Optimal. Leaf size=352 \[ \frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {\left (a^2 A+3 a b B-6 A b^2\right ) \log (\sin (c+d x))}{a^5 d}+\frac {b \left (-2 a^3 B+5 a^2 A b-3 a b^2 B+6 A b^3\right )}{2 a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {x \left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right )}{\left (a^2+b^2\right )^3}+\frac {b \left (a^5 (-B)+3 a^4 A b-6 a^3 b^2 B+11 a^2 A b^3-3 a b^4 B+6 A b^5\right )}{a^4 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {b^3 \left (-10 a^5 B+15 a^4 A b-9 a^3 b^2 B+17 a^2 A b^3-3 a b^4 B+6 A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left (a^2+b^2\right )^3}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2} \]
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Rubi [A] time = 1.25, antiderivative size = 352, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.161, Rules used = {3609, 3649, 3651, 3530, 3475} \[ \frac {b \left (11 a^2 A b^3+3 a^4 A b-6 a^3 b^2 B+a^5 (-B)-3 a b^4 B+6 A b^5\right )}{a^4 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {b \left (5 a^2 A b-2 a^3 B-3 a b^2 B+6 A b^3\right )}{2 a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {\left (a^2 A+3 a b B-6 A b^2\right ) \log (\sin (c+d x))}{a^5 d}-\frac {b^3 \left (17 a^2 A b^3+15 a^4 A b-9 a^3 b^2 B-10 a^5 B-3 a b^4 B+6 A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 d \left (a^2+b^2\right )^3}+\frac {x \left (3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3\right )}{\left (a^2+b^2\right )^3}+\frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2} \]
Antiderivative was successfully verified.
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Rule 3475
Rule 3530
Rule 3609
Rule 3649
Rule 3651
Rubi steps
\begin {align*} \int \frac {\cot ^3(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx &=-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}-\frac {\int \frac {\cot ^2(c+d x) \left (2 (2 A b-a B)+2 a A \tan (c+d x)+4 A b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{2 a}\\ &=\frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac {\int \frac {\cot (c+d x) \left (-2 \left (a^2 A-6 A b^2+3 a b B\right )-2 a^2 B \tan (c+d x)+6 b (2 A b-a B) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{2 a^2}\\ &=\frac {b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right )}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac {\int \frac {\cot (c+d x) \left (-4 \left (a^2+b^2\right ) \left (a^2 A-6 A b^2+3 a b B\right )+4 a^3 (A b-a B) \tan (c+d x)+4 b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{4 a^3 \left (a^2+b^2\right )}\\ &=\frac {b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right )}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^4 A b+11 a^2 A b^3+6 A b^5-a^5 B-6 a^3 b^2 B-3 a b^4 B\right )}{a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\cot (c+d x) \left (-4 \left (a^2+b^2\right )^2 \left (a^2 A-6 A b^2+3 a b B\right )+4 a^4 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+4 b \left (3 a^4 A b+11 a^2 A b^3+6 A b^5-a^5 B-6 a^3 b^2 B-3 a b^4 B\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{4 a^4 \left (a^2+b^2\right )^2}\\ &=\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}+\frac {b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right )}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^4 A b+11 a^2 A b^3+6 A b^5-a^5 B-6 a^3 b^2 B-3 a b^4 B\right )}{a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left (a^2 A-6 A b^2+3 a b B\right ) \int \cot (c+d x) \, dx}{a^5}-\frac {\left (b^3 \left (15 a^4 A b+17 a^2 A b^3+6 A b^5-10 a^5 B-9 a^3 b^2 B-3 a b^4 B\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^5 \left (a^2+b^2\right )^3}\\ &=\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) x}{\left (a^2+b^2\right )^3}-\frac {\left (a^2 A-6 A b^2+3 a b B\right ) \log (\sin (c+d x))}{a^5 d}-\frac {b^3 \left (15 a^4 A b+17 a^2 A b^3+6 A b^5-10 a^5 B-9 a^3 b^2 B-3 a b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^5 \left (a^2+b^2\right )^3 d}+\frac {b \left (5 a^2 A b+6 A b^3-2 a^3 B-3 a b^2 B\right )}{2 a^3 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {(2 A b-a B) \cot (c+d x)}{a^2 d (a+b \tan (c+d x))^2}-\frac {A \cot ^2(c+d x)}{2 a d (a+b \tan (c+d x))^2}+\frac {b \left (3 a^4 A b+11 a^2 A b^3+6 A b^5-a^5 B-6 a^3 b^2 B-3 a b^4 B\right )}{a^4 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 6.50, size = 320, normalized size = 0.91 \[ \frac {(3 A b-a B) \cot (c+d x)}{a^4 d}-\frac {A \cot ^2(c+d x)}{2 a^3 d}-\frac {\left (a^2 A+3 a b B-6 A b^2\right ) \log (\tan (c+d x))}{a^5 d}+\frac {b^3 (A b-a B)}{2 a^3 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {b^3 \left (-4 a^3 B+5 a^2 A b-2 a b^2 B+3 A b^3\right )}{a^4 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {b^3 \left (-10 a^5 B+15 a^4 A b-9 a^3 b^2 B+17 a^2 A b^3-3 a b^4 B+6 A b^5\right ) \log (a+b \tan (c+d x))}{a^5 d \left (a^2+b^2\right )^3}+\frac {(A+i B) \log (-\tan (c+d x)+i)}{2 d (a+i b)^3}+\frac {(A-i B) \log (\tan (c+d x)+i)}{2 d (a-i b)^3} \]
Warning: Unable to verify antiderivative.
