Optimal. Leaf size=120 \[ \frac {(3 c-2 d) d^2 x}{a^2}+\frac {(c-4 d) d^2 \cos (e+f x)}{3 a^2 f}-\frac {(c-d)^2 (c+6 d) \cos (e+f x)}{3 a^2 f (1+\sin (e+f x))}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a+a \sin (e+f x))^2} \]
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Rubi [A]
time = 0.26, antiderivative size = 120, 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 = {2844, 3047,
3102, 2814, 2727} \begin {gather*} \frac {d^2 (c-4 d) \cos (e+f x)}{3 a^2 f}+\frac {d^2 x (3 c-2 d)}{a^2}-\frac {(c+6 d) (c-d)^2 \cos (e+f x)}{3 a^2 f (\sin (e+f x)+1)}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a \sin (e+f x)+a)^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 2727
Rule 2814
Rule 2844
Rule 3047
Rule 3102
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^2} \, dx &=-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a+a \sin (e+f x))^2}-\frac {\int \frac {(c+d \sin (e+f x)) \left (-a \left (c^2+4 c d-2 d^2\right )+a (c-4 d) d \sin (e+f x)\right )}{a+a \sin (e+f x)} \, dx}{3 a^2}\\ &=-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a+a \sin (e+f x))^2}-\frac {\int \frac {-a c \left (c^2+4 c d-2 d^2\right )+\left (a c (c-4 d) d-a d \left (c^2+4 c d-2 d^2\right )\right ) \sin (e+f x)+a (c-4 d) d^2 \sin ^2(e+f x)}{a+a \sin (e+f x)} \, dx}{3 a^2}\\ &=\frac {(c-4 d) d^2 \cos (e+f x)}{3 a^2 f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a+a \sin (e+f x))^2}-\frac {\int \frac {-a^2 c \left (c^2+4 c d-2 d^2\right )-3 a^2 (3 c-2 d) d^2 \sin (e+f x)}{a+a \sin (e+f x)} \, dx}{3 a^3}\\ &=\frac {(3 c-2 d) d^2 x}{a^2}+\frac {(c-4 d) d^2 \cos (e+f x)}{3 a^2 f}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a+a \sin (e+f x))^2}+\frac {\left ((c-d)^2 (c+6 d)\right ) \int \frac {1}{a+a \sin (e+f x)} \, dx}{3 a}\\ &=\frac {(3 c-2 d) d^2 x}{a^2}+\frac {(c-4 d) d^2 \cos (e+f x)}{3 a^2 f}-\frac {(c-d)^2 (c+6 d) \cos (e+f x)}{3 f \left (a^2+a^2 \sin (e+f x)\right )}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{3 f (a+a \sin (e+f x))^2}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 212, normalized size = 1.77 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (2 (c-d)^3 \sin \left (\frac {1}{2} (e+f x)\right )-(c-d)^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+2 (c-d)^2 (c+8 d) \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+3 (3 c-2 d) d^2 (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3-3 d^3 \cos (e+f x) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3\right )}{3 a^2 f (1+\sin (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 157, normalized size = 1.31
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (c^{3}-3 c \,d^{2}+2 d^{3}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-2 c^{3}+6 c^{2} d -6 c \,d^{2}+2 d^{3}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (2 c^{3}-6 c^{2} d +6 c \,d^{2}-2 d^{3}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+2 d^{2} \left (-\frac {d}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (3 c -2 d \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{a^{2} f}\) | \(157\) |
default | \(\frac {-\frac {2 \left (c^{3}-3 c \,d^{2}+2 d^{3}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-2 c^{3}+6 c^{2} d -6 c \,d^{2}+2 d^{3}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {2 \left (2 c^{3}-6 c^{2} d +6 c \,d^{2}-2 d^{3}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+2 d^{2} \left (-\frac {d}{1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )}+\left (3 c -2 d \right ) \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )\right )}{a^{2} f}\) | \(157\) |
risch | \(\frac {3 d^{2} x c}{a^{2}}-\frac {2 d^{3} x}{a^{2}}-\frac {d^{3} {\mathrm e}^{i \left (f x +e \right )}}{2 a^{2} f}-\frac {d^{3} {\mathrm e}^{-i \left (f x +e \right )}}{2 a^{2} f}-\frac {2 i \left (-9 i c^{2} d \,{\mathrm e}^{2 i \left (f x +e \right )}+18 i c \,d^{2} {\mathrm e}^{2 i \left (f x +e \right )}-9 i d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+i c^{3}+6 i c^{2} d -15 i c \,d^{2}+8 i d^{3}+3 c^{3} {\mathrm e}^{i \left (f x +e \right )}+9 c^{2} d \,{\mathrm e}^{i \left (f x +e \right )}-27 c \,d^{2} {\mathrm e}^{i \left (f x +e \right )}+15 d^{3} {\mathrm e}^{i \left (f x +e \right )}\right )}{3 f \,a^{2} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{3}}\) | \(216\) |
norman | \(\frac {\frac {d^{2} \left (3 c -2 d \right ) x}{a}+\frac {\left (-2 c^{3}+6 c \,d^{2}-4 d^{3}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {\left (-2 c^{3}-6 c^{2} d +18 c \,d^{2}-16 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a f}+\frac {d^{2} \left (3 c -2 d \right ) x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {-4 c^{3}-6 c^{2} d +24 c \,d^{2}-20 d^{3}}{3 a f}+\frac {2 \left (-3 c^{3}-3 c^{2} d +15 c \,d^{2}-14 d^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {2 \left (-c^{3}-3 c^{2} d +9 c \,d^{2}-6 d^{3}\right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {2 \left (-3 c^{3}-9 c^{2} d +27 c \,d^{2}-22 d^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {2 \left (-5 c^{3}-3 c^{2} d +21 c \,d^{2}-20 d^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {2 \left (-3 c^{3}-9 c^{2} d +27 c \,d^{2}-20 d^{3}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {2 \left (-11 c^{3}-3 c^{2} d +39 c \,d^{2}-34 d^{3}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {3 d^{2} \left (3 c -2 d \right ) x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a}+\frac {6 d^{2} \left (3 c -2 d \right ) x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {10 d^{2} \left (3 c -2 d \right ) x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {12 d^{2} \left (3 c -2 d \right ) x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {12 d^{2} \left (3 c -2 d \right ) x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {10 d^{2} \left (3 c -2 d \right ) x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {6 d^{2} \left (3 c -2 d \right ) x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {3 d^{2} \left (3 c -2 d \right ) x \left (\tan ^{8}\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 )^{3} a \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}\) | \(646\) |
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. 641 vs.
