Optimal. Leaf size=207 \[ \frac {5 a^3 \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c+d)^3 \sqrt {c^2-d^2} f}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^3}+\frac {a^3 (c-d) (2 c+7 d) \cos (e+f x)}{6 d^2 (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^2+9 c d+22 d^2\right ) \cos (e+f x)}{6 d^2 (c+d)^3 f (c+d \sin (e+f x))} \]
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Rubi [A]
time = 0.33, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {2841, 3047,
3100, 2833, 12, 2739, 632, 210} \begin {gather*} \frac {5 a^3 \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f (c+d)^3 \sqrt {c^2-d^2}}-\frac {a^3 \left (2 c^2+9 c d+22 d^2\right ) \cos (e+f x)}{6 d^2 f (c+d)^3 (c+d \sin (e+f x))}+\frac {a^3 (c-d) (2 c+7 d) \cos (e+f x)}{6 d^2 f (c+d)^2 (c+d \sin (e+f x))^2}+\frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{3 d f (c+d) (c+d \sin (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 632
Rule 2739
Rule 2833
Rule 2841
Rule 3047
Rule 3100
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^3}{(c+d \sin (e+f x))^4} \, dx &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a \int \frac {(a+a \sin (e+f x)) (a (c-7 d)-2 a (c+2 d) \sin (e+f x))}{(c+d \sin (e+f x))^3} \, dx}{3 d (c+d)}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^3}-\frac {a \int \frac {a^2 (c-7 d)+\left (a^2 (c-7 d)-2 a^2 (c+2 d)\right ) \sin (e+f x)-2 a^2 (c+2 d) \sin ^2(e+f x)}{(c+d \sin (e+f x))^3} \, dx}{3 d (c+d)}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^3}+\frac {a^3 (c-d) (2 c+7 d) \cos (e+f x)}{6 d^2 (c+d)^2 f (c+d \sin (e+f x))^2}+\frac {a \int \frac {2 a^2 (c-d) d (c+11 d)+a^2 (c-d) \left (2 c^2+7 c d+15 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{6 (c-d) d^2 (c+d)^2}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^3}+\frac {a^3 (c-d) (2 c+7 d) \cos (e+f x)}{6 d^2 (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^2+9 c d+22 d^2\right ) \cos (e+f x)}{6 d^2 (c+d)^3 f (c+d \sin (e+f x))}-\frac {a \int -\frac {15 a^2 (c-d)^2 d^2}{c+d \sin (e+f x)} \, dx}{6 (c-d)^2 d^2 (c+d)^3}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^3}+\frac {a^3 (c-d) (2 c+7 d) \cos (e+f x)}{6 d^2 (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^2+9 c d+22 d^2\right ) \cos (e+f x)}{6 d^2 (c+d)^3 f (c+d \sin (e+f x))}+\frac {\left (5 a^3\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 (c+d)^3}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^3}+\frac {a^3 (c-d) (2 c+7 d) \cos (e+f x)}{6 d^2 (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^2+9 c d+22 d^2\right ) \cos (e+f x)}{6 d^2 (c+d)^3 f (c+d \sin (e+f x))}+\frac {\left (5 a^3\right ) \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 )}{(c+d)^3 f}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^3}+\frac {a^3 (c-d) (2 c+7 d) \cos (e+f x)}{6 d^2 (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^2+9 c d+22 d^2\right ) \cos (e+f x)}{6 d^2 (c+d)^3 f (c+d \sin (e+f x))}-\frac {\left (10 a^3\right ) \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 )}{(c+d)^3 f}\\ &=\frac {5 a^3 \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c+d)^3 \sqrt {c^2-d^2} f}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{3 d (c+d) f (c+d \sin (e+f x))^3}+\frac {a^3 (c-d) (2 c+7 d) \cos (e+f x)}{6 d^2 (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^2+9 c d+22 d^2\right ) \cos (e+f x)}{6 d^2 (c+d)^3 f (c+d \sin (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 1.50, size = 178, normalized size = 0.86 \begin {gather*} \frac {a^3 \cos (e+f x) \left (\frac {15 \tanh ^{-1}\left (\frac {\sqrt {c-d} \sqrt {1-\sin (e+f x)}}{\sqrt {-c-d} \sqrt {1+\sin (e+f x)}}\right )}{(-c-d)^{5/2} \sqrt {c-d} \sqrt {\cos ^2(e+f x)}}-\frac {(1+\sin (e+f x))^2}{(c+d \sin (e+f x))^3}-\frac {5 (1+\sin (e+f x))}{2 (c+d) (c+d \sin (e+f x))^2}-\frac {15}{2 (c+d)^2 (c+d \sin (e+f x))}\right )}{3 (c+d) f} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(465\) vs.