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fricas [B] time = 1.15, size = 1065, normalized size = 3.03 \[ -\frac {A a^{10} + 3 \, A a^{8} b^{2} + 3 \, A a^{6} b^{4} + A a^{4} b^{6} + {\left (A a^{8} b^{2} + 3 \, A a^{6} b^{4} - 9 \, B a^{5} b^{5} + 14 \, A a^{4} b^{6} - 3 \, B a^{3} b^{7} + 6 \, A a^{2} b^{8} + 2 \, {\left (B a^{8} b^{2} - 3 \, A a^{7} b^{3} - 3 \, B a^{6} b^{4} + A a^{5} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (A a^{9} b + B a^{8} b^{2} - 2 \, B a^{6} b^{4} + 6 \, B a^{4} b^{6} - 11 \, A a^{3} b^{7} + 3 \, B a^{2} b^{8} - 6 \, A a b^{9} + 2 \, {\left (B a^{9} b - 3 \, A a^{8} b^{2} - 3 \, B a^{7} b^{3} + A a^{6} b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{3} + {\left (A a^{10} + 4 \, B a^{9} b - 8 \, A a^{8} b^{2} + 12 \, B a^{7} b^{3} - 30 \, A a^{6} b^{4} + 23 \, B a^{5} b^{5} - 45 \, A a^{4} b^{6} + 9 \, B a^{3} b^{7} - 18 \, A a^{2} b^{8} + 2 \, {\left (B a^{10} - 3 \, A a^{9} b - 3 \, B a^{8} b^{2} + A a^{7} b^{3}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left ({\left (A a^{8} b^{2} + 3 \, B a^{7} b^{3} - 3 \, A a^{6} b^{4} + 9 \, B a^{5} b^{5} - 15 \, A a^{4} b^{6} + 9 \, B a^{3} b^{7} - 17 \, A a^{2} b^{8} + 3 \, B a b^{9} - 6 \, A b^{10}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (A a^{9} b + 3 \, B a^{8} b^{2} - 3 \, A a^{7} b^{3} + 9 \, B a^{6} b^{4} - 15 \, A a^{5} b^{5} + 9 \, B a^{4} b^{6} - 17 \, A a^{3} b^{7} + 3 \, B a^{2} b^{8} - 6 \, A a b^{9}\right )} \tan \left (d x + c\right )^{3} + {\left (A a^{10} + 3 \, B a^{9} b - 3 \, A a^{8} b^{2} + 9 \, B a^{7} b^{3} - 15 \, A a^{6} b^{4} + 9 \, B a^{5} b^{5} - 17 \, A a^{4} b^{6} + 3 \, B a^{3} b^{7} - 6 \, A a^{2} b^{8}\right )} \tan \left (d x + c\right )^{2}\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) - {\left ({\left (10 \, B a^{5} b^{5} - 15 \, A a^{4} b^{6} + 9 \, B a^{3} b^{7} - 17 \, A a^{2} b^{8} + 3 \, B a b^{9} - 6 \, A b^{10}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (10 \, B a^{6} b^{4} - 15 \, A a^{5} b^{5} + 9 \, B a^{4} b^{6} - 17 \, A a^{3} b^{7} + 3 \, B a^{2} b^{8} - 6 \, A a b^{9}\right )} \tan \left (d x + c\right )^{3} + {\left (10 \, B a^{7} b^{3} - 15 \, A a^{6} b^{4} + 9 \, B a^{5} b^{5} - 17 \, A a^{4} b^{6} + 3 \, B a^{3} b^{7} - 6 \, A a^{2} b^{8}\right )} \tan \left (d x + c\right )^{2}\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + 2 \, {\left (B a^{10} - 2 \, A a^{9} b + 3 \, B a^{8} b^{2} - 6 \, A a^{7} b^{3} + 3 \, B a^{6} b^{4} - 6 \, A a^{5} b^{5} + B a^{4} b^{6} - 2 \, A a^{3} b^{7}\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{11} b^{2} + 3 \, a^{9} b^{4} + 3 \, a^{7} b^{6} + a^{5} b^{8}\right )} d \tan \left (d x + c\right )^{4} + 2 \, {\left (a^{12} b + 3 \, a^{10} b^{3} + 3 \, a^{8} b^{5} + a^{6} b^{7}\right )} d \tan \left (d x + c\right )^{3} + {\left (a^{13} + 3 \, a^{11} b^{2} + 3 \, a^{9} b^{4} + a^{7} b^{6}\right )} d \tan \left (d x + c\right )^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 3.27, size = 812, normalized size = 2.31 \[ -\frac {\frac {4 \, {\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {4 \, {\left (10 \, B a^{5} b^{4} - 15 \, A a^{4} b^{5} + 9 \, B a^{3} b^{6} - 17 \, A a^{2} b^{7} + 3 \, B a b^{8} - 6 \, A b^{9}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}} - \frac {3 \, A a^{7} b^{2} \tan \left (d x + c\right )^{4} + 9 \, B a^{6} b^{3} \tan \left (d x + c\right )^{4} - 9 \, A a^{5} b^{4} \tan \left (d x + c\right )^{4} - 3 \, B a^{4} b^{5} \tan \left (d x + c\right )^{4} + 6 \, A a^{8} b \tan \left (d x + c\right )^{3} + 14 \, B a^{7} b^{2} \tan \left (d x + c\right )^{3} - 6 \, A a^{6} b^{3} \tan \left (d x + c\right )^{3} - 34 \, B a^{5} b^{4} \tan \left (d x + c\right )^{3} + 56 \, A a^{4} b^{5} \tan \left (d x + c\right )^{3} - 36 \, B a^{3} b^{6} \tan \left (d x + c\right )^{3} + 68 \, A a^{2} b^{7} \tan \left (d x + c\right )^{3} - 12 \, B a b^{8} \tan \left (d x + c\right )^{3} + 24 \, A b^{9} \tan \left (d x + c\right )^{3} + 3 \, A a^{9} \tan \left (d x + c\right )^{2} + B a^{8} b \tan \left (d x + c\right )^{2} + 13 \, A a^{7} b^{2} \tan \left (d x + c\right )^{2} - 45 \, B a^{6} b^{3} \tan \left (d x + c\right )^{2} + 88 \, A a^{5} b^{4} \tan \left (d x + c\right )^{2} - 52 \, B a^{4} b^{5} \tan \left (d x + c\right )^{2} + 102 \, A a^{3} b^{6} \tan \left (d x + c\right )^{2} - 18 \, B a^{2} b^{7} \tan \left (d x + c\right )^{2} + 36 \, A a b^{8} \tan \left (d x + c\right )^{2} - 4 \, B a^{9} \tan \left (d x + c\right ) + 8 \, A a^{8} b \tan \left (d x + c\right ) - 12 \, B a^{7} b^{2} \tan \left (d x + c\right ) + 24 \, A a^{6} b^{3} \tan \left (d x + c\right ) - 12 \, B a^{5} b^{4} \tan \left (d x + c\right ) + 24 \, A a^{4} b^{5} \tan \left (d x + c\right ) - 4 \, B a^{3} b^{6} \tan \left (d x + c\right ) + 8 \, A a^{2} b^{7} \tan \left (d x + c\right ) - 2 \, A a^{9} - 6 \, A a^{7} b^{2} - 6 \, A a^{5} b^{4} - 2 \, A a^{3} b^{6}}{{\left (a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}\right )} {\left (b \tan \left (d x + c\right )^{2} + a \tan \left (d x + c\right )\right )}^{2}} + \frac {4 \, {\left (A a^{2} + 3 \, B a b - 6 \, A b^{2}\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{5}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.76, size = 713, normalized size = 2.03 \[ \frac {6 \ln \left (\tan \left (d x +c \right )\right ) A \,b^{2}}{d \,a^{5}}-\frac {3 \ln \left (\tan \left (d x +c \right )\right ) B b}{d \,a^{4}}+\frac {3 A \arctan \left (\tan \left (d x +c \right )\right ) a^{2} b}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 B \arctan \left (\tan \left (d x +c \right )\right ) a \,b^{2}}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {A \arctan \left (\tan \left (d x +c \right )\right ) b^{3}}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {B \arctan \left (\tan \left (d x +c \right )\right ) a^{3}}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 A b}{d \,a^{4} \tan \left (d x +c \right )}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) A \,a^{3}}{2 d \left (a^{2}+b^{2}\right )^{3}}-\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{3} B}{2 d \left (a^{2}+b^{2}\right )^{3}}+\frac {10 \ln \left (a +b \tan \left (d x +c \right )\right ) b^{3} B}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{2} b B}{2 d \left (a^{2}+b^{2}\right )^{3}}-\frac {A \ln \left (\tan \left (d x +c \right )\right )}{a^{3} d}-\frac {A}{2 d \,a^{3} \tan \left (d x +c \right )^{2}}-\frac {B}{d \,a^{3} \tan \left (d x +c \right )}+\frac {5 b^{4} A}{d \,a^{2} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {b^{3} B}{2 d \,a^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {3 b^{6} A}{d \,a^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {4 b^{3} B}{d a \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) A