Optimal. Leaf size=260 \[ \frac {\left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} e}+\frac {C}{3 b e (a+b \cos (d+e x))^3}-\frac {(A b-a B) \sin (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cos (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cos (d+e x))^2}-\frac {\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cos (d+e x))} \]
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Rubi [A]
time = 0.35, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {4462, 2833, 12,
2738, 211, 2747, 32} \begin {gather*} -\frac {\left (-2 a^2 B+5 a A b-3 b^2 B\right ) \sin (d+e x)}{6 e \left (a^2-b^2\right )^2 (a+b \cos (d+e x))^2}-\frac {(A b-a B) \sin (d+e x)}{3 e \left (a^2-b^2\right ) (a+b \cos (d+e x))^3}+\frac {\left (2 a^3 A-4 a^2 b B+3 a A b^2-b^3 B\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{e (a-b)^{7/2} (a+b)^{7/2}}-\frac {\left (-2 a^3 B+11 a^2 A b-13 a b^2 B+4 A b^3\right ) \sin (d+e x)}{6 e \left (a^2-b^2\right )^3 (a+b \cos (d+e x))}+\frac {C}{3 b e (a+b \cos (d+e x))^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 32
Rule 211
Rule 2738
Rule 2747
Rule 2833
Rule 4462
Rubi steps
\begin {align*} \int \frac {A+B \cos (d+e x)+C \sin (d+e x)}{(a+b \cos (d+e x))^4} \, dx &=C \int \frac {\sin (d+e x)}{(a+b \cos (d+e x))^4} \, dx+\int \frac {A+B \cos (d+e x)}{(a+b \cos (d+e x))^4} \, dx\\ &=-\frac {(A b-a B) \sin (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cos (d+e x))^3}-\frac {\int \frac {-3 (a A-b B)+2 (A b-a B) \cos (d+e x)}{(a+b \cos (d+e x))^3} \, dx}{3 \left (a^2-b^2\right )}-\frac {C \text {Subst}\left (\int \frac {1}{(a+x)^4} \, dx,x,b \cos (d+e x)\right )}{b e}\\ &=\frac {C}{3 b e (a+b \cos (d+e x))^3}-\frac {(A b-a B) \sin (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cos (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cos (d+e x))^2}+\frac {\int \frac {2 \left (3 a^2 A+2 A b^2-5 a b B\right )-\left (5 a A b-2 a^2 B-3 b^2 B\right ) \cos (d+e x)}{(a+b \cos (d+e x))^2} \, dx}{6 \left (a^2-b^2\right )^2}\\ &=\frac {C}{3 b e (a+b \cos (d+e x))^3}-\frac {(A b-a B) \sin (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cos (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cos (d+e x))^2}-\frac {\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cos (d+e x))}-\frac {\int -\frac {3 \left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right )}{a+b \cos (d+e x)} \, dx}{6 \left (a^2-b^2\right )^3}\\ &=\frac {C}{3 b e (a+b \cos (d+e x))^3}-\frac {(A b-a B) \sin (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cos (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cos (d+e x))^2}-\frac {\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cos (d+e x))}+\frac {\left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \int \frac {1}{a+b \cos (d+e x)} \, dx}{2 \left (a^2-b^2\right )^3}\\ &=\frac {C}{3 b e (a+b \cos (d+e x))^3}-\frac {(A b-a B) \sin (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cos (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cos (d+e x))^2}-\frac {\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cos (d+e x))}+\frac {\left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (d+e x)\right )\right )}{\left (a^2-b^2\right )^3 e}\\ &=\frac {\left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2} e}+\frac {C}{3 b e (a+b \cos (d+e x))^3}-\frac {(A b-a B) \sin (d+e x)}{3 \left (a^2-b^2\right ) e (a+b \cos (d+e x))^3}-\frac {\left (5 a A b-2 a^2 B-3 b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^2 e (a+b \cos (d+e x))^2}-\frac {\left (11 a^2 A b+4 A b^3-2 a^3 B-13 a b^2 B\right ) \sin (d+e x)}{6 \left (a^2-b^2\right )^3 e (a+b \cos (d+e x))}\\ \end {align*}