\(2 (120) = 240\).
time = 0.50, size = 641, normalized size = 5.34 \begin {gather*} -\frac {2 \, {\left (2 \, d^{3} {\left (\frac {\frac {12 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {11 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {9 \, \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}} + 5}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {4 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {4 \, a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{2} \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^{2}}\right )} - 3 \, c d^{2} {\left (\frac {\frac {9 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 4}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {3 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{2}}\right )} + \frac {c^{3} {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 2\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}} + \frac {3 \, c^{2} d {\left (\frac {3 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + 1\right )}}{a^{2} + \frac {3 \, a^{2} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {3 \, a^{2} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a^{2} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}}\right )}}{3 \, 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. 318 vs.
\(2 (120) = 240\).
time = 0.38, size = 318, normalized size = 2.65 \begin {gather*} -\frac {3 \, d^{3} \cos \left (f x + e\right )^{3} - c^{3} + 3 \, c^{2} d - 3 \, c d^{2} + d^{3} + 6 \, {\left (3 \, c d^{2} - 2 \, d^{3}\right )} f x - {\left (c^{3} + 6 \, c^{2} d - 15 \, c d^{2} + 11 \, d^{3} + 3 \, {\left (3 \, c d^{2} - 2 \, d^{3}\right )} f x\right )} \cos \left (f x + e\right )^{2} - {\left (2 \, c^{3} + 3 \, c^{2} d - 12 \, c d^{2} + 13 \, d^{3} - 3 \, {\left (3 \, c d^{2} - 2 \, d^{3}\right )} f x\right )} \cos \left (f x + e\right ) - {\left (3 \, d^{3} \cos \left (f x + e\right )^{2} - c^{3} + 3 \, c^{2} d - 3 \, c d^{2} + d^{3} - 6 \, {\left (3 \, c d^{2} - 2 \, d^{3}\right )} f x + {\left (c^{3} + 6 \, c^{2} d - 15 \, c d^{2} + 14 \, d^{3} - 3 \, {\left (3 \, c d^{2} - 2 \, d^{3}\right )} f x\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{3 \, {\left (a^{2} f \cos \left (f x + e\right )^{2} - a^{2} f \cos \left (f x + e\right ) - 2 \, a^{2} f - {\left (a^{2} f \cos \left (f x + e\right ) + 2 \, a^{2} f\right )} \sin \left (f x + e\right )\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. 3585 vs.
\(2 (109) = 218\).
time = 4.99, size = 3585, normalized size = 29.88 \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 = 0.46, size = 209, normalized size = 1.74 \begin {gather*} \frac {\frac {3 \, {\left (3 \, c d^{2} - 2 \, d^{3}\right )} {\left (f x + e\right )}}{a^{2}} - \frac {6 \, d^{3}}{{\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )} a^{2}} - \frac {2 \, {\left (3 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 9 \, c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 6 \, d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 3 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 9 \, c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 27 \, c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 15 \, d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 2 \, c^{3} + 3 \, c^{2} d - 12 \, c d^{2} + 7 \, d^{3}\right )}}{a^{2} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{3}}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 8.39, size = 298, normalized size = 2.48 \begin {gather*} \frac {2\,d^2\,\mathrm {atan}\left (\frac {2\,d^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,c-2\,d\right )}{6\,c\,d^2-4\,d^3}\right )\,\left (3\,c-2\,d\right )}{a^2\,f}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (2\,c^3+6\,c^2\,d-18\,c\,d^2+12\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {10\,c^3}{3}+2\,c^2\,d-14\,c\,d^2+\frac {44\,d^3}{3}\right )-8\,c\,d^2+2\,c^2\,d+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,c^3-6\,c\,d^2+4\,d^3\right )+\frac {4\,c^3}{3}+\frac {20\,d^3}{3}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c^3+6\,c^2\,d-18\,c\,d^2+16\,d^3\right )}{f\,\left (a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+3\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+4\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+4\,a^2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+3\,a^2\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a^2\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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