Optimal. Leaf size=89 \[ -\frac {2 d \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{a (c-d) \sqrt {c^2-d^2} f}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x))} \]
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Rubi [A]
time = 0.09, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2826, 2727,
2739, 632, 210} \begin {gather*} -\frac {2 d \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{a f (c-d) \sqrt {c^2-d^2}}-\frac {\cos (e+f x)}{f (c-d) (a \sin (e+f x)+a)} \end {gather*}
Antiderivative was successfully verified.
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Rule 210
Rule 632
Rule 2727
Rule 2739
Rule 2826
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sin (e+f x)) (c+d \sin (e+f x))} \, dx &=\frac {\int \frac {1}{a+a \sin (e+f x)} \, dx}{c-d}-\frac {d \int \frac {1}{c+d \sin (e+f x)} \, dx}{a (c-d)}\\ &=-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x))}-\frac {(2 d) \text {Subst}\left (\int \frac {1}{c+2 d x+c x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{a (c-d) f}\\ &=-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x))}+\frac {(4 d) \text {Subst}\left (\int \frac {1}{-4 \left (c^2-d^2\right )-x^2} \, dx,x,2 d+2 c \tan \left (\frac {1}{2} (e+f x)\right )\right )}{a (c-d) f}\\ &=-\frac {2 d \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{a (c-d) \sqrt {c^2-d^2} f}-\frac {\cos (e+f x)}{(c-d) f (a+a \sin (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 114, normalized size = 1.28 \begin {gather*} \frac {\cos (e+f x) \left (\frac {2 d \tanh ^{-1}\left (\frac {\sqrt {c-d} \sqrt {1-\sin (e+f x)}}{\sqrt {-c-d} \sqrt {1+\sin (e+f x)}}\right )}{\sqrt {-c-d} \sqrt {c-d} \sqrt {\cos ^2(e+f x)}}+\frac {1}{1+\sin (e+f x)}\right )}{a (-c+d) f} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.44, size = 83, normalized size = 0.93
method | result | size |
derivativedivides | \(\frac {-\frac {2}{\left (c -d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {2 d \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right ) \sqrt {c^{2}-d^{2}}}}{a f}\) | \(83\) |
default | \(\frac {-\frac {2}{\left (c -d \right ) \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}-\frac {2 d \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c -d \right ) \sqrt {c^{2}-d^{2}}}}{a f}\) | \(83\) |
risch | \(-\frac {2}{f \left (c -d \right ) a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}-\frac {d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right )}{\sqrt {-c^{2}+d^{2}}\, \left (c -d \right ) f a}+\frac {d \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right )}{\sqrt {-c^{2}+d^{2}}\, \left (c -d \right ) f a}\) | \(184\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 209 vs.
\(2 (87) = 174\).
time = 0.39, size = 510, normalized size = 5.73 \begin {gather*} \left [\frac {\sqrt {-c^{2} + d^{2}} {\left (d \cos \left (f x + e\right ) + d \sin \left (f x + e\right ) + d\right )} \log \left (\frac {{\left (2 \, c^{2} - d^{2}\right )} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2} + 2 \, {\left (c \cos \left (f x + e\right ) \sin \left (f x + e\right ) + d \cos \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}}}{d^{2} \cos \left (f x + e\right )^{2} - 2 \, c d \sin \left (f x + e\right ) - c^{2} - d^{2}}\right ) - 2 \, c^{2} + 2 \, d^{2} - 2 \, {\left (c^{2} - d^{2}\right )} \cos \left (f x + e\right ) + 2 \, {\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}{2 \, {\left ({\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} f \cos \left (f x + e\right ) + {\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} f \sin \left (f x + e\right ) + {\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} f\right )}}, \frac {\sqrt {c^{2} - d^{2}} {\left (d \cos \left (f x + e\right ) + d \sin \left (f x + e\right ) + d\right )} \arctan \left (-\frac {c \sin \left (f x + e\right ) + d}{\sqrt {c^{2} - d^{2}} \cos \left (f x + e\right )}\right ) - c^{2} + d^{2} - {\left (c^{2} - d^{2}\right )} \cos \left (f x + e\right ) + {\left (c^{2} - d^{2}\right )} \sin \left (f x + e\right )}{{\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} f \cos \left (f x + e\right ) + {\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} f \sin \left (f x + e\right ) + {\left (a c^{3} - a c^{2} d - a c d^{2} + a d^{3}\right )} f}\right ] \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. 916 vs.
