Optimal. Leaf size=121 \[ \frac {3 d \left (2 c^2-2 c d+d^2\right ) x}{2 a}+\frac {2 d \left (c^2-3 c d+d^2\right ) \cos (e+f x)}{a f}+\frac {(2 c-3 d) d^2 \cos (e+f x) \sin (e+f x)}{2 a f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{f (a+a \sin (e+f x))} \]
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Rubi [A]
time = 0.08, antiderivative size = 121, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2846, 2813}
\begin {gather*} \frac {2 d \left (c^2-3 c d+d^2\right ) \cos (e+f x)}{a f}+\frac {3 d x \left (2 c^2-2 c d+d^2\right )}{2 a}+\frac {d^2 (2 c-3 d) \sin (e+f x) \cos (e+f x)}{2 a f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{f (a \sin (e+f x)+a)} \end {gather*}
Antiderivative was successfully verified.
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Rule 2813
Rule 2846
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^3}{a+a \sin (e+f x)} \, dx &=-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{f (a+a \sin (e+f x))}-\frac {d \int (-a (3 c-2 d)+a (2 c-3 d) \sin (e+f x)) (c+d \sin (e+f x)) \, dx}{a^2}\\ &=\frac {3 d \left (2 c^2-2 c d+d^2\right ) x}{2 a}+\frac {2 d \left (c^2-3 c d+d^2\right ) \cos (e+f x)}{a f}+\frac {(2 c-3 d) d^2 \cos (e+f x) \sin (e+f x)}{2 a f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{f (a+a \sin (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 192, normalized size = 1.59 \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 ) \left (6 \left (2 c^2-2 c d+d^2\right ) (e+f x)-4 (3 c-d) d \cos (e+f x)-d^2 \sin (2 (e+f x))\right )+\sin \left (\frac {1}{2} (e+f x)\right ) \left (2 \left (4 c^3+6 c^2 d (-2+e+f x)-6 c d^2 (-2+e+f x)+d^3 (-4+3 e+3 f x)\right )-4 (3 c-d) d^2 \cos (e+f x)-d^3 \sin (2 (e+f x))\right )\right )}{4 a f (1+\sin (e+f x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.32, size = 148, normalized size = 1.22
method | result | size |
derivativedivides | \(\frac {2 d \left (\frac {\frac {d^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\left (-3 c d +d^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}-3 c d +d^{2}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {3 \left (2 c^{2}-2 c d +d^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )-\frac {2 \left (c^{3}-3 c^{2} d +3 c \,d^{2}-d^{3}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a f}\) | \(148\) |
default | \(\frac {2 d \left (\frac {\frac {d^{2} \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}+\left (-3 c d +d^{2}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-\frac {d^{2} \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2}-3 c d +d^{2}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{2}}+\frac {3 \left (2 c^{2}-2 c d +d^{2}\right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2}\right )-\frac {2 \left (c^{3}-3 c^{2} d +3 c \,d^{2}-d^{3}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}}{a f}\) | \(148\) |
risch | \(\frac {3 d x \,c^{2}}{a}-\frac {3 d^{2} x c}{a}+\frac {3 d^{3} x}{2 a}-\frac {3 d^{2} {\mathrm e}^{i \left (f x +e \right )} c}{2 a f}+\frac {d^{3} {\mathrm e}^{i \left (f x +e \right )}}{2 a f}-\frac {3 d^{2} {\mathrm e}^{-i \left (f x +e \right )} c}{2 a f}+\frac {d^{3} {\mathrm e}^{-i \left (f x +e \right )}}{2 a f}-\frac {2 c^{3}}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}+\frac {6 c^{2} d}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}-\frac {6 c \,d^{2}}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}+\frac {2 d^{3}}{f a \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )}-\frac {d^{3} \sin \left (2 f x +2 e \right )}{4 a f}\) | \(235\) |
norman | \(\frac {\frac {-6 c \,d^{2}+d^{3}}{a f}+\frac {\left (-6 c \,d^{2}-d^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {\left (2 c^{3}-6 c^{2} d -2 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {\left (2 c^{3}-6 c^{2} d +6 c \,d^{2}-3 d^{3}\right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {\left (6 c^{3}-18 c^{2} d +6 c \,d^{2}-5 d^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {3 d \left (2 c^{2}-2 c d +d^{2}\right ) x}{2 a}-\frac {12 c \,d^{2} \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {3 \left (2 c^{3}-6 c^{2} d +4 c \,d^{2}-2 d^{3}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {3 d \left (2 c^{2}-2 c d +d^{2}\right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 a}+\frac {9 d \left (2 c^{2}-2 c d +d^{2}\right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {9 d \left (2 c^{2}-2 c d +d^{2}\right ) x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {9 d \left (2 c^{2}-2 c d +d^{2}\right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {9 d \left (2 c^{2}-2 c d +d^{2}\right ) x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {3 d \left (2 c^{2}-2 c d +d^{2}\right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}+\frac {3 d \left (2 c^{2}-2 c d +d^{2}\right ) x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 a}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}\) | \(493\) |
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. 463 vs.
