Optimal. Leaf size=192 \[ \frac {a \left (2 c^2-2 c d+d^2\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c+d) \left (c^2-d^2\right )^{5/2} f}-\frac {a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac {a (2 c-3 d) \cos (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a (c-4 d) (2 c-d) \cos (e+f x)}{6 (c-d)^2 (c+d)^3 f (c+d \sin (e+f x))} \]
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Rubi [A]
time = 0.23, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {2833, 12, 2739,
632, 210} \begin {gather*} \frac {a \left (2 c^2-2 c d+d^2\right ) \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f (c+d) \left (c^2-d^2\right )^{5/2}}-\frac {a (c-4 d) (2 c-d) \cos (e+f x)}{6 f (c-d)^2 (c+d)^3 (c+d \sin (e+f x))}-\frac {a (2 c-3 d) \cos (e+f x)}{6 f (c-d) (c+d)^2 (c+d \sin (e+f x))^2}-\frac {a \cos (e+f x)}{3 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
Rubi steps
\begin {align*} \int \frac {a+a \sin (e+f x)}{(c+d \sin (e+f x))^4} \, dx &=-\frac {a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac {\int \frac {-3 a (c-d)-2 a (c-d) \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{3 \left (c^2-d^2\right )}\\ &=-\frac {a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac {a (2 c-3 d) \cos (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sin (e+f x))^2}+\frac {\int \frac {2 a (3 c-2 d) (c-d)+a (2 c-3 d) (c-d) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{6 \left (c^2-d^2\right )^2}\\ &=-\frac {a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac {a (2 c-3 d) \cos (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a (c-4 d) (2 c-d) \cos (e+f x)}{6 (c-d)^2 (c+d)^3 f (c+d \sin (e+f x))}-\frac {\int -\frac {3 a (c-d) \left (2 c^2-2 c d+d^2\right )}{c+d \sin (e+f x)} \, dx}{6 \left (c^2-d^2\right )^3}\\ &=-\frac {a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac {a (2 c-3 d) \cos (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a (c-4 d) (2 c-d) \cos (e+f x)}{6 (c-d)^2 (c+d)^3 f (c+d \sin (e+f x))}+\frac {\left (a \left (2 c^2-2 c d+d^2\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 (c-d)^2 (c+d)^3}\\ &=-\frac {a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac {a (2 c-3 d) \cos (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a (c-4 d) (2 c-d) \cos (e+f x)}{6 (c-d)^2 (c+d)^3 f (c+d \sin (e+f x))}+\frac {\left (a \left (2 c^2-2 c d+d^2\right )\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)^2 (c+d)^3 f}\\ &=-\frac {a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac {a (2 c-3 d) \cos (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a (c-4 d) (2 c-d) \cos (e+f x)}{6 (c-d)^2 (c+d)^3 f (c+d \sin (e+f x))}-\frac {\left (2 a \left (2 c^2-2 c d+d^2\right )\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)^2 (c+d)^3 f}\\ &=\frac {a \left (2 c^2-2 c d+d^2\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{(c-d)^2 (c+d)^3 \sqrt {c^2-d^2} f}-\frac {a \cos (e+f x)}{3 (c+d) f (c+d \sin (e+f x))^3}-\frac {a (2 c-3 d) \cos (e+f x)}{6 (c-d) (c+d)^2 f (c+d \sin (e+f x))^2}-\frac {a (c-4 d) (2 c-d) \cos (e+f x)}{6 (c-d)^2 (c+d)^3 f (c+d \sin (e+f x))}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.79, size = 428, normalized size = 2.23 \begin {gather*} \frac {a (1+\sin (e+f x)) \left (\frac {24 \left (2 c^2-2 c d+d^2\right ) \tan ^{-1}\left (\frac {\sec \left (\frac {f x}{2}\right ) (\cos (e)-i \sin (e)) \left (d \cos \left (e+\frac {f x}{2}\right )+c \sin \left (\frac {f x}{2}\right )\right )}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}\right ) (\cos (e)-i \sin (e))}{\sqrt {c^2-d^2} \sqrt {(\cos (e)-i \sin (e))^2}}+\frac {2 c \left (4 c^4-18 c^3 d+14 c^2 d^2-27 c d^3+12 d^4\right ) \cot (e)-d \csc (e) \left (3 d \left (4 c^3-16 c^2 d+6 c d^2+d^3\right ) \cos (e+2 f x)-3 d^2 \left (2 c^2-2 c d+d^2\right ) \cos (3 e+2 f x)-24 c^4 \sin (f x)+78 c^3 d \sin (f x)-24 c^2 d^2 \sin (f x)+12 c d^3 \sin (f x)-12 d^4 \sin (f x)+30 c^3 d \sin (2 e+f x)-30 c^2 d^2 \sin (2 e+f x)+15 c d^3 \sin (2 e+f x)+2 c^2 d^2 \sin (2 e+3 f x)-9 c d^3 \sin (2 e+3 f x)+4 d^4 \sin (2 e+3 f x)\right )}{d (c+d \sin (e+f x))^3}\right )}{24 (c-d)^2 (c+d)^3 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2} \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. \(640\) vs.
