Optimal. Leaf size=142 \[ \frac {d^3 x}{a^3}-\frac {(c-d)^2 (2 c+7 d) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \left (2 c^2+11 c d+29 d^2\right ) \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a+a \sin (e+f x))^3} \]
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Rubi [A]
time = 0.23, antiderivative size = 142, 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,
3098, 2814, 2727} \begin {gather*} -\frac {(c-d) \left (2 c^2+11 c d+29 d^2\right ) \cos (e+f x)}{15 f \left (a^3 \sin (e+f x)+a^3\right )}+\frac {d^3 x}{a^3}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a \sin (e+f x)+a)^3}-\frac {(c-d)^2 (2 c+7 d) \cos (e+f x)}{15 a 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 3098
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^3}{(a+a \sin (e+f x))^3} \, dx &=-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a+a \sin (e+f x))^3}-\frac {\int \frac {(c+d \sin (e+f x)) \left (-a \left (2 c^2+5 c d-2 d^2\right )-5 a d^2 \sin (e+f x)\right )}{(a+a \sin (e+f x))^2} \, dx}{5 a^2}\\ &=-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a+a \sin (e+f x))^3}-\frac {\int \frac {-a c \left (2 c^2+5 c d-2 d^2\right )+\left (-5 a c d^2-a d \left (2 c^2+5 c d-2 d^2\right )\right ) \sin (e+f x)-5 a d^3 \sin ^2(e+f x)}{(a+a \sin (e+f x))^2} \, dx}{5 a^2}\\ &=-\frac {(c-d)^2 (2 c+7 d) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a+a \sin (e+f x))^3}+\frac {\int \frac {a^2 \left (2 c^3+9 c^2 d+18 c d^2-14 d^3\right )+15 a^2 d^3 \sin (e+f x)}{a+a \sin (e+f x)} \, dx}{15 a^4}\\ &=\frac {d^3 x}{a^3}-\frac {(c-d)^2 (2 c+7 d) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a+a \sin (e+f x))^3}+\frac {\left ((c-d) \left (2 c^2+11 c d+29 d^2\right )\right ) \int \frac {1}{a+a \sin (e+f x)} \, dx}{15 a^2}\\ &=\frac {d^3 x}{a^3}-\frac {(c-d)^2 (2 c+7 d) \cos (e+f x)}{15 a f (a+a \sin (e+f x))^2}-\frac {(c-d) \left (2 c^2+11 c d+29 d^2\right ) \cos (e+f x)}{15 f \left (a^3+a^3 \sin (e+f x)\right )}-\frac {(c-d) \cos (e+f x) (c+d \sin (e+f x))^2}{5 f (a+a \sin (e+f x))^3}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in
optimal.
time = 7.78, size = 1366, normalized size = 9.62 \begin {gather*} -\frac {\cos (e+f x) \left (9450 \sqrt {2} (c+d)^3 \sin ^{-1}\left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right )-4725 (c+d)^3 \sqrt {1+\cos (2 e+2 f x)}+142 (c+d)^3 \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1-\sin (e+f x))^{9/2}+60 (c+d)^3 \, _3F_2\left (\frac {3}{2},2,\frac {9}{2};1,\frac {11}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1-\sin (e+f x))^{9/2}+8 (c+d)^3 \, _4F_3\left (\frac {3}{2},2,2,\frac {9}{2};1,1,\frac {11}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1-\sin (e+f x))^{9/2}+9450 \sqrt {2} d (c+d)^2 \sin ^{-1}\left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right ) (-1+\sin (e+f x))+5670 \sqrt {2} d^2 (c+d) \sin ^{-1}\left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right ) (-1+\sin (e+f x))^2+1350 \sqrt {2} d^3 \sin ^{-1}\left (\frac {\sqrt {1-\sin (e+f x)}}{\sqrt {2}}\right ) (-1+\sin (e+f x))^3-1575 \sqrt {2} (c+d)^3 (1-\sin (e+f x))^{3/2} \sqrt {1+\sin (e+f x)}-630 \sqrt {2} (c+d)^3 (1-\sin (e+f x))^{5/2} \sqrt {1+\sin (e+f x)}+4725 \sqrt {2} (c+d)^2 \sqrt {\cos ^2(e+f x)} (d-d \sin (e+f x))-282 (c+d)^2 \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1-\sin (e+f x))^{9/2} (d-d \sin (e+f x))-156 (c+d)^2 \, _3F_2\left (\frac {3}{2},2,\frac {9}{2};1,\frac {11}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1-\sin (e+f x))^{9/2} (d-d \sin (e+f x))-24 (c+d)^2 \, _4F_3\left (\frac {3}{2},2,2,\frac {9}{2};1,1,\frac {11}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1-\sin (e+f x))^{9/2} (d-d \sin (e+f x))+1575 \sqrt {2} (c+d)^2 (1-\sin (e+f x))^{3/2} \sqrt {1+\sin (e+f x)} (d-d \sin (e+f x))+630 \sqrt {2} (c+d)^2 (1-\sin (e+f x))^{5/2} \sqrt {1+\sin (e+f x)} (d-d \sin (e+f x))-2835 \sqrt {2} (c+d) \sqrt {\cos ^2(e+f x)} (d-d \sin (e+f x))^2+186 (c+d) \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1-\sin (e+f x))^{9/2} (d-d \sin (e+f x))^2+132 (c+d) \, _3F_2\left (\frac {3}{2},2,\frac {9}{2};1,\frac {11}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1-\sin (e+f x))^{9/2} (d-d \sin (e+f x))^2+24 (c+d) \, _4F_3\left (\frac {3}{2},2,2,\frac {9}{2};1,1,\frac {11}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1-\sin (e+f x))^{9/2} (d-d \sin (e+f x))^2-945 \sqrt {2} (c+d) (1-\sin (e+f x))^{3/2} \sqrt {1+\sin (e+f x)} (d-d \sin (e+f x))^2-378 \sqrt {2} (c+d) (1-\sin (e+f x))^{5/2} \sqrt {1+\sin (e+f x)} (d-d \sin (e+f x))^2+675 \sqrt {2} \sqrt {\cos ^2(e+f x)} (d-d \sin (e+f x))^3-46 \, _2F_1\left (\frac {3}{2},\frac {9}{2};\frac {11}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1-\sin (e+f x))^{9/2} (d-d \sin (e+f x))^3-36 \, _3F_2\left (\frac {3}{2},2,\frac {9}{2};1,\frac {11}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1-\sin (e+f x))^{9/2} (d-d \sin (e+f x))^3-8 \, _4F_3\left (\frac {3}{2},2,2,\frac {9}{2};1,1,\frac {11}{2};\frac {1}{2} (1-\sin (e+f x))\right ) (1-\sin (e+f x))^{9/2} (d-d \sin (e+f x))^3+225 \sqrt {2} (1-\sin (e+f x))^{3/2} \sqrt {1+\sin (e+f x)} (d-d \sin (e+f x))^3+90 \sqrt {2} (1-\sin (e+f x))^{5/2} \sqrt {1+\sin (e+f x)} (d-d \sin (e+f x))^3\right )}{2160 \sqrt {2} a^3 f (1-\sin (e+f x))^{7/2} \sqrt {1+\sin (e+f x)}} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [A]
time = 0.47, size = 194, normalized size = 1.37
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (c^{3}-d^{3}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-4 c^{3}+6 c^{2} d -2 d^{3}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-8 c^{3}+24 c^{2} d -24 c \,d^{2}+8 d^{3}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 c^{3}-12 c^{2} d +12 c \,d^{2}-4 d^{3}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 \left (8 c^{3}-18 c^{2} d +12 c \,d^{2}-2 d^{3}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+2 d^{3} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f \,a^{3}}\) | \(194\) |
default | \(\frac {-\frac {2 \left (c^{3}-d^{3}\right )}{\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1}-\frac {-4 c^{3}+6 c^{2} d -2 d^{3}}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{2}}-\frac {-8 c^{3}+24 c^{2} d -24 c \,d^{2}+8 d^{3}}{2 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{4}}-\frac {2 \left (4 c^{3}-12 c^{2} d +12 c \,d^{2}-4 d^{3}\right )}{5 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}-\frac {2 \left (8 c^{3}-18 c^{2} d +12 c \,d^{2}-2 d^{3}\right )}{3 \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{3}}+2 d^{3} \arctan \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{f \,a^{3}}\) | \(194\) |