\(2 (123) = 246\).
time = 0.53, size = 463, normalized size = 3.83 \begin {gather*} \frac {d^{3} {\left (\frac {\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {5 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 4}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {2 \, a \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {2 \, a \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {a \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {3 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a}\right )} - 6 \, c 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 )} + 6 \, c^{2} 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 {2 \, c^{3}}{a + \frac {a \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}}}{f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.34, size = 244, normalized size = 2.02 \begin {gather*} \frac {d^{3} \cos \left (f x + e\right )^{3} - 2 \, c^{3} + 6 \, c^{2} d - 6 \, c d^{2} + 2 \, d^{3} + 3 \, {\left (2 \, c^{2} d - 2 \, c d^{2} + d^{3}\right )} f x - 2 \, {\left (3 \, c d^{2} - d^{3}\right )} \cos \left (f x + e\right )^{2} - {\left (2 \, c^{3} - 6 \, c^{2} d + 12 \, c d^{2} - 3 \, d^{3} - 3 \, {\left (2 \, c^{2} d - 2 \, c d^{2} + d^{3}\right )} f x\right )} \cos \left (f x + e\right ) - {\left (d^{3} \cos \left (f x + e\right )^{2} - 2 \, c^{3} + 6 \, c^{2} d - 6 \, c d^{2} + 2 \, d^{3} - 3 \, {\left (2 \, c^{2} d - 2 \, c d^{2} + d^{3}\right )} f x + {\left (6 \, c d^{2} - d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{2 \, {\left (a f \cos \left (f x + e\right ) + a f \sin \left (f x + e\right ) + a 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. 3602 vs.
\(2 (107) = 214\).
time = 2.50, size = 3602, normalized size = 29.77 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 172, normalized size = 1.42 \begin {gather*} \frac {\frac {3 \, {\left (2 \, c^{2} d - 2 \, c d^{2} + d^{3}\right )} {\left (f x + e\right )}}{a} - \frac {4 \, {\left (c^{3} - 3 \, c^{2} d + 3 \, c d^{2} - d^{3}\right )}}{a {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}} + \frac {2 \, {\left (d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 6 \, c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, c d^{2} + 2 \, d^{3}\right )}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )}^{2} a}}{2 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.43, size = 282, normalized size = 2.33 \begin {gather*} \frac {3\,d\,\mathrm {atan}\left (\frac {3\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c^2-2\,c\,d+d^2\right )}{6\,c^2\,d-6\,c\,d^2+3\,d^3}\right )\,\left (2\,c^2-2\,c\,d+d^2\right )}{a\,f}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (6\,c\,d^2-d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,c^3-6\,c^2\,d+6\,c\,d^2-3\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (4\,c^3-12\,c^2\,d+18\,c\,d^2-5\,d^3\right )+12\,c\,d^2-6\,c^2\,d+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (6\,c\,d^2-3\,d^3\right )+2\,c^3-4\,d^3}{f\,\left (a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+2\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+2\,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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