Optimal. Leaf size=331 \[ \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {a \left (-3 a^3 C+a^2 b B-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {\left (a^3 (-C)+3 a^2 b B+3 a b^2 C-b^3 B\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {x \left (a^3 B+3 a^2 b C-3 a b^2 B-b^3 C\right )}{\left (a^2+b^2\right )^3}-\frac {\left (-3 a^4 C+a^3 b B-6 a^2 b^2 C+3 a b^3 B-b^4 C\right ) \tan (c+d x)}{b^3 d \left (a^2+b^2\right )^2}+\frac {a^2 \left (-3 a^5 C+a^4 b B-9 a^3 b^2 C+3 a^2 b^3 B-10 a b^4 C+6 b^5 B\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^3} \]
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Rubi [A] time = 0.86, antiderivative size = 331, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3632, 3605, 3645, 3647, 3626, 3617, 31, 3475} \[ \frac {a (b B-a C) \tan ^3(c+d x)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {a \left (a^2 b B-3 a^3 C-7 a b^2 C+5 b^3 B\right ) \tan ^2(c+d x)}{2 b^2 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {\left (-6 a^2 b^2 C+a^3 b B-3 a^4 C+3 a b^3 B-b^4 C\right ) \tan (c+d x)}{b^3 d \left (a^2+b^2\right )^2}+\frac {a^2 \left (3 a^2 b^3 B-9 a^3 b^2 C+a^4 b B-3 a^5 C-10 a b^4 C+6 b^5 B\right ) \log (a+b \tan (c+d x))}{b^4 d \left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b B+a^3 (-C)+3 a b^2 C-b^3 B\right ) \log (\cos (c+d x))}{d \left (a^2+b^2\right )^3}+\frac {x \left (3 a^2 b C+a^3 B-3 a b^2 B-b^3 C\right )}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 31
Rule 3475
Rule 3605
Rule 3617
Rule 3626
Rule 3632
Rule 3645
Rule 3647
Rubi steps
\begin {align*} \int \frac {\tan ^3(c+d x) \left (B \tan (c+d x)+C \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx &=\int \frac {\tan ^4(c+d x) (B+C \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx\\ &=\frac {a (b B-a C) \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 (b B-a C)+2 b (b B-a C) \tan (c+d x)-\left (a b B-3 a^2 C-2 b^2 C\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 (b B-a C) \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 b B+5 b^3 B-3 a^3 C-7 a b^2 C\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 b B+5 b^3 B-3 a^3 C-7 a b^2 C\right )-2 b^2 \left (a^2 B-b^2 B+2 a b C\right ) \tan (c+d x)-2 \left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\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 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (b B-a C) \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 b B+5 b^3 B-3 a^3 C-7 a b^2 C\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 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right )-2 b^3 \left (2 a b B-a^2 C+b^2 C\right ) \tan (c+d x)+2 \left (a^2+b^2\right )^2 (b B-3 a C) \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 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) x}{\left (a^2+b^2\right )^3}-\frac {\left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (b B-a C) \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 b B+5 b^3 B-3 a^3 C-7 a b^2 C\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 b B-b^3 B-a^3 C+3 a b^2 C\right ) \int \tan (c+d x) \, dx}{\left (a^2+b^2\right )^3}+\frac {\left (a^2 \left (a^4 b B+3 a^2 b^3 B+6 b^5 B-3 a^5 C-9 a^3 b^2 C-10 a b^4 C\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 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}-\frac {\left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (b B-a C) \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 b B+5 b^3 B-3 a^3 C-7 a b^2 C\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 b B+3 a^2 b^3 B+6 b^5 B-3 a^5 C-9 a^3 b^2 C-10 a b^4 C\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 B-3 a b^2 B+3 a^2 b C-b^3 C\right ) x}{\left (a^2+b^2\right )^3}+\frac {\left (3 a^2 b B-b^3 B-a^3 C+3 a b^2 C\right ) \log (\cos (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac {a^2 \left (a^4 b B+3 a^2 b^3 B+6 b^5 B-3 a^5 C-9 a^3 b^2 C-10 a b^4 C\right ) \log (a+b \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3 d}-\frac {\left (a^3 b B+3 a b^3 B-3 a^4 C-6 a^2 b^2 C-b^4 C\right ) \tan (c+d x)}{b^3 \left (a^2+b^2\right )^2 d}+\frac {a (b B-a C) \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 b B+5 b^3 B-3 a^3 C-7 a b^2 