Optimal. Leaf size=280 \[ \frac {(A b-3 a B) x}{b^4}-\frac {a \left (2 a^4 A b-5 a^2 A b^3+6 A b^5-6 a^5 B+15 a^3 b^2 B-12 a b^4 B\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^4 (a+b)^{5/2} d}-\frac {\left (a A b-3 a^2 B+2 b^2 B\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {a (A b-a B) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a^2 \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))} \]
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Rubi [A]
time = 0.81, antiderivative size = 280, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {3068, 3110,
3102, 2814, 2738, 211} \begin {gather*} \frac {a (A b-a B) \sin (c+d x) \cos ^2(c+d x)}{2 b d \left (a^2-b^2\right ) (a+b \cos (c+d x))^2}-\frac {\left (-3 a^2 B+a A b+2 b^2 B\right ) \sin (c+d x)}{2 b^3 d \left (a^2-b^2\right )}-\frac {a^2 \left (-3 a^3 B+a^2 A b+6 a b^2 B-4 A b^3\right ) \sin (c+d x)}{2 b^3 d \left (a^2-b^2\right )^2 (a+b \cos (c+d x))}-\frac {a \left (-6 a^5 B+2 a^4 A b+15 a^3 b^2 B-5 a^2 A b^3-12 a b^4 B+6 A b^5\right ) \text {ArcTan}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{b^4 d (a-b)^{5/2} (a+b)^{5/2}}+\frac {x (A b-3 a B)}{b^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 211
Rule 2738
Rule 2814
Rule 3068
Rule 3102
Rule 3110
Rubi steps
\begin {align*} \int \frac {\cos ^3(c+d x) (A+B \cos (c+d x))}{(a+b \cos (c+d x))^3} \, dx &=\frac {a (A b-a B) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {\int \frac {\cos (c+d x) \left (-2 a (A b-a B)+2 b (A b-a B) \cos (c+d x)+\left (a A b-3 a^2 B+2 b^2 B\right ) \cos ^2(c+d x)\right )}{(a+b \cos (c+d x))^2} \, dx}{2 b \left (a^2-b^2\right )}\\ &=\frac {a (A b-a B) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a^2 \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac {\int \frac {-a b \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right )-\left (a^2-b^2\right ) \left (a^2 A b-2 A b^3-3 a^3 B+4 a b^2 B\right ) \cos (c+d x)+b \left (a^2-b^2\right ) \left (a A b-3 a^2 B+2 b^2 B\right ) \cos ^2(c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^3 \left (a^2-b^2\right )^2}\\ &=-\frac {\left (a A b-3 a^2 B+2 b^2 B\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {a (A b-a B) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a^2 \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac {\int \frac {-a b^2 \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right )-2 b \left (a^2-b^2\right )^2 (A b-3 a B) \cos (c+d x)}{a+b \cos (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {(A b-3 a B) x}{b^4}-\frac {\left (a A b-3 a^2 B+2 b^2 B\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {a (A b-a B) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a^2 \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac {\left (a \left (2 a^4 A b-5 a^2 A b^3+6 A b^5-6 a^5 B+15 a^3 b^2 B-12 a b^4 B\right )\right ) \int \frac {1}{a+b \cos (c+d x)} \, dx}{2 b^4 \left (a^2-b^2\right )^2}\\ &=\frac {(A b-3 a B) x}{b^4}-\frac {\left (a A b-3 a^2 B+2 b^2 B\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {a (A b-a B) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a^2 \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}-\frac {\left (a \left (2 a^4 A b-5 a^2 A b^3+6 A b^5-6 a^5 B+15 a^3 b^2 B-12 a b^4 B\right )\right ) \text {Subst}\left (\int \frac {1}{a+b+(a-b) x^2} \, dx,x,\tan \left (\frac {1}{2} (c+d x)\right )\right )}{b^4 \left (a^2-b^2\right )^2 d}\\ &=\frac {(A b-3 a B) x}{b^4}-\frac {a \left (2 a^4 A b-5 a^2 A b^3+6 A b^5-6 a^5 B+15 a^3 b^2 B-12 a b^4 B\right ) \tan ^{-1}\left (\frac {\sqrt {a-b} \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a+b}}\right )}{(a-b)^{5/2} b^4 (a+b)^{5/2} d}-\frac {\left (a A b-3 a^2 B+2 b^2 B\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right ) d}+\frac {a (A b-a B) \cos ^2(c+d x) \sin (c+d x)}{2 b \left (a^2-b^2\right ) d (a+b \cos (c+d x))^2}-\frac {a^2 \left (a^2 A b-4 A b^3-3 a^3 B+6 a b^2 B\right ) \sin (c+d x)}{2 b^3 \left (a^2-b^2\right )^2 d (a+b \cos (c+d x))}\\ \end {align*}
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Mathematica [A]
time = 2.22, size = 232, normalized size = 0.83 \begin {gather*} \frac {2 (A b-3 a B) (c+d x)-\frac {2 a \left (-2 a^4 A b+5 a^2 A b^3-6 A b^5+6 a^5 B-15 a^3 b^2 B+12 a b^4 B\right ) \tanh ^{-1}\left (\frac {(a-b) \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2+b^2}}\right )}{\left (-a^2+b^2\right )^{5/2}}+2 b B \sin (c+d x)+\frac {a^3 b (A b-a B) \sin (c+d x)}{(a-b) (a+b) (a+b \cos (c+d x))^2}+\frac {a^2 b \left (-3 a^2 A b+6 A b^3+5 a^3 B-8 a b^2 B\right ) \sin (c+d x)}{(a-b)^2 (a+b)^2 (a+b \cos (c+d x))}}{2 b^4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.66, size = 341, normalized size = 1.22
method | result | size |
derivativedivides | \(\frac {-\frac {2 a \left (\frac {\frac {\left (2 A \,a^{2} b -A a \,b^{2}-6 A \,b^{3}-4 a^{3} B +a^{2} b B +8 B a \,b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (2 A \,a^{2} b +A a \,b^{2}-6 A \,b^{3}-4 a^{3} B -a^{2} b B +8 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (2 A \,a^{4} b -5 A \,a^{2} b^{3}+6 A \,b^{5}-6 B \,a^{5}+15 B \,a^{3} b^{2}-12 B a \,b^{4}\right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{4}}+\frac {\frac {2 B b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+2 \left (A b -3 a B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}}}{d}\) | \(341\) |
default | \(\frac {-\frac {2 a \left (\frac {\frac {\left (2 A \,a^{2} b -A a \,b^{2}-6 A \,b^{3}-4 a^{3} B +a^{2} b B +8 B a \,b^{2}\right ) a b \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 \left (a -b \right ) \left (a^{2}+2 a b +b^{2}\right )}+\frac {b a \left (2 A \,a^{2} b +A a \,b^{2}-6 A \,b^{3}-4 a^{3} B -a^{2} b B +8 B a \,b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 \left (a +b \right ) \left (a -b \right )^{2}}}{\left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )^{2}}+\frac {\left (2 A \,a^{4} b -5 A \,a^{2} b^{3}+6 A \,b^{5}-6 B \,a^{5}+15 B \,a^{3} b^{2}-12 B a \,b^{4}\right ) \arctan \left (\frac {\tan \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (a -b \right )}{\sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{2 \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \sqrt {\left (a -b \right ) \left (a +b \right )}}\right )}{b^{4}}+\frac {\frac {2 B b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}+2 \left (A b -3 a B \right ) \arctan \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{b^{4}}}{d}\) | \(341\) |
risch | \(\text {Expression too large to display}\) | \(1411\) |
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. 746 vs.