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Mathematica [A]
time = 1.24, size = 302, normalized size = 1.16 \begin {gather*} \frac {\frac {24 \left (2 a^3 A+3 a A b^2-4 a^2 b B-b^3 B\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (d+e x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{7/2}}+\frac {-8 a^6 C+24 a^4 b^2 C-24 a^2 b^4 C+8 b^6 C-3 b \left (-24 a^4 A b+3 a^2 A b^3-4 A b^5+8 a^5 B+14 a^3 b^2 B+3 a b^4 B\right ) \sin (d+e x)+6 b^2 \left (9 a^3 A b+a A b^3-2 a^4 B-9 a^2 b^2 B+b^4 B\right ) \sin (2 (d+e x))+11 a^2 A b^4 \sin (3 (d+e x))+4 A b^6 \sin (3 (d+e x))-2 a^3 b^3 B \sin (3 (d+e x))-13 a b^5 B \sin (3 (d+e x))}{b \left (-a^2+b^2\right )^3 (a+b \cos (d+e x))^3}}{24 e} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.53, size = 456, normalized size = 1.75
method | result | size |
derivativedivides | \(\frac {\frac {-\frac {\left (6 A \,a^{2} b +3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B -2 a^{2} b B -6 B a \,b^{2}-b^{3} B \right ) \left (\tan ^{5}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{\left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 C \left (\tan ^{4}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{a -b}-\frac {4 \left (9 A \,a^{2} b +A \,b^{3}-3 a^{3} B -7 B a \,b^{2}\right ) \left (\tan ^{3}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {4 a C \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{a^{2}-2 a b +b^{2}}-\frac {\left (6 A \,a^{2} b -3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B +2 a^{2} b B -6 B a \,b^{2}+b^{3} B \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{\left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}-\frac {2 \left (3 a^{2}+b^{2}\right ) C}{3 \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )+a +b \right )^{3}}+\frac {\left (2 A \,a^{3}+3 A a \,b^{2}-4 a^{2} b B -b^{3} B \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{e}\) | \(456\) |
default | \(\frac {\frac {-\frac {\left (6 A \,a^{2} b +3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B -2 a^{2} b B -6 B a \,b^{2}-b^{3} B \right ) \left (\tan ^{5}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{\left (a -b \right ) \left (a^{3}+3 a^{2} b +3 b^{2} a +b^{3}\right )}-\frac {2 C \left (\tan ^{4}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{a -b}-\frac {4 \left (9 A \,a^{2} b +A \,b^{3}-3 a^{3} B -7 B a \,b^{2}\right ) \left (\tan ^{3}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{3 \left (a^{2}-2 a b +b^{2}\right ) \left (a^{2}+2 a b +b^{2}\right )}-\frac {4 a C \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )}{a^{2}-2 a b +b^{2}}-\frac {\left (6 A \,a^{2} b -3 A a \,b^{2}+2 A \,b^{3}-2 a^{3} B +2 a^{2} b B -6 B a \,b^{2}+b^{3} B \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{\left (a +b \right ) \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}-\frac {2 \left (3 a^{2}+b^{2}\right ) C}{3 \left (a^{3}-3 a^{2} b +3 b^{2} a -b^{3}\right )}}{\left (a \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {e x}{2}+\frac {d}{2}\right )\right )+a +b \right )^{3}}+\frac {\left (2 A \,a^{3}+3 A a \,b^{2}-4 a^{2} b B -b^{3} B \right ) \arctan \left (\frac {\left (a -b \right ) \tan \left (\frac {e x}{2}+\frac {d}{2}\right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{\left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}}{e}\) | \(456\) |
risch | \(\text {Expression too large to display}\) | \(1282\) |
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. 647 vs.
\(2 (246) = 492\).
