Optimal. Leaf size=289 \[ \frac {5 a^3 (4 c-3 d) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{4 (c-d) (c+d)^4 \sqrt {c^2-d^2} f}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 (c+d)^3 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 (c-d) d^2 (c+d)^4 f (c+d \sin (e+f x))} \]
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Rubi [A]
time = 0.48, antiderivative size = 289, normalized size of antiderivative = 1.00, number of steps
used = 9, 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 (4 c-3 d) \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{4 f (c-d) (c+d)^4 \sqrt {c^2-d^2}}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 f (c+d)^3 (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 d^2 f (c-d) (c+d)^4 (c+d \sin (e+f x))}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 f (c+d)^2 (c+d \sin (e+f x))^3}+\frac {(c-d) \cos (e+f x) \left (a^3 \sin (e+f x)+a^3\right )}{4 d f (c+d) (c+d \sin (e+f x))^4} \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))^5} \, dx &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}-\frac {a \int \frac {(a+a \sin (e+f x)) (a (c-9 d)-2 a (c+3 d) \sin (e+f x))}{(c+d \sin (e+f x))^4} \, dx}{4 d (c+d)}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}-\frac {a \int \frac {a^2 (c-9 d)+\left (a^2 (c-9 d)-2 a^2 (c+3 d)\right ) \sin (e+f x)-2 a^2 (c+3 d) \sin ^2(e+f x)}{(c+d \sin (e+f x))^4} \, dx}{4 d (c+d)}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}+\frac {a \int \frac {3 a^2 (c-d) d (c+15 d)+2 a^2 (c-d) \left (c^2+5 c d+18 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{12 (c-d) d^2 (c+d)^2}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 (c+d)^3 f (c+d \sin (e+f x))^2}-\frac {a \int \frac {-2 a^2 (c-d)^2 d (c+36 d)-a^2 (c-d)^2 \left (2 c^2+12 c d+45 d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{24 (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 )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 (c+d)^3 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 (c-d) d^2 (c+d)^4 f (c+d \sin (e+f x))}+\frac {a \int \frac {15 a^2 (4 c-3 d) (c-d)^2 d^2}{c+d \sin (e+f x)} \, dx}{24 (c-d)^3 d^2 (c+d)^4}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 (c+d)^3 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 (c-d) d^2 (c+d)^4 f (c+d \sin (e+f x))}+\frac {\left (5 a^3 (4 c-3 d)\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{8 (c-d) (c+d)^4}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 (c+d)^3 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 (c-d) d^2 (c+d)^4 f (c+d \sin (e+f x))}+\frac {\left (5 a^3 (4 c-3 d)\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 )}{4 (c-d) (c+d)^4 f}\\ &=\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 (c+d)^3 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 (c-d) d^2 (c+d)^4 f (c+d \sin (e+f x))}-\frac {\left (5 a^3 (4 c-3 d)\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 )}{2 (c-d) (c+d)^4 f}\\ &=\frac {5 a^3 (4 c-3 d) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{4 (c-d) (c+d)^4 \sqrt {c^2-d^2} f}+\frac {(c-d) \cos (e+f x) \left (a^3+a^3 \sin (e+f x)\right )}{4 d (c+d) f (c+d \sin (e+f x))^4}+\frac {a^3 (c-d) (2 c+9 d) \cos (e+f x)}{12 d^2 (c+d)^2 f (c+d \sin (e+f x))^3}-\frac {a^3 \left (2 c^2+12 c d+45 d^2\right ) \cos (e+f x)}{24 d^2 (c+d)^3 f (c+d \sin (e+f x))^2}-\frac {a^3 \left (2 c^3+12 c^2 d+43 c d^2-72 d^3\right ) \cos (e+f x)}{24 (c-d) d^2 (c+d)^4 f (c+d \sin (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 2.01, size = 240, normalized size = 0.83 \begin {gather*} \frac {a^3 \cos (e+f x) \left (-\frac {d (1+\sin (e+f x))^3}{(c+d \sin (e+f x))^4}-\frac {(4 c-3 d) \left (-\frac {5 \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)^{7/2} \sqrt {c-d}}-\frac {\sqrt {\cos ^2(e+f x)} \left (22 c^2+9 c d+2 d^2+\left (9 c^2+48 c d+9 d^2\right ) \sin (e+f x)+\left (2 c^2+9 c d+22 d^2\right ) \sin ^2(e+f x)\right )}{6 (c+d)^3 (c+d \sin (e+f x))^3}\right )}{\sqrt {\cos ^2(e+f x)}}\right )}{4 (-c+d) (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. \(921\) vs.
