Optimal. Leaf size=331 \[ \frac {a (A b-a B) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {a \left (-3 a^3 B+a^2 A b-7 a b^2 B+5 A b^3\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {\left (a^3 (-B)+3 a^2 A b+3 a b^2 B-A b^3\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {x \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )}{\left (a^2+b^2\right )^3}-\frac {\left (-3 a^4 B+a^3 A b-6 a^2 b^2 B+3 a A b^3-b^4 B\right ) \tan (c+d x)}{b^3 d \left (a^2+b^2\right )^2}+\frac {a^2 \left (-3 a^5 B+a^4 A b-9 a^3 b^2 B+3 a^2 A b^3-10 a b^4 B+6 A b^5\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^3} \]
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Rubi [A] time = 0.80, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.226, Rules used = {3605, 3645, 3647, 3626, 3617, 31, 3475} \[ \frac {a (A b-a B) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b-3 a^3 B-7 a b^2 B+5 A b^3\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {\left (a^3 A b-6 a^2 b^2 B-3 a^4 B+3 a A b^3-b^4 B\right ) \tan (c+d x)}{b^3 d \left (a^2+b^2\right )^2}+\frac {a^2 \left (3 a^2 A b^3+a^4 A b-9 a^3 b^2 B-3 a^5 B-10 a b^4 B+6 A b^5\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^3}+\frac {\left (3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {x \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3475
Rule 3605
Rule 3617
Rule 3626
Rule 3645
Rule 3647
Rubi steps
\begin {align*} \int \frac {\tan ^4(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx &=\frac {a (A b-a B) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {\int \frac {\tan ^2(c+d x) \left (-3 a (A b-a B)+2 b (A b-a B) \tan (c+d x)-\left (a A b-3 a^2 B-2 b^2 B\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 b \left (a^2+b^2\right )}\\ &=\frac {a (A b-a B) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {\tan (c+d x) \left (-2 a \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right )-2 b^2 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)-2 \left (a^3 A b+3 a A b^3-3 a^4 B-6 a^2 b^2 B-b^4 B\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 b^2 \left (a^2+b^2\right )^2}\\ &=-\frac {\left (a^3 A b+3 a A b^3-3 a^4 B-6 a^2 b^2 B-b^4 B\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\int \frac {2 a \left (a^3 A b+3 a A b^3-3 a^4 B-6 a^2 b^2 B-b^4 B\right )-2 b^3 \left (2 a A b-a^2 B+b^2 B\right ) \tan (c+d x)+2 \left (a^2+b^2\right )^2 (A b-3 a B) \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{2 b^3 \left (a^2+b^2\right )^2}\\ &=\frac {\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) x}{\left (a^2+b^2\right )^3}-\frac {\left (a^3 A b+3 a A b^3-3 a^4 B-6 a^2 b^2 B-b^4 B\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\left (3 a^2 A b-A b^3-a^3 B+3 a b^2 B\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^3}+\frac {\left (a^2 \left (a^4 A b+3 a^2 A b^3+6 A b^5-3 a^5 B-9 a^3 b^2 B-10 a b^4 B\right )\right ) \int \frac {1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b^3 \left (a^2+b^2\right )^3}\\ &=\frac {\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) x}{\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 ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac {\left (a^3 A b+3 a A b^3-3 a^4 B-6 a^2 b^2 B-b^4 B\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac {\left (a^2 \left (a^4 A b+3 a^2 A b^3+6 A b^5-3 a^5 B-9 a^3 b^2 B-10 a b^4 B\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^4 \left (a^2+b^2\right )^3 d}\\ &=\frac {\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) x}{\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 ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a^2 \left (a^4 A b+3 a^2 A b^3+6 A b^5-3 a^5 B-9 a^3 b^2 B-10 a b^4 B\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 A b+3 a A b^3-3 a^4 B-6 a^2 b^2 B-b^4 B\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (A b-a B) \tan ^3(c+d x)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac {a \left (a^2 A b+5 A b^3-3 a^3 B-7 a b^2 B\right ) \tan ^2(c+d x)}{2 b^2 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 6.86, size = 1146, normalized size = 3.46 \[ \frac {(a B-A b) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) (A+B \tan (c+d x)) a^4}{2 (a-i b)^2 (a+i b)^2 b^2 