Optimal. Leaf size=305 \[ -\frac {\left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2} f}+\frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))} \]
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Rubi [A]
time = 0.41, antiderivative size = 305, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {2869, 2833, 12,
2739, 632, 210} \begin {gather*} -\frac {\left (-\left (a^2 \left (2 c^3+3 c d^2\right )\right )+2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )\right ) \text {ArcTan}\left (\frac {c \tan \left (\frac {1}{2} (e+f x)\right )+d}{\sqrt {c^2-d^2}}\right )}{f \left (c^2-d^2\right )^{7/2}}+\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-\left (b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right )\right ) \cos (e+f x)}{6 d f \left (c^2-d^2\right )^3 (c+d \sin (e+f x))}+\frac {(b c-a d)^2 \cos (e+f x)}{3 d f \left (c^2-d^2\right ) (c+d \sin (e+f x))^3}-\frac {\left (5 a c d+b \left (c^2-6 d^2\right )\right ) (b c-a d) \cos (e+f x)}{6 d f \left (c^2-d^2\right )^2 (c+d \sin (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 210
Rule 632
Rule 2739
Rule 2833
Rule 2869
Rubi steps
\begin {align*} \int \frac {(a+b \sin (e+f x))^2}{(c+d \sin (e+f x))^4} \, dx &=\frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}+\frac {\int \frac {3 d \left (\left (a^2+b^2\right ) c-2 a b d\right )+\left (4 a b c d-2 a^2 d^2+b^2 \left (c^2-3 d^2\right )\right ) \sin (e+f x)}{(c+d \sin (e+f x))^3} \, dx}{3 d \left (c^2-d^2\right )}\\ &=\frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}-\frac {\int \frac {2 d \left (10 a b c d-a^2 \left (3 c^2+2 d^2\right )-b^2 \left (2 c^2+3 d^2\right )\right )-(b c-a d) \left (b c^2+5 a c d-6 b d^2\right ) \sin (e+f x)}{(c+d \sin (e+f x))^2} \, dx}{6 d \left (c^2-d^2\right )^2}\\ &=\frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}+\frac {\int -\frac {3 d \left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\right )\right )}{c+d \sin (e+f x)} \, dx}{6 d \left (c^2-d^2\right )^3}\\ &=\frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}-\frac {\left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\right )\right ) \int \frac {1}{c+d \sin (e+f x)} \, dx}{2 \left (c^2-d^2\right )^3}\\ &=\frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}-\frac {\left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c 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 )}{\left (c^2-d^2\right )^3 f}\\ &=\frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}+\frac {\left (2 \left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\right )\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 )}{\left (c^2-d^2\right )^3 f}\\ &=-\frac {\left (2 a b d \left (4 c^2+d^2\right )-b^2 c \left (c^2+4 d^2\right )-a^2 \left (2 c^3+3 c d^2\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2} f}+\frac {(b c-a d)^2 \cos (e+f x)}{3 d \left (c^2-d^2\right ) f (c+d \sin (e+f x))^3}-\frac {(b c-a d) \left (5 a c d+b \left (c^2-6 d^2\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^2 f (c+d \sin (e+f x))^2}+\frac {\left (a^2 d^2 \left (11 c^2+4 d^2\right )-a b \left (4 c^3 d+26 c d^3\right )-b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (e+f x)}{6 d \left (c^2-d^2\right )^3 f (c+d \sin (e+f x))}\\ \end {align*}
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Mathematica [A]
time = 1.49, size = 346, normalized size = 1.13 \begin {gather*} \frac {\frac {12 \left (-2 a b d \left (4 c^2+d^2\right )+b^2 c \left (c^2+4 d^2\right )+a^2 \left (2 c^3+3 c d^2\right )\right ) \tan ^{-1}\left (\frac {d+c \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right )}{\left (c^2-d^2\right )^{7/2}}+\frac {\cos (e+f x) \left (-24 a b c^5+36 a^2 c^4 d+25 b^2 c^4 d-44 a b c^3 d^2+a^2 c^2 d^3+14 b^2 c^2 d^3-22 a b c d^4+8 a^2 d^5+6 b^2 d^5+d \left (-a^2 d^2 \left (11 c^2+4 d^2\right )+a b \left (4 c^3 d+26 c d^3\right )+b^2 \left (c^4-10 c^2 d^2-6 d^4\right )\right ) \cos (2 (e+f x))-6 \left (-a^2 c d^2 \left (9 c^2+d^2\right )-2 a b d \left (-2 c^4-9 c^2 d^2+d^4\right )+b^2 \left (c^5-9 c^3 d^2-2 c d^4\right )\right ) \sin (e+f x)\right )}{\left (c^2-d^2\right )^3 (c+d \sin (e+f x))^3}}{12 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. \(932\) vs.