\(2(274)=548\).
time = 1.40, size = 922, normalized size = 3.19 Too large to display
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. 977 vs.
\(2 (284) = 568\).
time = 0.44, size = 2044, normalized size = 7.07 \begin {gather*} \text {Too large to display} \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. 1338 vs.
\(2 (284) = 568\).
time = 0.53, size = 1338, normalized size = 4.63 \begin {gather*} \frac {\frac {15 \, {\left (4 \, a^{3} c - 3 \, a^{3} d\right )} {\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 )}}{{\left (c^{5} + 3 \, c^{4} d + 2 \, c^{3} d^{2} - 2 \, c^{2} d^{3} - 3 \, c d^{4} - d^{5}\right )} \sqrt {c^{2} - d^{2}}} + \frac {36 \, a^{3} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 117 \, a^{3} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 48 \, a^{3} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 48 \, a^{3} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 72 \, a^{3} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} + 24 \, a^{3} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{7} - 72 \, a^{3} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 132 \, a^{3} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} - 675 \, a^{3} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 360 \, a^{3} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 288 \, a^{3} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 72 \, a^{3} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{6} + 36 \, a^{3} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 813 \, a^{3} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 288 \, a^{3} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 892 \, a^{3} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 552 \, a^{3} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 664 \, a^{3} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 384 \, a^{3} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 96 \, a^{3} c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 264 \, a^{3} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 108 \, a^{3} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 2001 \, a^{3} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 936 \, a^{3} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 202 \, a^{3} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 864 \, a^{3} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 440 \, a^{3} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 192 \, a^{3} c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 48 \, a^{3} d^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 36 \, a^{3} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 1299 \, a^{3} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 576 \, a^{3} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 1036 \, a^{3} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 1176 \, a^{3} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 664 \, a^{3} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 384 \, a^{3} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 96 \, a^{3} c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 280 \, a^{3} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, a^{3} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1289 \, a^{3} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 960 \, a^{3} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 552 \, a^{3} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 288 \, a^{3} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 72 \, a^{3} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 36 \, a^{3} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 587 \, a^{3} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 336 \, a^{3} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 248 \, a^{3} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 120 \, a^{3} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 24 \, a^{3} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 88 \, a^{3} c^{8} + 36 \, a^{3} c^{7} d + 37 \, a^{3} c^{6} d^{2} + 24 \, a^{3} c^{5} d^{3} + 6 \, a^{3} c^{4} d^{4}}{{\left (c^{9} + 3 \, c^{8} d + 2 \, c^{7} d^{2} - 2 \, c^{6} d^{3} - 3 \, c^{5} d^{4} - c^{4} d^{5}\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 )}^{4}}}{12 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 10.10, size = 1231, normalized size = 4.26 \begin {gather*} -\frac {\frac {-88\,a^3\,c^4+36\,a^3\,c^3\,d+37\,a^3\,c^2\,d^2+24\,a^3\,c\,d^3+6\,a^3\,d^4}{12\,\left (-c^5-3\,c^4\,d-2\,c^3\,d^2+2\,c^2\,d^3+3\,c\,d^4+d^5\right )}+\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\,\left (12\,c^5-39\,c^4\,d-16\,c^3\,d^2+16\,c^2\,d^3+24\,c\,d^4+8\,d^5\right )}{4\,c\,\left (-c^5-3\,c^4\,d-2\,c^3\,d^2+2\,c^2\,d^3+3\,c\,d^4+d^5\right )}+\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (-24\,c^6+44\,c^5\,d-225\,c^4\,d^2+120\,c^2\,d^4+96\,c\,d^5+24\,d^6\right )}{4\,c^2\,\left (-c^5-3\,c^4\,d-2\,c^3\,d^2+2\,c^2\,d^3+3\,c\,d^4+d^5\right )}+\frac {a^3\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (-36\,c^5-587\,c^4\,d+336\,c^3\,d^2+248\,c^2\,d^3+120\,c\,d^4+24\,d^5\right )}{12\,c\,\left (-c^5-3\,c^4\,d-2\,c^3\,d^2+2\,c^2\,d^3+3\,c\,d^4+d^5\right )}+\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (36\,c^7-813\,c^6\,d+288\,c^5\,d^2-892\,c^4\,d^3+552\,c^3\,d^4+664\,c^2\,d^5+384\,c\,d^6+96\,d^7\right )}{12\,c^3\,\left (-c^5-3\,c^4\,d-2\,c^3\,d^2+2\,c^2\,d^3+3\,c\,d^4+d^5\right )}+\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (-36\,c^7-1299\,c^6\,d+576\,c^5\,d^2-1036\,c^4\,d^3+1176\,c^3\,d^4+664\,c^2\,d^5+384\,c\,d^6+96\,d^7\right )}{12\,c^3\,\left (-c^5-3\,c^4\,d-2\,c^3\,d^2+2\,c^2\,d^3+3\,c\,d^4+d^5\right )}+\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (-280\,c^6+12\,c^5\,d-1289\,c^4\,d^2+960\,c^3\,d^3+552\,c^2\,d^4+288\,c\,d^5+72\,d^6\right )}{12\,c^2\,\left (-c^5-3\,c^4\,d-2\,c^3\,d^2+2\,c^2\,d^3+3\,c\,d^4+d^5\right )}+\frac {a^3\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (3\,c^4+24\,c^2\,d^2+8\,d^4\right )\,\left (-88\,c^4+36\,c^3\,d+37\,c^2\,d^2+24\,c\,d^3+6\,d^4\right )}{12\,c^4\,\left (-c^5-3\,c^4\,d-2\,c^3\,d^2+2\,c^2\,d^3+3\,c\,d^4+d^5\right )}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (6\,c^4+48\,c^2\,d^2+16\,d^4\right )+c^4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8+c^4+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (4\,c^4+24\,c^2\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (4\,c^4+24\,c^2\,d^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (24\,c^3\,d+32\,c\,d^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (24\,c^3\,d+32\,c\,d^3\right )+8\,c^3\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )+8\,c^3\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^7\right )}-\frac {5\,a^3\,\mathrm {atan}\left (\frac {4\,\left (\frac {5\,a^3\,\left (4\,c-3\,d\right )\,\left (-8\,c^5\,d-24\,c^4\,d^2-16\,c^3\,d^3+16\,c^2\,d^4+24\,c\,d^5+8\,d^6\right )}{32\,{\left (c+d\right )}^{9/2}\,{\left (c-d\right )}^{3/2}\,\left (-c^5-3\,c^4\,d-2\,c^3\,d^2+2\,c^2\,d^3+3\,c\,d^4+d^5\right )}+\frac {5\,a^3\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (4\,c-3\,d\right )}{4\,{\left (c+d\right )}^{9/2}\,{\left (c-d\right )}^{3/2}}\right )\,\left (-c^5-3\,c^4\,d-2\,c^3\,d^2+2\,c^2\,d^3+3\,c\,d^4+d^5\right )}{20\,a^3\,c-15\,a^3\,d}\right )\,\left (4\,c-3\,d\right )}{4\,f\,{\left (c+d\right )}^{9/2}\,{\left (c-d\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
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