\(2(196)=392\).
time = 0.89, size = 466, normalized size = 2.25
method | result | size |
derivativedivides | \(\frac {2 a^{3} \left (\frac {\frac {\left (3 c^{3}-6 c^{2} d -6 c \,d^{2}-2 d^{3}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}-\frac {\left (6 c^{4}-3 c^{3} d +30 c^{2} d^{2}+18 d^{3} c +4 d^{4}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c^{2} \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}-\frac {d \left (66 c^{4}+27 c^{3} d +50 c^{2} d^{2}+18 d^{3} c +4 d^{4}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c^{3} \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}-\frac {\left (8 c^{4}+6 c^{3} d +30 c^{2} d^{2}+9 d^{3} c +2 d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{2} \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}-\frac {\left (3 c^{3}+38 c^{2} d +12 c \,d^{2}+2 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}-\frac {22 c^{2}+9 c d +2 d^{2}}{6 \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{3}}+\frac {5 \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right ) \sqrt {c^{2}-d^{2}}}\right )}{f}\) | \(466\) |
default | \(\frac {2 a^{3} \left (\frac {\frac {\left (3 c^{3}-6 c^{2} d -6 c \,d^{2}-2 d^{3}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}-\frac {\left (6 c^{4}-3 c^{3} d +30 c^{2} d^{2}+18 d^{3} c +4 d^{4}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{2 c^{2} \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}-\frac {d \left (66 c^{4}+27 c^{3} d +50 c^{2} d^{2}+18 d^{3} c +4 d^{4}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c^{3} \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}-\frac {\left (8 c^{4}+6 c^{3} d +30 c^{2} d^{2}+9 d^{3} c +2 d^{4}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{2} \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}-\frac {\left (3 c^{3}+38 c^{2} d +12 c \,d^{2}+2 d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 c \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}-\frac {22 c^{2}+9 c d +2 d^{2}}{6 \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}}{\left (c \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+2 d \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+c \right )^{3}}+\frac {5 \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{2 \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right ) \sqrt {c^{2}-d^{2}}}\right )}{f}\) | \(466\) |
risch | \(\frac {i a^{3} \left (8 c^{5} {\mathrm e}^{3 i \left (f x +e \right )}-22 i d^{5}+9 d^{5} {\mathrm e}^{5 i \left (f x +e \right )}-9 d^{5} {\mathrm e}^{i \left (f x +e \right )}-54 i c^{3} d^{2} {\mathrm e}^{4 i \left (f x +e \right )}-90 i c^{2} d^{3} {\mathrm e}^{4 i \left (f x +e \right )}+9 i c \,d^{4} {\mathrm e}^{4 i \left (f x +e \right )}+12 i c^{4} d \,{\mathrm e}^{2 i \left (f x +e \right )}+180 i c^{2} d^{3} {\mathrm e}^{2 i \left (f x +e \right )}+54 i c^{3} d^{2} {\mathrm e}^{2 i \left (f x +e \right )}+36 i {\mathrm e}^{2 i \left (f x +e \right )} c \,d^{4}-12 i c^{4} d \,{\mathrm e}^{4 i \left (f x +e \right )}-2 i c^{2} d^{3}-9 i c \,d^{4}-6 c^{3} d^{2} {\mathrm e}^{i \left (f x +e \right )}+36 c^{4} d \,{\mathrm e}^{3 i \left (f x +e \right )}+100 c^{3} d^{2} {\mathrm e}^{3 i \left (f x +e \right )}+54 c^{2} d^{3} {\mathrm e}^{3 i \left (f x +e \right )}+132 c \,d^{4} {\mathrm e}^{3 i \left (f x +e \right )}-36 c^{2} d^{3} {\mathrm e}^{i \left (f x +e \right )}-114 c \,d^{4} {\mathrm e}^{i \left (f x +e \right )}+48 i d^{5} {\mathrm e}^{2 i \left (f x +e \right )}-18 i d^{5} {\mathrm e}^{4 i \left (f x +e \right )}-18 c^{2} d^{3} {\mathrm e}^{5 i \left (f x +e \right )}-18 c \,d^{4} {\mathrm e}^{5 i \left (f x +e \right )}-6 d^{2} c^{3} {\mathrm e}^{5 i \left (f x +e \right )}\right )}{3 \left (c +d \right )^{3} \left (-i d \,{\mathrm e}^{2 i \left (f x +e \right )}+i d +2 c \,{\mathrm e}^{i \left (f x +e \right )}\right )^{3} f \,d^{3}}-\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}-c^{2}+d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right )}{2 \sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{3} f}+\frac {5 a^{3} \ln \left ({\mathrm e}^{i \left (f x +e \right )}+\frac {i c \sqrt {-c^{2}+d^{2}}+c^{2}-d^{2}}{\sqrt {-c^{2}+d^{2}}\, d}\right )}{2 \sqrt {-c^{2}+d^{2}}\, \left (c +d \right )^{3} f}\) | \(592\) |
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. 523 vs.