\(2(294)=588\).
time = 0.91, size = 933, normalized size = 3.06
method | result | size |
derivativedivides | \(\frac {\frac {\frac {\left (9 a^{2} c^{4} d^{2}-6 a^{2} c^{2} d^{4}+2 a^{2} d^{6}-8 a b \,c^{5} d -2 a b \,c^{3} d^{3}+b^{2} c^{6}+4 b^{2} c^{4} d^{2}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {\left (6 a^{2} c^{6} d +27 a^{2} c^{4} d^{3}-12 a^{2} c^{2} d^{5}+4 a^{2} d^{7}-4 a b \,c^{7}-28 a b \,c^{5} d^{2}-22 a b \,c^{3} d^{4}+4 a b c \,d^{6}+5 b^{2} c^{6} d +20 b^{2} c^{4} d^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{\left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) c^{2}}+\frac {2 d \left (54 a^{2} c^{6} d +21 a^{2} c^{4} d^{3}-4 a^{2} c^{2} d^{5}+4 a^{2} d^{7}-36 a b \,c^{7}-84 a b \,c^{5} d^{2}-34 a b \,c^{3} d^{4}+4 a b c \,d^{6}+39 b^{2} c^{6} d +32 b^{2} c^{4} d^{3}+4 b^{2} c^{2} d^{5}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c^{3} \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {2 \left (6 a^{2} c^{6} d +20 a^{2} c^{4} d^{3}-3 a^{2} c^{2} d^{5}+2 a^{2} d^{7}-4 a b \,c^{7}-20 a b \,c^{5} d^{2}-28 a b \,c^{3} d^{4}+2 a b c \,d^{6}+4 b^{2} c^{6} d +17 b^{2} c^{4} d^{3}+4 b^{2} c^{2} d^{5}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{2} \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {\left (27 a^{2} c^{4} d^{2}-4 a^{2} c^{2} d^{4}+2 a^{2} d^{6}-16 a b \,c^{5} d -38 a b \,c^{3} d^{3}+4 a b c \,d^{5}-b^{2} c^{6}+22 b^{2} c^{4} d^{2}+4 b^{2} c^{2} d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {2 \left (18 a^{2} c^{4} d -5 a^{2} c^{2} d^{3}+2 a^{2} d^{5}-12 a b \,c^{5}-20 a b \,c^{3} d^{2}+2 a b c \,d^{4}+13 b^{2} c^{4} d +2 b^{2} c^{2} d^{3}\right )}{6 c^{6}-18 c^{4} d^{2}+18 c^{2} d^{4}-6 d^{6}}}{\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 {\left (2 a^{2} c^{3}+3 a^{2} c \,d^{2}-8 a b \,c^{2} d -2 a b \,d^{3}+b^{2} c^{3}+4 b^{2} c \,d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) \sqrt {c^{2}-d^{2}}}}{f}\) | \(933\) |
default | \(\frac {\frac {\frac {\left (9 a^{2} c^{4} d^{2}-6 a^{2} c^{2} d^{4}+2 a^{2} d^{6}-8 a b \,c^{5} d -2 a b \,c^{3} d^{3}+b^{2} c^{6}+4 b^{2} c^{4} d^{2}\right ) \left (\tan ^{5}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {\left (6 a^{2} c^{6} d +27 a^{2} c^{4} d^{3}-12 a^{2} c^{2} d^{5}+4 a^{2} d^{7}-4 a b \,c^{7}-28 a b \,c^{5} d^{2}-22 a b \,c^{3} d^{4}+4 a b c \,d^{6}+5 b^{2} c^{6} d +20 b^{2} c^{4} d^{3}\right ) \left (\tan ^{4}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{\left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) c^{2}}+\frac {2 d \left (54 a^{2} c^{6} d +21 a^{2} c^{4} d^{3}-4 a^{2} c^{2} d^{5}+4 a^{2} d^{7}-36 a b \,c^{7}-84 a b \,c^{5} d^{2}-34 a b \,c^{3} d^{4}+4 a b c \,d^{6}+39 b^{2} c^{6} d +32 b^{2} c^{4} d^{3}+4 b^{2} c^{2} d^{5}\right ) \left (\tan ^{3}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{3 c^{3} \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {2 \left (6 a^{2} c^{6} d +20 a^{2} c^{4} d^{3}-3 a^{2} c^{2} d^{5}+2 a^{2} d^{7}-4 a b \,c^{7}-20 a b \,c^{5} d^{2}-28 a b \,c^{3} d^{4}+2 a b c \,d^{6}+4 b^{2} c^{6} d +17 b^{2} c^{4} d^{3}+4 b^{2} c^{2} d^{5}\right ) \left (\tan ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )}{c^{2} \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {\left (27 a^{2} c^{4} d^{2}-4 a^{2} c^{2} d^{4}+2 a^{2} d^{6}-16 a b \,c^{5} d -38 a b \,c^{3} d^{3}+4 a b c \,d^{5}-b^{2} c^{6}+22 b^{2} c^{4} d^{2}+4 b^{2} c^{2} d^{4}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{c \left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right )}+\frac {2 \left (18 a^{2} c^{4} d -5 a^{2} c^{2} d^{3}+2 a^{2} d^{5}-12 a b \,c^{5}-20 a b \,c^{3} d^{2}+2 a b c \,d^{4}+13 b^{2} c^{4} d +2 b^{2} c^{2} d^{3}\right )}{6 c^{6}-18 c^{4} d^{2}+18 c^{2} d^{4}-6 d^{6}}}{\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 {\left (2 a^{2} c^{3}+3 a^{2} c \,d^{2}-8 a b \,c^{2} d -2 a b \,d^{3}+b^{2} c^{3}+4 b^{2} c \,d^{2}\right ) \arctan \left (\frac {2 c \tan \left (\frac {f x}{2}+\frac {e}{2}\right )+2 d}{2 \sqrt {c^{2}-d^{2}}}\right )}{\left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) \sqrt {c^{2}-d^{2}}}}{f}\) | \(933\) |