a \,b^{2}}{2 d \left (a^{2}+b^{2}\right )^{3}}+\frac {9 b^{5} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d \,a^{2} \left (a^{2}+b^{2}\right )^{3}}-\frac {6 b^{8} \ln \left (a +b \tan \left (d x +c \right )\right ) A}{d \,a^{5} \left (a^{2}+b^{2}\right )^{3}}-\frac {15 b^{4} \ln \left (a +b \tan \left (d x +c \right )\right ) A}{d a \left (a^{2}+b^{2}\right )^{3}}-\frac {17 b^{6} \ln \left (a +b \tan \left (d x +c \right )\right ) A}{d \,a^{3} \left (a^{2}+b^{2}\right )^{3}}-\frac {2 b^{5} B}{d \,a^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {b^{4} A}{2 d \,a^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {3 b^{7} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d \,a^{4} \left (a^{2}+b^{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.84, size = 541, normalized size = 1.54 \[ -\frac {\frac {2 \, {\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (10 \, B a^{5} b^{3} - 15 \, A a^{4} b^{4} + 9 \, B a^{3} b^{5} - 17 \, A a^{2} b^{6} + 3 \, B a b^{7} - 6 \, A b^{8}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{11} + 3 \, a^{9} b^{2} + 3 \, a^{7} b^{4} + a^{5} b^{6}} - \frac {{\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {A a^{7} + 2 \, A a^{5} b^{2} + A a^{3} b^{4} + 2 \, {\left (B a^{5} b^{2} - 3 \, A a^{4} b^{3} + 6 \, B a^{3} b^{4} - 11 \, A a^{2} b^{5} + 3 \, B a b^{6} - 6 \, A b^{7}\right )} \tan \left (d x + c\right )^{3} + {\left (4 \, B a^{6} b - 11 \, A a^{5} b^{2} + 17 \, B a^{4} b^{3} - 33 \, A a^{3} b^{4} + 9 \, B a^{2} b^{5} - 18 \, A a b^{6}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (B a^{7} - 2 \, A a^{6} b + 2 \, B a^{5} b^{2} - 4 \, A a^{4} b^{3} + B a^{3} b^{4} - 2 \, A a^{2} b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{8} b^{2} + 2 \, a^{6} b^{4} + a^{4} b^{6}\right )} \tan \left (d x + c\right )^{4} + 2 \, {\left (a^{9} b + 2 \, a^{7} b^{3} + a^{5} b^{5}\right )} \tan \left (d x + c\right )^{3} + {\left (a^{10} + 2 \, a^{8} b^{2} + a^{6} b^{4}\right )} \tan \left (d x + c\right )^{2}} + \frac {2 \, {\left (A a^{2} + 3 \, B a b - 6 \, A b^{2}\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{5}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 12.87, size = 434, normalized size = 1.23 \[ \frac {\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (2\,A\,b-B\,a\right )}{a^2}-\frac {A}{2\,a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^3\,\left (-B\,a^5\,b^2+3\,A\,a^4\,b^3-6\,B\,a^3\,b^4+11\,A\,a^2\,b^5-3\,B\,a\,b^6+6\,A\,b^7\right )}{a^4\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (-4\,B\,a^5\,b+11\,A\,a^4\,b^2-17\,B\,a^3\,b^3+33\,A\,a^2\,b^4-9\,B\,a\,b^5+18\,A\,b^6\right )}{2\,a^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,{\mathrm {tan}\left (c+d\,x\right )}^2+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^3+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^4\right )}+\frac {\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (\frac {A}{a^3}-\frac {A\,a+3\,B\,b}{{\left (a^2+b^2\right )}^2}-\frac {6\,A\,b^2}{a^5}+\frac {3\,B\,b}{a^4}+\frac {4\,b^2\,\left (A\,a+B\,b\right )}{{\left (a^2+b^2\right )}^3}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (A\,a^2+3\,B\,a\,b-6\,A\,b^2\right )}{a^5\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: AttributeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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