C\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 C-b B) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x)) (B+C \tan (c+d x)) a^4}{2 (a-i b)^2 (a+i b)^2 b^2 d (B \cos (c+d x)+C \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 C \sin (c+d x) a^5-b B \sin (c+d x) a^4+5 b^2 C \sin (c+d x) a^3-4 b^3 B \sin (c+d x) a^2\right ) (B+C \tan (c+d x))}{(a-i b)^2 (a+i b)^2 b^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {C \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 \tan (c+d x) (B+C \tan (c+d x))}{b^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {\left (B a^3+3 b C a^2-3 b^2 B a-b^3 C\right ) (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{(a-i b)^3 (a+i b)^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {\left (6 a^2 B b^{13}+6 i a^3 B b^{12}-10 a^3 C b^{12}+15 a^4 B b^{11}-10 i a^4 C b^{11}+15 i a^5 B b^{10}-29 a^5 C b^{10}+13 a^6 B b^9-29 i a^6 C b^9+13 i a^7 B b^8-31 a^7 C b^8+5 a^8 B b^7-31 i a^8 C b^7+5 i a^9 B b^6-15 a^9 C b^6+a^{10} B b^5-15 i a^{10} C b^5+i a^{11} B b^4-3 a^{11} C b^4-3 i a^{12} C b^3\right ) (c+d x) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{(a-i b)^6 (a+i b)^5 b^7 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}-\frac {i \left (-3 C a^7+b B a^6-9 b^2 C a^5+3 b^3 B a^4-10 b^4 C a^3+6 b^5 B 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 (B+C \tan (c+d x))}{b^4 \left (a^2+b^2\right )^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {(3 a C-b B) \log (\cos (c+d x)) \sec ^2(c+d x) (a \cos (c+d x)+b \sin (c+d x))^3 (B+C \tan (c+d x))}{b^4 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3}+\frac {\left (-3 C a^7+b B a^6-9 b^2 C a^5+3 b^3 B a^4-10 b^4 C a^3+6 b^5 B 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 (B+C \tan (c+d x))}{2 b^4 \left (a^2+b^2\right )^3 d (B \cos (c+d x)+C \sin (c+d x)) (a+b \tan (c+d x))^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.93, size = 890, normalized size = 2.69 \[ -\frac {3 \, C a^{7} b^{2} - B a^{6} b^{3} + 9 \, C a^{5} b^{4} - 7 \, B a^{4} b^{5} - 2 \, {\left (C a^{6} b^{3} + 3 \, C a^{4} b^{5} + 3 \, C a^{2} b^{7} + C b^{9}\right )} \tan \left (d x + c\right )^{3} - 2 \, {\left (B a^{5} b^{4} + 3 \, C a^{4} b^{5} - 3 \, B a^{3} b^{6} - C a^{2} b^{7}\right )} d x - {\left (9 \, C a^{7} b^{2} - 3 \, B a^{6} b^{3} + 23 \, C a^{5} b^{4} - 9 \, B a^{4} b^{5} + 12 \, C a^{3} b^{6} + 4 \, C a b^{8} + 2 \, {\left (B a^{3} b^{6} + 3 \, C a^{2} b^{7} - 3 \, B a b^{8} - C b^{9}\right )} d x\right )} \tan \left (d x + c\right )^{2} + {\left (3 \, C a^{9} - B a^{8} b + 9 \, C a^{7} b^{2} - 3 \, B a^{6} b^{3} + 10 \, C a^{5} b^{4} - 6 \, B a^{4} b^{5} + {\left (3 \, C a^{7} b^{2} - B a^{6} b^{3} + 9 \, C a^{5} b^{4} - 3 \, B a^{4} b^{5} + 10 \, C a^{3} b^{6} - 6 \, B a^{2} b^{7}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, C a^{8} b - B a^{7} b^{2} + 9 \, C a^{6} b^{3} - 3 \, B a^{5} b^{4} + 10 \, C a^{4} b^{5} - 6 \, B 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 \, C a^{9} - B a^{8} b + 9 \, C a^{7} b^{2} - 3 \, B a^{6} b^{3} + 9 \, C a^{5} b^{4} - 3 \, B a^{4} b^{5} + 3 \, C a^{3} b^{6} - B a^{2} b^{7} + {\left (3 \, C a^{7} b^{2} - B a^{6} b^{3} + 9 \, C a^{5} b^{4} - 3 \, B a^{4} b^{5} + 9 \, C a^{3} b^{6} - 3 \, B a^{2} b^{7} + 3 \, C a b^{8} - B b^{9}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (3 \, C a^{8} b - B a^{7} b^{2} + 9 \, C a^{6} b^{3} - 3 \, B a^{5} b^{4} + 9 \, C a^{4} b^{5} - 3 \, B a^{3} b^{6} + 3 \, C a^{2} b^{7} - B 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 \, C a^{8} b - B a^{7} b^{2} + 6 \, C a^{6} b^{3} - 3 \, B a^{5} b^{4} - 2 \, C a^{4} b^{5} + 4 \, B a^{3} b^{6} + C a^{2} b^{7} + 2 \, {\left (B a^{4} b^{5} + 3 \, C a^{3} b^{6} - 3 \, B a^{2} b^{7} - C 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 = 4.34, size = 505, normalized size = 1.53 \[ \frac {\frac {2 \, {\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (3 \, C a^{7} - B a^{6} b + 9 \, C a^{5} b^{2} - 3 \, B a^{4} b^{3} + 10 \, C a^{3} b^{4} - 6 \, B a^{2} b^{5}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac {2 \, C \tan \left (d x + c\right )}{b^{3}} + \frac {9 \, C a^{7} b^{2} \tan \left (d x + c\right )^{2} - 3 \, B a^{6} b^{3} \tan \left (d x + c\right )^{2} + 27 \, C a^{5} b^{4} \tan \left (d x + c\right )^{2} - 9 \, B a^{4} b^{5} \tan \left (d x + c\right )^{2} + 30 \, C a^{3} b^{6} \tan \left (d x + c\right )^{2} - 18 \, B a^{2} b^{7} \tan \left (d x + c\right )^{2} + 12 \, C a^{8} b \tan \left (d x + c\right ) - 2 \, B a^{7} b^{2} \tan \left (d x + c\right ) + 38 \, C a^{6} b^{3} \tan \left (d x + c\right ) - 6 \, B a^{5} b^{4} \tan \left (d x + c\right ) + 50 \, C a^{4} b^{5} \tan \left (d x + c\right ) - 28 \, B a^{3} b^{6} \tan \left (d x + c\right ) + 4 \, C a^{9} + 13 \, C a^{7} b^{2} + B a^{6} b^{3} + 21 \, C a^{5} b^{4} - 11 \, B a^{4} b^{5}}{{\left (a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.28, size = 619, normalized size = 1.87 \[ \frac {C \tan \left (d x +c \right )}{d \,b^{3}}+\frac {a^{6} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{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 ) B}{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 ) B}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {3 a^{7} \ln \left (a +b \tan \left (d x +c \right )\right ) C}{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 ) C}{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 ) C}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {a^{4} B}{2 d \,b^{3} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {a^{5} C}{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} B}{d \,b^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {4 a^{3} B}{d b \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {3 a^{6} C}{d \,b^{4} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {5 a^{4} C}{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^{2} b B}{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 {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) C \,a^{3}}{2 d \left (a^{2}+b^{2}\right )^{3}}-\frac {3 \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) C a \,b^{2}}{2 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 B \arctan \left (\tan \left (d x +c \right )\right ) a \,b^{2}}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {3 C \arctan \left (\tan \left (d x +c \right )\right ) a^{2} b}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {C \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 = 1.04, size = 389, normalized size = 1.18 \[ \frac {\frac {2 \, {\left (B a^{3} + 3 \, C a^{2} b - 3 \, B a b^{2} - C b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {2 \, {\left (3 \, C a^{7} - B a^{6} b + 9 \, C a^{5} b^{2} - 3 \, B a^{4} b^{3} + 10 \, C a^{3} b^{4} - 6 \, B a^{2} b^{5}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} b^{4} + 3 \, a^{4} b^{6} + 3 \, a^{2} b^{8} + b^{10}} + \frac {{\left (C a^{3} - 3 \, B a^{2} b - 3 \, C a b^{2} + B b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {5 \, C a^{7} - 3 \, B a^{6} b + 9 \, C a^{5} b^{2} - 7 \, B a^{4} b^{3} + 2 \, {\left (3 \, C a^{6} b - 2 \, B a^{5} b^{2} + 5 \, C a^{4} b^{3} - 4 \, B a^{3} b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b^{4} + 2 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{4} b^{6} + 2 \, a^{2} b^{8} + b^{10}\right )} \tan \left (d x + c\right )^{2} + 2 \, {\left (a^{5} b^{5} + 2 \, a^{3} b^{7} + a b^{9}\right )} \tan \left (d x + c\right )} + \frac {2 \, C \tan \left (d x + c\right )}{b^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 10.43, size = 335, normalized size = 1.01 \[ \frac {C\,\mathrm {tan}\left (c+d\,x\right )}{b^3\,d}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-C+B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}+\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B-C\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\frac {5\,C\,a^7-3\,B\,a^6\,b+9\,C\,a^5\,b^2-7\,B\,a^4\,b^3}{2\,b\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (3\,C\,a^6-2\,B\,a^5\,b+5\,C\,a^4\,b^2-4\,B\,a^3\,b^3\right )}{a^4+2\,a^2\,b^2+b^4}}{d\,\left (a^2\,b^3+2\,a\,b^4\,\mathrm {tan}\left (c+d\,x\right )+b^5\,{\mathrm {tan}\left (c+d\,x\right )}^2\right )}+\frac {a^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-3\,C\,a^5+B\,a^4\,b-9\,C\,a^3\,b^2+3\,B\,a^2\,b^3-10\,C\,a\,b^4+6\,B\,b^5\right )}{b^4\,d\,{\left (a^2+b^2\right )}^3} \]
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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