risch | \(\text {Expression too large to display}\) | \(1942\) |
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. 832 vs.
\(2 (298) = 596\).
time = 0.44, size = 1753, normalized size = 5.75 \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. 1316 vs.
\(2 (298) = 596\).
time = 0.58, size = 1316, normalized size = 4.31 \begin {gather*} \frac {\frac {3 \, {\left (2 \, a^{2} c^{3} + b^{2} c^{3} - 8 \, a b c^{2} d + 3 \, a^{2} c d^{2} + 4 \, b^{2} c d^{2} - 2 \, a b d^{3}\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^{6} - 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} - d^{6}\right )} \sqrt {c^{2} - d^{2}}} + \frac {3 \, b^{2} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 24 \, a b c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 27 \, a^{2} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 12 \, b^{2} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 6 \, a b c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 18 \, a^{2} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 6 \, a^{2} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, a b c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 18 \, a^{2} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 15 \, b^{2} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 84 \, a b c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 81 \, a^{2} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 60 \, b^{2} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 66 \, a b c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 36 \, a^{2} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 12 \, a b c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} + 12 \, a^{2} c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{4} - 72 \, a b c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 108 \, a^{2} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 78 \, b^{2} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 168 \, a b c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 42 \, a^{2} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 64 \, b^{2} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 68 \, a b c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 8 \, a^{2} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8 \, b^{2} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8 \, a b c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 8 \, a^{2} d^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 24 \, a b c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 36 \, a^{2} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, b^{2} c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 120 \, a b c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 120 \, a^{2} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 102 \, b^{2} c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 168 \, a b c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 18 \, a^{2} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 24 \, b^{2} c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, a b