^{2} - 2 \, c d + d^{2}\right )} \sin \left (f x + e\right )}{a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a 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. 940 vs.
\(2 (46) = 92\).
time = 1.39, size = 940, normalized size = 15.16 \begin {gather*} \begin {cases} - \frac {2 c^{2} \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} - \frac {2 c^{2}}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} + \frac {2 c d f x \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} + \frac {2 c d f x \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} + \frac {2 c d f x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} + \frac {2 c d f x}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} + \frac {4 c d \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} + \frac {4 c d}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} - \frac {d^{2} f x \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} - \frac {d^{2} f x \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} - \frac {d^{2} f x \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} - \frac {d^{2} f x}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} - \frac {2 d^{2} \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} - \frac {2 d^{2} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} - \frac {4 d^{2}}{a f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} & \text {for}\: f \neq 0 \\\frac {x \left (c + d \sin {\left (e \right )}\right )^{2}}{a \sin {\left (e \right )} + a} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs.
\(2 (65) = 130\).
time = 0.41, size = 143, normalized size = 2.31 \begin {gather*} \frac {\frac {{\left (2 \, c d - d^{2}\right )} {\left (f x + e\right )}}{a} - \frac {2 \, {\left (c^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 2 \, c d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c^{2} - 2 \, c d + 2 \, d^{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 )} a}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.38, size = 124, normalized size = 2.00 \begin {gather*} -\frac {d^2\,f\,x-2\,c\,d\,f\,x}{a\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (2\,c^2-4\,c\,d+2\,d^2\right )-4\,c\,d+2\,c^2+4\,d^2+2\,d^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{f\,\left (a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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