time = 0.45, size = 1368, normalized size = 5.26 \begin {gather*} \left [\frac {4 \, C a^{8} - 16 \, C a^{6} b^{2} + 24 \, C a^{4} b^{4} - 16 \, C a^{2} b^{6} + 4 \, C b^{8} - 3 \, {\left (2 \, A a^{6} b - 4 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} - B a^{3} b^{4} + {\left (2 \, A a^{3} b^{4} - 4 \, B a^{2} b^{5} + 3 \, A a b^{6} - B b^{7}\right )} \cos \left (x e + d\right )^{3} + 3 \, {\left (2 \, A a^{4} b^{3} - 4 \, B a^{3} b^{4} + 3 \, A a^{2} b^{5} - B a b^{6}\right )} \cos \left (x e + d\right )^{2} + 3 \, {\left (2 \, A a^{5} b^{2} - 4 \, B a^{4} b^{3} + 3 \, A a^{3} b^{4} - B a^{2} b^{5}\right )} \cos \left (x e + d\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (x e + d\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (x e + d\right )^{2} + 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (x e + d\right ) + b\right )} \sin \left (x e + d\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (x e + d\right )^{2} + 2 \, a b \cos \left (x e + d\right ) + a^{2}}\right ) + 2 \, {\left (6 \, B a^{7} b - 18 \, A a^{6} b^{2} + 4 \, B a^{5} b^{3} + 23 \, A a^{4} b^{4} - 11 \, B a^{3} b^{5} - 7 \, A a^{2} b^{6} + B a b^{7} + 2 \, A b^{8} + {\left (2 \, B a^{5} b^{3} - 11 \, A a^{4} b^{4} + 11 \, B a^{3} b^{5} + 7 \, A a^{2} b^{6} - 13 \, B a b^{7} + 4 \, A b^{8}\right )} \cos \left (x e + d\right )^{2} + 3 \, {\left (2 \, B a^{6} b^{2} - 9 \, A a^{5} b^{3} + 7 \, B a^{4} b^{4} + 8 \, A a^{3} b^{5} - 10 \, B a^{2} b^{6} + A a b^{7} + B b^{8}\right )} \cos \left (x e + d\right )\right )} \sin \left (x e + d\right )}{12 \, {\left ({\left (a^{8} b^{4} - 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} - 4 \, a^{2} b^{10} + b^{12}\right )} \cos \left (x e + d\right )^{3} e + 3 \, {\left (a^{9} b^{3} - 4 \, a^{7} b^{5} + 6 \, a^{5} b^{7} - 4 \, a^{3} b^{9} + a b^{11}\right )} \cos \left (x e + d\right )^{2} e + 3 \, {\left (a^{10} b^{2} - 4 \, a^{8} b^{4} + 6 \, a^{6} b^{6} - 4 \, a^{4} b^{8} + a^{2} b^{10}\right )} \cos \left (x e + d\right ) e + {\left (a^{11} b - 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} - 4 \, a^{5} b^{7} + a^{3} b^{9}\right )} e\right )}}, \frac {2 \, C a^{8} - 8 \, C a^{6} b^{2} + 12 \, C a^{4} b^{4} - 8 \, C a^{2} b^{6} + 2 \, C b^{8} + 3 \, {\left (2 \, A a^{6} b - 4 \, B a^{5} b^{2} + 3 \, A a^{4} b^{3} - B a^{3} b^{4} + {\left (2 \, A a^{3} b^{4} - 4 \, B a^{2} b^{5} + 3 \, A a b^{6} - B b^{7}\right )} \cos \left (x e + d\right )^{3} + 3 \, {\left (2 \, A a^{4} b^{3} - 4 \, B a^{3} b^{4} + 3 \, A a^{2} b^{5} - B a b^{6}\right )} \cos \left (x e + d\right )^{2} + 3 \, {\left (2 \, A a^{5} b^{2} - 4 \, B a^{4} b^{3} + 3 \, A a^{3} b^{4} - B a^{2} b^{5}\right )} \cos \left (x e + d\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (x e + d\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (x e + d\right )}\right ) + {\left (6 \, B a^{7} b - 18 \, A a^{6} b^{2} + 4 \, B a^{5} b^{3} + 23 \, A a^{4} b^{4} - 11 \, B a^{3} b^{5} - 7 \, A a^{2} b^{6} + B a b^{7} + 2 \, A b^{8} + {\left (2 \, B a^{5} b^{3} - 11 \, A a^{4} b^{4} + 11 \, B a^{3} b^{5} + 7 \, A a^{2} b^{6} - 13 \, B a b^{7} + 4 \, A b^{8}\right )} \cos \left (x e + d\right )^{2} + 3 \, {\left (2 \, B a^{6} b^{2} - 9 \, A a^{5} b^{3} + 7 \, B a^{4} b^{4} + 8 \, A a^{3} b^{5} - 10 \, B a^{2} b^{6} + A a b^{7} + B b^{8}\right )} \cos \left (x e + d\right )\right )} \sin \left (x e + d\right )}{6 \, {\left ({\left (a^{8} b^{4} - 4 \, a^{6} b^{6} + 6 \, a^{4} b^{8} - 4 \, a^{2} b^{10} + b^{12}\right )} \cos \left (x e + d\right )^{3} e + 3 \, {\left (a^{9} b^{3} - 4 \, a^{7} b^{5} + 6 \, a^{5} b^{7} - 4 \, a^{3} b^{9} + a b^{11}\right )} \cos \left (x e + d\right )^{2} e + 3 \, {\left (a^{10} b^{2} - 4 \, a^{8} b^{4} + 6 \, a^{6} b^{6} - 4 \, a^{4} b^{8} + a^{2} b^{10}\right )} \cos \left (x e + d\right ) e + {\left (a^{11} b - 4 \, a^{9} b^{3} + 6 \, a^{7} b^{5} - 4 \, a^{5} b^{7} + a^{3} b^{9}\right )} e\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. 960 vs.