\(2 (68) = 136\).
time = 119.14, size = 916, normalized size = 10.29 \begin {gather*} \begin {cases} \frac {\tilde {\infty } x}{\left (a \sin {\left (e \right )} + a\right ) \sin {\left (e \right )}} & \text {for}\: c = 0 \wedge d = 0 \wedge f = 0 \\\frac {2 \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a d f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} - a d f} & \text {for}\: c = - d \\- \frac {6 \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a d f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a d f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a d f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a d f} - \frac {6 \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{3 a d f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a d f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a d f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a d f} - \frac {4}{3 a d f \tan ^{3}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a d f \tan ^{2}{\left (\frac {e}{2} + \frac {f x}{2} \right )} + 9 a d f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + 3 a d f} & \text {for}\: c = d \\\frac {x}{\left (c + d \sin {\left (e \right )}\right ) \left (a \sin {\left (e \right )} + a\right )} & \text {for}\: f = 0 \\\frac {\frac {\log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} \right )} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} + \frac {\log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} \right )}}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f} + \frac {2}{a f \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a f}}{d} & \text {for}\: c = 0 \\- \frac {d \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + \frac {d}{c} - \frac {\sqrt {- c^{2} + d^{2}}}{c} \right )} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a c f \sqrt {- c^{2} + d^{2}} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a c f \sqrt {- c^{2} + d^{2}} - a d f \sqrt {- c^{2} + d^{2}} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - a d f \sqrt {- c^{2} + d^{2}}} - \frac {d \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + \frac {d}{c} - \frac {\sqrt {- c^{2} + d^{2}}}{c} \right )}}{a c f \sqrt {- c^{2} + d^{2}} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a c f \sqrt {- c^{2} + d^{2}} - a d f \sqrt {- c^{2} + d^{2}} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - a d f \sqrt {- c^{2} + d^{2}}} + \frac {d \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + \frac {d}{c} + \frac {\sqrt {- c^{2} + d^{2}}}{c} \right )} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )}}{a c f \sqrt {- c^{2} + d^{2}} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a c f \sqrt {- c^{2} + d^{2}} - a d f \sqrt {- c^{2} + d^{2}} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - a d f \sqrt {- c^{2} + d^{2}}} + \frac {d \log {\left (\tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + \frac {d}{c} + \frac {\sqrt {- c^{2} + d^{2}}}{c} \right )}}{a c f \sqrt {- c^{2} + d^{2}} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a c f \sqrt {- c^{2} + d^{2}} - a d f \sqrt {- c^{2} + d^{2}} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - a d f \sqrt {- c^{2} + d^{2}}} - \frac {2 \sqrt {- c^{2} + d^{2}}}{a c f \sqrt {- c^{2} + d^{2}} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} + a c f \sqrt {- c^{2} + d^{2}} - a d f \sqrt {- c^{2} + d^{2}} \tan {\left (\frac {e}{2} + \frac {f x}{2} \right )} - a d f \sqrt {- c^{2} + d^{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.46, size = 100, normalized size = 1.12 \begin {gather*} -\frac {2 \, {\left (\frac {{\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (c\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + d}{\sqrt {c^{2} - d^{2}}}\right )\right )} d}{{\left (a c - a d\right )} \sqrt {c^{2} - d^{2}}} + \frac {1}{{\left (a c - a d\right )} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}}\right )}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 6.99, size = 121, normalized size = 1.36 \begin {gather*} \frac {2\,d\,\mathrm {atan}\left (\frac {\frac {d\,\left (2\,a\,d^2-2\,a\,c\,d\right )}{a\,\sqrt {c+d}\,{\left (c-d\right )}^{3/2}}-\frac {2\,c\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (a\,c-a\,d\right )}{a\,\sqrt {c+d}\,{\left (c-d\right )}^{3/2}}}{2\,d}\right )}{a\,f\,\sqrt {c+d}\,{\left (c-d\right )}^{3/2}}-\frac {2}{f\,\left (a+a\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (c-d\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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