\(2 (204) = 408\).
time = 0.41, size = 1135, normalized size = 5.48 \begin {gather*} \left [-\frac {2 \, {\left (2 \, a^{3} c^{4} + 9 \, a^{3} c^{3} d + 20 \, a^{3} c^{2} d^{2} - 9 \, a^{3} c d^{3} - 22 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )^{3} - 6 \, {\left (3 \, a^{3} c^{4} + 16 \, a^{3} c^{3} d - 16 \, a^{3} c d^{3} - 3 \, a^{3} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 15 \, {\left (3 \, a^{3} c d^{2} \cos \left (f x + e\right )^{2} - a^{3} c^{3} - 3 \, a^{3} c d^{2} + {\left (a^{3} d^{3} \cos \left (f x + e\right )^{2} - 3 \, a^{3} c^{2} d - a^{3} d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c^{2} + d^{2}} \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 ) - 12 \, {\left (4 \, a^{3} c^{4} + 3 \, a^{3} c^{3} d - 3 \, a^{3} c d^{3} - 4 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )}{12 \, {\left (3 \, {\left (c^{6} d^{2} + 3 \, c^{5} d^{3} + 2 \, c^{4} d^{4} - 2 \, c^{3} d^{5} - 3 \, c^{2} d^{6} - c d^{7}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{8} + 3 \, c^{7} d + 5 \, c^{6} d^{2} + 7 \, c^{5} d^{3} + 3 \, c^{4} d^{4} - 7 \, c^{3} d^{5} - 9 \, c^{2} d^{6} - 3 \, c d^{7}\right )} f + {\left ({\left (c^{5} d^{3} + 3 \, c^{4} d^{4} + 2 \, c^{3} d^{5} - 2 \, c^{2} d^{6} - 3 \, c d^{7} - d^{8}\right )} f \cos \left (f x + e\right )^{2} - {\left (3 \, c^{7} d + 9 \, c^{6} d^{2} + 7 \, c^{5} d^{3} - 3 \, c^{4} d^{4} - 7 \, c^{3} d^{5} - 5 \, c^{2} d^{6} - 3 \, c d^{7} - d^{8}\right )} f\right )} \sin \left (f x + e\right )\right )}}, -\frac {{\left (2 \, a^{3} c^{4} + 9 \, a^{3} c^{3} d + 20 \, a^{3} c^{2} d^{2} - 9 \, a^{3} c d^{3} - 22 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (3 \, a^{3} c^{4} + 16 \, a^{3} c^{3} d - 16 \, a^{3} c d^{3} - 3 \, a^{3} d^{4}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) + 15 \, {\left (3 \, a^{3} c d^{2} \cos \left (f x + e\right )^{2} - a^{3} c^{3} - 3 \, a^{3} c d^{2} + {\left (a^{3} d^{3} \cos \left (f x + e\right )^{2} - 3 \, a^{3} c^{2} d - a^{3} d^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {c^{2} - d^{2}} \arctan \left (-\frac {c \sin \left (f x + e\right ) + d}{\sqrt {c^{2} - d^{2}} \cos \left (f x + e\right )}\right ) - 6 \, {\left (4 \, a^{3} c^{4} + 3 \, a^{3} c^{3} d - 3 \, a^{3} c d^{3} - 4 \, a^{3} d^{4}\right )} \cos \left (f x + e\right )}{6 \, {\left (3 \, {\left (c^{6} d^{2} + 3 \, c^{5} d^{3} + 2 \, c^{4} d^{4} - 2 \, c^{3} d^{5} - 3 \, c^{2} d^{6} - c d^{7}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{8} + 3 \, c^{7} d + 5 \, c^{6} d^{2} + 7 \, c^{5} d^{3} + 3 \, c^{4} d^{4} - 7 \, c^{3} d^{5} - 9 \, c^{2} d^{6} - 3 \, c d^{7}\right )} f + {\left ({\left (c^{5} d^{3} + 3 \, c^{4} d^{4} + 2 \, c^{3} d^{5} - 2 \, c^{2} d^{6} - 3 \, c d^{7} - d^{8}\right )} f \cos \left (f x + e\right )^{2} - {\left (3 \, c^{7} d + 9 \, c^{6} d^{2} + 7 \, c^{5} d^{3} - 3 \, c^{4} d^{4} - 7 \, c^{3} d^{5} - 5 \, c^{2} d^{6} - 3 \, c d^{7} - d^{8}\right )} f\right )} \sin \left (f x + e\right )\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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. 667 vs.