c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 12 \, a^{2} c d^{7} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 3 \, b^{2} c^{8} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 48 \, a b c^{7} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 81 \, a^{2} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 66 \, b^{2} c^{6} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 114 \, a b c^{5} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, a^{2} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, b^{2} c^{4} d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, a b c^{3} d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, a^{2} c^{2} d^{6} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 12 \, a b c^{8} + 18 \, a^{2} c^{7} d + 13 \, b^{2} c^{7} d - 20 \, a b c^{6} d^{2} - 5 \, a^{2} c^{5} d^{3} + 2 \, b^{2} c^{5} d^{3} + 2 \, a b c^{4} d^{4} + 2 \, a^{2} c^{3} d^{5}}{{\left (c^{9} - 3 \, c^{7} d^{2} + 3 \, c^{5} d^{4} - c^{3} d^{6}\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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time = 11.16, size = 1220, normalized size = 4.00 \begin {gather*} \frac {\frac {18\,a^2\,c^4\,d-5\,a^2\,c^2\,d^3+2\,a^2\,d^5-12\,a\,b\,c^5-20\,a\,b\,c^3\,d^2+2\,a\,b\,c\,d^4+13\,b^2\,c^4\,d+2\,b^2\,c^2\,d^3}{3\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (6\,a^2\,c^6\,d+27\,a^2\,c^4\,d^3-12\,a^2\,c^2\,d^5+4\,a^2\,d^7-4\,a\,b\,c^7-28\,a\,b\,c^5\,d^2-22\,a\,b\,c^3\,d^4+4\,a\,b\,c\,d^6+5\,b^2\,c^6\,d+20\,b^2\,c^4\,d^3\right )}{c^2\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}+\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (27\,a^2\,c^4\,d^2-4\,a^2\,c^2\,d^4+2\,a^2\,d^6-16\,a\,b\,c^5\,d-38\,a\,b\,c^3\,d^3+4\,a\,b\,c\,d^5-b^2\,c^6+22\,b^2\,c^4\,d^2+4\,b^2\,c^2\,d^4\right )}{c\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}+\frac {2\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (6\,a^2\,c^6\,d+20\,a^2\,c^4\,d^3-3\,a^2\,c^2\,d^5+2\,a^2\,d^7-4\,a\,b\,c^7-20\,a\,b\,c^5\,d^2-28\,a\,b\,c^3\,d^4+2\,a\,b\,c\,d^6+4\,b^2\,c^6\,d+17\,b^2\,c^4\,d^3+4\,b^2\,c^2\,d^5\right )}{c^2\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (9\,a^2\,c^4\,d^2-6\,a^2\,c^2\,d^4+2\,a^2\,d^6-8\,a\,b\,c^5\,d-2\,a\,b\,c^3\,d^3+b^2\,c^6+4\,b^2\,c^4\,d^2\right )}{c\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}+\frac {2\,d\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (3\,c^2+2\,d^2\right )\,\left (18\,a^2\,c^4\,d-5\,a^2\,c^2\,d^3+2\,a^2\,d^5-12\,a\,b\,c^5-20\,a\,b\,c^3\,d^2+2\,a\,b\,c\,d^4+13\,b^2\,c^4\,d+2\,b^2\,c^2\,d^3\right )}{3\,c^3\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\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 )}+\frac {\mathrm {atan}\left (\frac {\left (\frac {c\,\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,a^2\,c^3+3\,a^2\,c\,d^2-8\,a\,b\,c^2\,d-2\,a\,b\,d^3+b^2\,c^3+4\,b^2\,c\,d^2\right )}{{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{7/2}}+\frac {\left (2\,c^6\,d-6\,c^4\,d^3+6\,c^2\,d^5-2\,d^7\right )\,\left (2\,a^2\,c^3+3\,a^2\,c\,d^2-8\,a\,b\,c^2\,d-2\,a\,b\,d^3+b^2\,c^3+4\,b^2\,c\,d^2\right )}{2\,{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{7/2}\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}\right )\,\left (c^6-3\,c^4\,d^2+3\,c^2\,d^4-d^6\right )}{2\,a^2\,c^3+3\,a^2\,c\,d^2-8\,a\,b\,c^2\,d-2\,a\,b\,d^3+b^2\,c^3+4\,b^2\,c\,d^2}\right )\,\left (2\,a^2\,c^3+3\,a^2\,c\,d^2-8\,a\,b\,c^2\,d-2\,a\,b\,d^3+b^2\,c^3+4\,b^2\,c\,d^2\right )}{f\,{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{7/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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