\(2(181)=362\).
time = 0.74, size = 641, normalized size = 3.34 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. 642 vs.
\(2 (188) = 376\).
time = 0.40, size = 1373, normalized size = 7.15 \begin {gather*} \left [-\frac {2 \, {\left (2 \, a c^{4} d^{2} - 9 \, a c^{3} d^{3} + 2 \, a c^{2} d^{4} + 9 \, a c d^{5} - 4 \, a d^{6}\right )} \cos \left (f x + e\right )^{3} - 6 \, {\left (2 \, a c^{5} d - 7 \, a c^{4} d^{2} + 8 \, a c^{2} d^{4} - 2 \, a c d^{5} - a d^{6}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 3 \, {\left (2 \, a c^{5} - 2 \, a c^{4} d + 7 \, a c^{3} d^{2} - 6 \, a c^{2} d^{3} + 3 \, a c d^{4} - 3 \, {\left (2 \, a c^{3} d^{2} - 2 \, a c^{2} d^{3} + a c d^{4}\right )} \cos \left (f x + e\right )^{2} + {\left (6 \, a c^{4} d - 6 \, a c^{3} d^{2} + 5 \, a c^{2} d^{3} - 2 \, a c d^{4} + a d^{5} - {\left (2 \, a c^{2} d^{3} - 2 \, a c d^{4} + a d^{5}\right )} \cos \left (f x + e\right )^{2}\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 (a c^{6} - 2 \, a c^{5} d - a c^{4} d^{2} + a c^{3} d^{3} + a c^{2} d^{4} + a c d^{5} - a d^{6}\right )} \cos \left (f x + e\right )}{12 \, {\left (3 \, {\left (c^{8} d^{2} + c^{7} d^{3} - 3 \, c^{6} d^{4} - 3 \, c^{5} d^{5} + 3 \, c^{4} d^{6} + 3 \, c^{3} d^{7} - c^{2} d^{8} - c d^{9}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{10} + c^{9} d - 6 \, c^{6} d^{4} - 6 \, c^{5} d^{5} + 8 \, c^{4} d^{6} + 8 \, c^{3} d^{7} - 3 \, c^{2} d^{8} - 3 \, c d^{9}\right )} f + {\left ({\left (c^{7} d^{3} + c^{6} d^{4} - 3 \, c^{5} d^{5} - 3 \, c^{4} d^{6} + 3 \, c^{3} d^{7} + 3 \, c^{2} d^{8} - c d^{9} - d^{10}\right )} f \cos \left (f x + e\right )^{2} - {\left (3 \, c^{9} d + 3 \, c^{8} d^{2} - 8 \, c^{7} d^{3} - 8 \, c^{6} d^{4} + 6 \, c^{5} d^{5} + 6 \, c^{4} d^{6} - c d^{9} - d^{10}\right )} f\right )} \sin \left (f x + e\right )\right )}}, -\frac {{\left (2 \, a c^{4} d^{2} - 9 \, a c^{3} d^{3} + 2 \, a c^{2} d^{4} + 9 \, a c d^{5} - 4 \, a d^{6}\right )} \cos \left (f x + e\right )^{3} - 3 \, {\left (2 \, a c^{5} d - 7 \, a c^{4} d^{2} + 8 \, a c^{2} d^{4} - 2 \, a c d^{5} - a d^{6}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 3 \, {\left (2 \, a c^{5} - 2 \, a c^{4} d + 7 \, a c^{3} d^{2} - 6 \, a c^{2} d^{3} + 3 \, a c d^{4} - 3 \, {\left (2 \, a c^{3} d^{2} - 2 \, a c^{2} d^{3} + a c d^{4}\right )} \cos \left (f x + e\right )^{2} + {\left (6 \, a c^{4} d - 6 \, a c^{3} d^{2} + 5 \, a c^{2} d^{3} - 2 \, a c d^{4} + a d^{5} - {\left (2 \, a c^{2} d^{3} - 2 \, a c d^{4} + a d^{5}\right )} \cos \left (f x + e\right )^{2}\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 (a c^{6} - 2 \, a c^{5} d - a c^{4} d^{2} + a c^{3} d^{3} + a c^{2} d^{4} + a c d^{5} - a d^{6}\right )} \cos \left (f x + e\right )}{6 \, {\left (3 \, {\left (c^{8} d^{2} + c^{7} d^{3} - 3 \, c^{6} d^{4} - 3 \, c^{5} d^{5} + 3 \, c^{4} d^{6} + 3 \, c^{3} d^{7} - c^{2} d^{8} - c d^{9}\right )} f \cos \left (f x + e\right )^{2} - {\left (c^{10} + c^{9} d - 6 \, c^{6} d^{4} - 6 \, c^{5} d^{5} + 8 \, c^{4} d^{6} + 8 \, c^{3} d^{7} - 3 \, c^{2} d^{8} - 3 \, c d^{9}\right )} f + {\left ({\left (c^{7} d^{3} + c^{6} d^{4} - 3 \, c^{5} d^{5} - 3 \, c^{4} d^{6} + 3 \, c^{3} d^{7} + 3 \, c^{2} d^{8} - c d^{9} - d^{10}\right )} f \cos \left (f x + e\right )^{2} - {\left (3 \, c^{9} d + 3 \, c^{8} d^{2} - 8 \, c^{7} d^{3} - 8 \, c^{6} d^{4} + 6 \, c^{5} d^{5} + 6 \, c^{4} d^{6} - c d^{9} - d^{10}\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. 808 vs.