\(2 (204) = 408\).
time = 0.54, size = 667, normalized size = 3.22 \begin {gather*} \frac {\frac {15 \, {\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 )} a^{3}}{{\left (c^{3} + 3 \, c^{2} d + 3 \, c d^{2} + d^{3}\right )} \sqrt {c^{2} - d^{2}}} + \frac {9 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 18 \, a^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 18 \, a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 6 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 18 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 9 \, a^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 90 \, a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 54 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 12 \, a^{3} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 132 \, a^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 54 \, a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 100 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 36 \, a^{3} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 8 \, a^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 48 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 36 \, a^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 180 \, a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 54 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 12 \, a^{3} c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 9 \, a^{3} c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 114 \, a^{3} c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 36 \, a^{3} c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a^{3} c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 22 \, a^{3} c^{5} - 9 \, a^{3} c^{4} d - 2 \, a^{3} c^{3} d^{2}}{{\left (c^{6} + 3 \, c^{5} d + 3 \, c^{4} d^{2} + c^{3} d^{3}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 2 \, d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + c\right )}^{3}}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.36, size = 649, normalized size = 3.14 \begin {gather*} \frac {5\,a^3\,\mathrm {atan}\left (\frac {\left (\frac {5\,a^3\,\left (2\,c^3\,d+6\,c^2\,d^2+6\,c\,d^3+2\,d^4\right )}{2\,{\left (c+d\right )}^{7/2}\,\sqrt {c-d}\,\left (c^3+3\,c^2\,d+3\,c\,d^2+d^3\right )}+\frac {5\,a^3\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}{{\left (c+d\right )}^{7/2}\,\sqrt {c-d}}\right )\,\left (c^3+3\,c^2\,d+3\,c\,d^2+d^3\right )}{5\,a^3}\right )}{f\,{\left (c+d\right )}^{7/2}\,\sqrt {c-d}}-\frac {\frac {22\,a^3\,c^2+9\,a^3\,c\,d+2\,a^3\,d^2}{3\,\left (c^3+3\,c^2\,d+3\,c\,d^2+d^3\right )}+\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (-3\,c^3+6\,c^2\,d+6\,c\,d^2+2\,d^3\right )}{c\,\left (c^3+3\,c^2\,d+3\,c\,d^2+d^3\right )}+\frac {a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (3\,c^3+38\,c^2\,d+12\,c\,d^2+2\,d^3\right )}{c\,\left (c^3+3\,c^2\,d+3\,c\,d^2+d^3\right )}+\frac {2\,a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (8\,c^4+6\,c^3\,d+30\,c^2\,d^2+9\,c\,d^3+2\,d^4\right )}{c^2\,\left (c^3+3\,c^2\,d+3\,c\,d^2+d^3\right )}+\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (6\,c^4-3\,c^3\,d+30\,c^2\,d^2+18\,c\,d^3+4\,d^4\right )}{c^2\,\left (c^3+3\,c^2\,d+3\,c\,d^2+d^3\right )}+\frac {2\,a^3\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,c^2+2\,d^2\right )\,\left (22\,c^2+9\,c\,d+2\,d^2\right )}{3\,c^3\,\left (c^3+3\,c^2\,d+3\,c\,d^2+d^3\right )}}{f\,\left (c^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (3\,c^3+12\,c\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (3\,c^3+12\,c\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (12\,c^2\,d+8\,d^3\right )+c^3+6\,c^2\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+6\,c^2\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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