\(2 (268) = 536\).
time = 0.46, size = 1561, normalized size = 5.58 \begin {gather*} \left [-\frac {4 \, {\left (3 \, B a^{7} b^{2} - A a^{6} b^{3} - 9 \, B a^{5} b^{4} + 3 \, A a^{4} b^{5} + 9 \, B a^{3} b^{6} - 3 \, A a^{2} b^{7} - 3 \, B a b^{8} + A b^{9}\right )} d x \cos \left (d x + c\right )^{2} + 8 \, {\left (3 \, B a^{8} b - A a^{7} b^{2} - 9 \, B a^{6} b^{3} + 3 \, A a^{5} b^{4} + 9 \, B a^{4} b^{5} - 3 \, A a^{3} b^{6} - 3 \, B a^{2} b^{7} + A a b^{8}\right )} d x \cos \left (d x + c\right ) + 4 \, {\left (3 \, B a^{9} - A a^{8} b - 9 \, B a^{7} b^{2} + 3 \, A a^{6} b^{3} + 9 \, B a^{5} b^{4} - 3 \, A a^{4} b^{5} - 3 \, B a^{3} b^{6} + A a^{2} b^{7}\right )} d x - {\left (6 \, B a^{8} - 2 \, A a^{7} b - 15 \, B a^{6} b^{2} + 5 \, A a^{5} b^{3} + 12 \, B a^{4} b^{4} - 6 \, A a^{3} b^{5} + {\left (6 \, B a^{6} b^{2} - 2 \, A a^{5} b^{3} - 15 \, B a^{4} b^{4} + 5 \, A a^{3} b^{5} + 12 \, B a^{2} b^{6} - 6 \, A a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, B a^{7} b - 2 \, A a^{6} b^{2} - 15 \, B a^{5} b^{3} + 5 \, A a^{4} b^{4} + 12 \, B a^{3} b^{5} - 6 \, A a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {-a^{2} + b^{2}} \log \left (\frac {2 \, a b \cos \left (d x + c\right ) + {\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt {-a^{2} + b^{2}} {\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right ) - 2 \, {\left (6 \, B a^{8} b - 2 \, A a^{7} b^{2} - 17 \, B a^{6} b^{3} + 7 \, A a^{5} b^{4} + 13 \, B a^{4} b^{5} - 5 \, A a^{3} b^{6} - 2 \, B a^{2} b^{7} + 2 \, {\left (B a^{6} b^{3} - 3 \, B a^{4} b^{5} + 3 \, B a^{2} b^{7} - B b^{9}\right )} \cos \left (d x + c\right )^{2} + {\left (9 \, B a^{7} b^{2} - 3 \, A a^{6} b^{3} - 25 \, B a^{5} b^{4} + 9 \, A a^{4} b^{5} + 20 \, B a^{3} b^{6} - 6 \, A a^{2} b^{7} - 4 \, B a b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{4 \, {\left ({\left (a^{6} b^{6} - 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} - b^{12}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{5} - 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} - a b^{11}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b^{4} - 3 \, a^{6} b^{6} + 3 \, a^{4} b^{8} - a^{2} b^{10}\right )} d\right )}}, -\frac {2 \, {\left (3 \, B a^{7} b^{2} - A a^{6} b^{3} - 9 \, B a^{5} b^{4} + 3 \, A a^{4} b^{5} + 9 \, B a^{3} b^{6} - 3 \, A a^{2} b^{7} - 3 \, B a b^{8} + A b^{9}\right )} d x \cos \left (d x + c\right )^{2} + 4 \, {\left (3 \, B a^{8} b - A a^{7} b^{2} - 9 \, B a^{6} b^{3} + 3 \, A a^{5} b^{4} + 9 \, B a^{4} b^{5} - 3 \, A a^{3} b^{6} - 3 \, B a^{2} b^{7} + A a b^{8}\right )} d x \cos \left (d x + c\right ) + 2 \, {\left (3 \, B a^{9} - A a^{8} b - 9 \, B a^{7} b^{2} + 3 \, A a^{6} b^{3} + 9 \, B a^{5} b^{4} - 3 \, A a^{4} b^{5} - 3 \, B a^{3} b^{6} + A a^{2} b^{7}\right )} d x - {\left (6 \, B a^{8} - 2 \, A a^{7} b - 15 \, B a^{6} b^{2} + 5 \, A a^{5} b^{3} + 12 \, B a^{4} b^{4} - 6 \, A a^{3} b^{5} + {\left (6 \, B a^{6} b^{2} - 2 \, A a^{5} b^{3} - 15 \, B a^{4} b^{4} + 5 \, A a^{3} b^{5} + 12 \, B a^{2} b^{6} - 6 \, A a b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (6 \, B a^{7} b - 2 \, A a^{6} b^{2} - 15 \, B a^{5} b^{3} + 5 \, A a^{4} b^{4} + 12 \, B a^{3} b^{5} - 6 \, A