risch | \(\frac {d^{3} x}{a^{3}}-\frac {2 \left (-45 d \,c^{2} {\mathrm e}^{2 i \left (f x +e \right )}+21 c \,d^{2}-45 i c^{2} d \,{\mathrm e}^{i \left (f x +e \right )}-60 i c \,d^{2} {\mathrm e}^{i \left (f x +e \right )}+45 c \,d^{2} {\mathrm e}^{4 i \left (f x +e \right )}+90 i c \,d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+115 i d^{3} {\mathrm e}^{i \left (f x +e \right )}-135 i d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+9 c^{2} d +2 c^{3}-120 d^{2} {\mathrm e}^{2 i \left (f x +e \right )} c -10 i c^{3} {\mathrm e}^{i \left (f x +e \right )}+45 i c^{2} d \,{\mathrm e}^{3 i \left (f x +e \right )}-45 d^{3} {\mathrm e}^{4 i \left (f x +e \right )}+185 d^{3} {\mathrm e}^{2 i \left (f x +e \right )}-20 c^{3} {\mathrm e}^{2 i \left (f x +e \right )}-32 d^{3}\right )}{15 f \,a^{3} \left ({\mathrm e}^{i \left (f x +e \right )}+i\right )^{5}}\) | \(251\) |
norman | \(\frac {\frac {d^{3} x}{a}+\frac {d^{3} x \left (\tan ^{11}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {\left (-2 c^{3}+2 d^{3}\right ) \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {\left (-4 c^{3}-6 c^{2} d +10 d^{3}\right ) \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {\left (-12 c^{3}-24 c^{2} d -12 c \,d^{2}+48 d^{3}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {-14 c^{3}-18 c^{2} d -12 c \,d^{2}+44 d^{3}}{15 a f}+\frac {5 d^{3} x \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{a}+\frac {13 d^{3} x \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {25 d^{3} x \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {38 d^{3} x \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {46 d^{3} x \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {46 d^{3} x \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {38 d^{3} x \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {25 d^{3} x \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {13 d^{3} x \left (\tan ^{9}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {5 d^{3} x \left (\tan ^{10}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a}+\frac {\left (-8 c^{3}-18 c^{2} d -12 c \,d^{2}+38 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{3 a f}+\frac {2 \left (-10 c^{3}-18 c^{2} d -6 c \,d^{2}+34 d^{3}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{a f}+\frac {2 \left (-22 c^{3}-36 c^{2} d -6 c \,d^{2}+64 d^{3}\right ) \left (\tan ^{7}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {\left (-34 c^{3}-18 c^{2} d -24 c \,d^{2}+76 d^{3}\right ) \left (\tan ^{8}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 a f}+\frac {2 \left (-52 c^{3}-54 c^{2} d -66 c \,d^{2}+172 d^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{5 a f}+\frac {\left (-122 c^{3}-144 c^{2} d -156 c \,d^{2}+422 d^{3}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f}+\frac {2 \left (-172 c^{3}-144 c^{2} d -186 c \,d^{2}+502 d^{3}\right ) \left (\tan ^{6}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{15 a f}}{\left (1+\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )^{3} a^{2} \left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )^{5}}\) | \(687\) |
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. 852 vs.