\(2 (188) = 376\).
time = 0.50, size = 808, normalized size = 4.21 \begin {gather*} \frac {\frac {3 \, {\left (2 \, a c^{2} - 2 \, a c d + a d^{2}\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} + c^{4} d - 2 \, c^{3} d^{2} - 2 \, c^{2} d^{3} + c d^{4} + d^{5}\right )} \sqrt {c^{2} - d^{2}}} - \frac {12 \, a c^{6} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 15 \, a c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, a c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a c^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 12 \, a c^{6} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 30 \, a c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 51 \, a c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 18 \, a c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 18 \, a c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 12 \, a c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 36 \, a c^{6} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 72 \, a c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 12 \, a c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 30 \, a c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 4 \, a c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 12 \, a c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8 \, a d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 12 \, a c^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 24 \, a c^{6} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, a c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 84 \, a c^{4} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 18 \, a c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, a c d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, a c^{6} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 57 \, a c^{5} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, a c^{3} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a c^{2} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a c^{7} - 12 \, a c^{6} d - 2 \, a c^{5} d^{2} + 3 \, a c^{4} d^{3} + 2 \, a c^{3} d^{4}}{{\left (c^{8} + c^{7} d - 2 \, c^{6} d^{2} - 2 \, c^{5} d^{3} + c^{4} d^{4} + c^{3} 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 )}^{3}}}{3 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 9.71, size = 877, normalized size = 4.57 \begin {gather*} \frac {a\,\mathrm {atan}\left (\frac {\left (\frac {a\,c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c^2-2\,c\,d+d^2\right )}{{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{5/2}}+\frac {a\,\left (2\,c^2-2\,c\,d+d^2\right )\,\left (2\,c^5\,d+2\,c^4\,d^2-4\,c^3\,d^3-4\,c^2\,d^4+2\,c\,d^5+2\,d^6\right )}{2\,{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{5/2}\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}\right )\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}{2\,a\,c^2-2\,a\,c\,d+a\,d^2}\right )\,\left (2\,c^2-2\,c\,d+d^2\right )}{f\,{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{5/2}}-\frac {\frac {6\,a\,c^4-12\,a\,c^3\,d-2\,a\,c^2\,d^2+3\,a\,c\,d^3+2\,a\,d^4}{3\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}+\frac {a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (2\,c^6-4\,c^5\,d+10\,c^4\,d^2-17\,c^3\,d^3-6\,c^2\,d^4+6\,c\,d^5+4\,d^6\right )}{c^2\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}+\frac {2\,a\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (2\,c^6-4\,c^5\,d+6\,c^4\,d^2-14\,c^3\,d^3+3\,c\,d^5+2\,d^6\right )}{c^2\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}+\frac {a\,d\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (8\,c^4-19\,c^3\,d+4\,c\,d^3+2\,d^4\right )}{c\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}+\frac {a\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (4\,c^4-5\,c^3\,d-4\,c^2\,d^2+2\,c\,d^3+2\,d^4\right )}{c\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\right )}+\frac {2\,a\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,c^2+2\,d^2\right )\,\left (6\,c^4-12\,c^3\,d-2\,c^2\,d^2+3\,c\,d^3+2\,d^4\right )}{3\,c^3\,\left (c^5+c^4\,d-2\,c^3\,d^2-2\,c^2\,d^3+c\,d^4+d^5\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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