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {\sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^2 \left (2 B \sin (c+d x) a^5-A b \sin (c+d x) a^4+5 b^2 B \sin (c+d x) a^3-4 A b^3 \sin (c+d x) a^2\right ) (A+B \tan (c+d x))}{(a-i b)^2 (a+i b)^2 b^3 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {B \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \tan (c+d x) (A+B \tan (c+d x))}{b^3 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {\left (A a^3+3 b B a^2-3 A b^2 a-b^3 B\right ) (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (A+B \tan (c+d x))}{(a-i b)^3 (a+i b)^3 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {\left (6 a^2 A b^{13}+6 i a^3 A b^{12}-10 a^3 B b^{12}+15 a^4 A b^{11}-10 i a^4 B b^{11}+15 i a^5 A b^{10}-29 a^5 B b^{10}+13 a^6 A b^9-29 i a^6 B b^9+13 i a^7 A b^8-31 a^7 B b^8+5 a^8 A b^7-31 i a^8 B b^7+5 i a^9 A b^6-15 a^9 B b^6+a^{10} A b^5-15 i a^{10} B b^5+i a^{11} A b^4-3 a^{11} B b^4-3 i a^{12} B b^3\right ) (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (A+B \tan (c+d x))}{(a-i b)^6 (a+i b)^5 b^7 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}-\frac {i \left (-3 B a^7+A b a^6-9 b^2 B a^5+3 A b^3 a^4-10 b^4 B a^3+6 A b^5 a^2\right ) \tan ^{-1}(\tan (c+d x)) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (A+B \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {(3 a B-A b) \log (\cos (c+d x)) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (A+B \tan (c+d x))}{b^4 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {\left (-3 B a^7+A b a^6-9 b^2 B a^5+3 A b^3 a^4-10 b^4 B a^3+6 A b^5 a^2\right ) \log \left ((a \cos (c+d x)+b \sin (c+d x))^2\right ) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (A+B \tan (c+d x))}{2 b^4 \left (a^2+b^2\right )^3 d (A \cos (c+d x)+B \sin (c+d x)) (a+b \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.94, size = 890, normalized size = 2.69 \[ -\frac {3 \, B a^{7} b^{2} - A a^{6} b^{3} + 9 \, B a^{5} b^{4} - 7 \, A a^{4} b^{5} - 2 \, {\left (B a^{6} b^{3} + 3 \, B a^{4} b^{5} + 3 \, B a^{2} b^{7} + B b^{9}\right )} \tan \left (d x + c\right )^{3} - 2 \, {\left (A a^{5} b^{4} + 3 \, B a^{4} b^{5} - 3 \, A a^{3} b^{6} - B a^{2} b^{7}\right )} d x - {\left (9 \, B a^{7} b^{2} - 3 \, A a^{6} b^{3} + 23 \, B a^{5} b^{4} - 9 \, A a^{4} b^{5} + 12 \, B a^{3} b^{6} + 4 \, B a b^{8} + 2 \, {\left (A a^{3} b^{6} + 3 \, B a^{2} b^{7} - 3 \, A a b^{8} - B b^{9}\right )} d x\right )} \tan \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} + 10 \, B a^{5} b^{4} - 6 \, A a^{4} b^{5} + {\left (3 \, B a^{7} b^{2} - A a^{6} b^{3} + 9 \, B a^{5} b^{4} - 3 \, A a^{4} b^{5} + 10 \, B a^{3} b^{6} - 6 \, A a^{2} b^{7}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, B a^{8} b - A a^{7} b^{2} + 9 \, B a^{6} b^{3} - 3 \, A a^{5} b^{4} + 10 \, B a^{4} b^{5} - 6 \, A a^{3} b^{6}\right )} \tan \left (d x + c\right )\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 ) - {\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} + {\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 )} \tan \left (d x + c\right )^{2} + 2 \, {\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 )} \tan \left (d x + c\right )\right )} \log \left (\frac {1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \, {\left (3 \, B a^{8} b - A a^{7} b^{2} + 6 \, B a^{6} b^{3} - 3 \, A a^{5} b^{4} - 2 \, B a^{4} b^{5} + 4 \, A a^{3} b^{6} + B a^{2} b^{7} + 2 \, {\left (A a^{4} b^{5} + 3 \, B a^{3} b^{6} - 3 \, A a^{2} b^{7} - B a b^{8}\right )} d x\right )} \tan \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 \tan \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 \tan \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 )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 1.90, size = 505, normalized size = 1.53 \[ \frac {\frac {2 \, {\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (3 \, B a^{7} - A a^{6} b + 9 \, B a^{5} b^{2} - 3 \, A a^{4} b^{3} + 10 \, B a^{3} b^{4} - 6 \, A a^{2} b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac {2 \, B \tan \left (d x + c\right )}{b^{3}} + \frac {9 \, B a^{7} b^{2} \tan \left (d x + c\right )^{2} - 3 \, A a^{6} b^{3} \tan \left (d x + c\right )^{2} + 27 \, B a^{5} b^{4} \tan \left (d x + c\right )^{2} - 9 \, A a^{4} b^{5} \tan \left (d x + c\right )^{2} + 30 \, B a^{3} b^{6} \tan \left (d x + c\right )^{2} - 18 \, A a^{2} b^{7} \tan \left (d x + c\right )^{2} + 12 \, B a^{8} b \tan \left (d x + c\right ) - 2 \, A a^{7} b^{2} \tan \left (d x + c\right ) + 38 \, B a^{6} b^{3} \tan \left (d x + c\right ) - 6 \, A a^{5} b^{4} \tan \left (d x + c\right ) + 50 \, B a^{4} b^{5} \tan \left (d x + c\right ) - 28 \, A a^{3} b^{6} \tan \left (d x + c\right ) + 4 \, B a^{9} + 13 \, B a^{7} b^{2} + A a^{6} b^{3} + 21 \, B a^{5} b^{4} - 11 \, A a^{4} b^{5}}{{\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.25, size = 619, normalized size = 1.87 \[ \frac {B \tan \left (d x +c \right )}{d \,b^{3}}+\frac {a^{6} \ln \left (a +b \tan \left (d x +c \right )\right ) A}{d \,b^{3} \left (a^{2}+b^{2}\right )^{3}}+\frac {3 a^{4} \ln \left (a +b \tan \left (d x +c \right )\right ) A}{d b \left (a^{2}+b^{2}\right )^{3}}+\frac {6 b \,a^{2} \ln \left (a +b \tan \left (d x +c \right )\right ) A}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {3 a^{7} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d \,b^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {9 a^{5} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d \,b^{2} \left (a^{2}+b^{2}\right )^{3}}-\frac {10 a^{3} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {a^{4} A}{2 d \,b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{5} B}{2 d \,b^{4} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {2 a^{5} A}{d \,b^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {4 a^{3} A}{d b \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {3 a^{6} B}{d \,b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {5 a^{4} B}{d \,b^{2} \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^{2} b}{2 d \left (a^{2}+b^{2}\right )^{3}}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) A \,b^{3}}{2 d \left (a^{2}+b^{2}\right )^{3}}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a^{3} B}{2 d \left (a^{2}+b^{2}\right )^{3}}-\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) B a \,b^{2}}{2 d \left (a^{2}+b^{2}\right )^{3}}+\frac {A \arctan \left (\tan \left (d x +c \right )\right ) a^{3}}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {3 A \arctan \left (\tan \left (d x +c \right )\right ) a \,b^{2}}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 B \arctan \left (\tan \left (d x +c \right )\right ) a^{2} b}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {B \arctan \left (\tan \left (d x +c \right )\right ) b^{3}}{d \left (a^{2}+b^{2}\right )^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.83, size = 389, normalized size = 1.18 \[ \frac {\frac {2 \, {\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (3 \, B a^{7} - A a^{6} b + 9 \, B a^{5} b^{2} - 3 \, A a^{4} b^{3} + 10 \, B a^{3} b^{4} - 6 \, A a^{2} b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac {{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {5 \, B a^{7} - 3 \, A a^{6} b + 9 \, B a^{5} b^{2} - 7 \, A a^{4} b^{3} + 2 \, {\left (3 \, B a^{6} b - 2 \, A a^{5} b^{2} + 5 \, B a^{4} b^{3} - 4 \, A a^{3} b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b^{4} + 2 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{4} b^{6} + 2 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} + 2 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )} + \frac {2 \, B \tan \left (d x + c\right )}{b^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 7.87, size = 335, normalized size = 1.01 \[ \frac {B\,\mathrm {tan}\left (c+d\,x\right )}{b^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\frac {5\,B\,a^7-3\,A\,a^6\,b+9\,B\,a^5\,b^2-7\,A\,a^4\,b^3}{2\,b\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (3\,B\,a^6-2\,A\,a^5\,b+5\,B\,a^4\,b^2-4\,A\,a^3\,b^3\right )}{a^4+2\,a^2\,b^2+b^4}}{d\,\left (a^2\,b^3+2\,a\,b^4\,\mathrm {tan}\left (c+d\,x\right )+b^5\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {a^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-3\,B\,a^5+A\,a^4\,b-9\,B\,a^3\,b^2+3\,A\,a^2\,b^3-10\,B\,a\,b^4+6\,A\,b^5\right )}{b^4\,d\,{\left (a^2+b^2\right )}^3} \]
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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