\(2 (143) = 286\).
time = 0.52, size = 852, normalized size = 6.00 \begin {gather*} \frac {2 \, {\left (d^{3} {\left (\frac {\frac {95 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {145 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {75 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 22}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} + \frac {15 \, \arctan \left (\frac {\sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1}\right )}{a^{3}}\right )} - \frac {c^{3} {\left (\frac {20 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {40 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {30 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {15 \, \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + 7\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} - \frac {6 \, c d^{2} {\left (\frac {5 \, \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}} - \frac {9 \, c^{2} d {\left (\frac {5 \, \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 {5 \, \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + 1\right )}}{a^{3} + \frac {5 \, a^{3} \sin \left (f x + e\right )}{\cos \left (f x + e\right ) + 1} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (f x + e\right )^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{3} \sin \left (f x + e\right )^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {a^{3} \sin \left (f x + e\right )^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}}}\right )}}{15 \, 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. 364 vs.
\(2 (143) = 286\).
time = 0.39, size = 364, normalized size = 2.56 \begin {gather*} -\frac {60 \, d^{3} f x - {\left (15 \, d^{3} f x - 2 \, c^{3} - 9 \, c^{2} d - 21 \, c d^{2} + 32 \, d^{3}\right )} \cos \left (f x + e\right )^{3} - 3 \, c^{3} + 9 \, c^{2} d - 9 \, c d^{2} + 3 \, d^{3} - {\left (45 \, d^{3} f x + 4 \, c^{3} + 18 \, c^{2} d - 3 \, c d^{2} - 19 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (10 \, d^{3} f x - 3 \, c^{3} - 6 \, c^{2} d - 9 \, c d^{2} + 18 \, d^{3}\right )} \cos \left (f x + e\right ) + {\left (60 \, d^{3} f x + 3 \, c^{3} - 9 \, c^{2} d + 9 \, c d^{2} - 3 \, d^{3} - {\left (15 \, d^{3} f x + 2 \, c^{3} + 9 \, c^{2} d + 21 \, c d^{2} - 32 \, d^{3}\right )} \cos \left (f x + e\right )^{2} + 3 \, {\left (10 \, d^{3} f x - 2 \, c^{3} - 9 \, c^{2} d - 6 \, c d^{2} + 17 \, d^{3}\right )} \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )}{15 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 3 \, a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - 2 \, a^{3} f \cos \left (f x + e\right ) - 4 \, a^{3} 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. 2640 vs.
\(2 (126) = 252\).
time = 10.00, size = 2640, normalized size = 18.59 \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 = 280, normalized size = 1.97 \begin {gather*} \frac {\frac {15 \, {\left (f x + e\right )} d^{3}}{a^{3}} - \frac {2 \, {\left (15 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 15 \, d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 45 \, c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 75 \, d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 40 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 45 \, c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 60 \, c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 145 \, d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 20 \, c^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 45 \, c^{2} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 30 \, c d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 95 \, d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 7 \, c^{3} + 9 \, c^{2} d + 6 \, c d^{2} - 22 \, d^{3}\right )}}{a^{3} {\left (\tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}^{5}}}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.90, size = 240, normalized size = 1.69 \begin {gather*} \frac {d^3\,x}{a^3}-\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,c^3-2\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {16\,c^3}{3}+6\,c^2\,d+8\,c\,d^2-\frac {58\,d^3}{3}\right )+\frac {4\,c\,d^2}{5}+\frac {6\,c^2\,d}{5}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (4\,c^3+6\,c^2\,d-10\,d^3\right )+\frac {14\,c^3}{15}-\frac {44\,d^3}{15}+\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (\frac {8\,c^3}{3}+6\,c^2\,d+4\,c\,d^2-\frac {38\,d^3}{3}\right )}{f\,\left (a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5+5\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+10\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3+10\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2+5\,a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+a^3\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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