a^{2} b^{6}\right )} \cos \left (d x + c\right )\right )} \sqrt {a^{2} - b^{2}} \arctan \left (-\frac {a \cos \left (d x + c\right ) + b}{\sqrt {a^{2} - b^{2}} \sin \left (d x + c\right )}\right ) - {\left (6 \, B a^{8} b - 2 \, A a^{7} b^{2} - 17 \, B a^{6} b^{3} + 7 \, A a^{5} b^{4} + 13 \, B a^{4} b^{5} - 5 \, A a^{3} b^{6} - 2 \, B a^{2} b^{7} + 2 \, {\left (B a^{6} b^{3} - 3 \, B a^{4} b^{5} + 3 \, B a^{2} b^{7} - B b^{9}\right )} \cos \left (d x + c\right )^{2} + {\left (9 \, B a^{7} b^{2} - 3 \, A a^{6} b^{3} - 25 \, B a^{5} b^{4} + 9 \, A a^{4} b^{5} + 20 \, B a^{3} b^{6} - 6 \, A a^{2} b^{7} - 4 \, B a b^{8}\right )} \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )}{2 \, {\left ({\left (a^{6} b^{6} - 3 \, a^{4} b^{8} + 3 \, a^{2} b^{10} - b^{12}\right )} d \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b^{5} - 3 \, a^{5} b^{7} + 3 \, a^{3} b^{9} - a b^{11}\right )} d \cos \left (d x + c\right ) + {\left (a^{8} b^{4} - 3 \, a^{6} b^{6} + 3 \, a^{4} b^{8} - a^{2} b^{10}\right )} d\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. 543 vs.
\(2 (268) = 536\).
time = 0.54, size = 543, normalized size = 1.94 \begin {gather*} -\frac {\frac {{\left (6 \, B a^{6} - 2 \, A a^{5} b - 15 \, B a^{4} b^{2} + 5 \, A a^{3} b^{3} + 12 \, B a^{2} b^{4} - 6 \, A a b^{5}\right )} {\left (\pi \left \lfloor \frac {d x + c}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (-2 \, a + 2 \, b\right ) + \arctan \left (-\frac {a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\sqrt {a^{2} - b^{2}}}\right )\right )}}{{\left (a^{4} b^{4} - 2 \, a^{2} b^{6} + b^{8}\right )} \sqrt {a^{2} - b^{2}}} - \frac {4 \, B a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 2 \, A a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 5 \, B a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 3 \, A a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 7 \, B a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 5 \, A a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 8 \, B a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 6 \, A a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4 \, B a^{6} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 2 \, A a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, B a^{5} b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 3 \, A a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 7 \, B a^{4} b^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 5 \, A a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 8 \, B a^{3} b^{3} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 6 \, A a^{2} b^{4} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (a^{4} b^{3} - 2 \, a^{2} b^{5} + b^{7}\right )} {\left (a \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - b \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a + b\right )}^{2}} + \frac {{\left (3 \, B a - A b\right )} {\left (d x + c\right )}}{b^{4}} - \frac {2 \, B \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 1\right )} b^{3}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 7.66, size = 2500, normalized size = 8.93 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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