\(2 (246) = 492\).
time = 0.55, size = 960, normalized size = 3.69 \begin {gather*} -\frac {1}{3} \, {\left (\frac {3 \, {\left (2 \, A a^{3} - 4 \, B a^{2} b + 3 \, A a b^{2} - B b^{3}\right )} {\left (\pi \left \lfloor \frac {x e + d}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt {a^{2} - b^{2}}} - \frac {6 \, B a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 18 \, A a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 6 \, B a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 27 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 12 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 6 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 27 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 3 \, A a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 12 \, B a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 6 \, A b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} + 3 \, B b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{5} - 6 \, C a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} - 6 \, C a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} + 12 \, C a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} + 12 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} - 6 \, C a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} - 6 \, C b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{4} + 12 \, B a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 36 \, A a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 16 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 32 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 28 \, B a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} + 4 \, A b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{3} - 12 \, C a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - 24 \, C a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 24 \, C a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 12 \, C a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + 6 \, B a^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 18 \, A a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 6 \, B a^{4} b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 27 \, A a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 12 \, B a^{3} b^{2} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 6 \, A a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 27 \, B a^{2} b^{3} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 3 \, A a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) + 12 \, B a b^{4} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 6 \, A b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 3 \, B b^{5} \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right ) - 6 \, C a^{5} - 18 \, C a^{4} b - 20 \, C a^{3} b^{2} - 12 \, C a^{2} b^{3} - 6 \, C a b^{4} - 2 \, C b^{5}}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} {\left (a \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} - b \tan \left (\frac {1}{2} \, x e + \frac {1}{2} \, d\right )^{2} + a + b\right )}^{3}}\right )} e^{\left (-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.22, size = 502, normalized size = 1.93 \begin {gather*} \frac {\mathrm {atan}\left (\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (2\,a-2\,b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}{2\,\sqrt {a+b}\,{\left (a-b\right )}^{7/2}}\right )\,\left (2\,A\,a^3-4\,B\,a^2\,b+3\,A\,a\,b^2-B\,b^3\right )}{e\,{\left (a+b\right )}^{7/2}\,{\left (a-b\right )}^{7/2}}-\frac {\frac {2\,C\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4}{a-b}+\frac {2\,C\,\left (3\,a^2+b^2\right )}{3\,{\left (a-b\right )}^3}+\frac {4\,C\,a\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2}{{\left (a-b\right )}^2}+\frac {4\,{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^3\,\left (-3\,B\,a^3+9\,A\,a^2\,b-7\,B\,a\,b^2+A\,b^3\right )}{3\,{\left (a+b\right )}^2\,\left (a^2-2\,a\,b+b^2\right )}-\frac {{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^5\,\left (2\,B\,a^3-2\,A\,b^3+B\,b^3-3\,A\,a\,b^2-6\,A\,a^2\,b+6\,B\,a\,b^2+2\,B\,a^2\,b\right )}{{\left (a+b\right )}^3\,\left (a-b\right )}+\frac {\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )\,\left (2\,A\,b^3-2\,B\,a^3+B\,b^3-3\,A\,a\,b^2+6\,A\,a^2\,b-6\,B\,a\,b^2+2\,B\,a^2\,b\right )}{\left (a+b\right )\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )}}{e\,\left (3\,a\,b^2-{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^4\,\left (-3\,a^3+3\,a^2\,b+3\,a\,b^2-3\,b^3\right )-{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^2\,\left (-3\,a^3-3\,a^2\,b+3\,a\,b^2+3\,b^3\right )+3\,a^2\,b+a^3+b^3+{\mathrm {tan}\left (\frac {d}{2}+\frac {e\,x}{2}\right )}^6\,\left (a^3-3\,a^2\,b+3\,a\,b^2-b^3\right )\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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