Optimal. Leaf size=287 \[ -\frac {(3 A b-a B) \log (\sin (c+d x))}{a^4 d}-\frac {b \left (2 a^2 A-a b B+3 A b^2\right )}{2 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {x \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )}{\left (a^2+b^2\right )^3}-\frac {b \left (a^4 A-3 a^3 b B+6 a^2 A b^2-a b^3 B+3 A b^4\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {b^2 \left (-6 a^5 B+10 a^4 A b-3 a^3 b^2 B+9 a^2 A b^3-a b^4 B+3 A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^3}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^2} \]
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Rubi [A] time = 0.88, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps used = 6, 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} \[ -\frac {b \left (6 a^2 A b^2+a^4 A-3 a^3 b B-a b^3 B+3 A b^4\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac {b \left (2 a^2 A-a b B+3 A b^2\right )}{2 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac {b^2 \left (9 a^2 A b^3+10 a^4 A b-3 a^3 b^2 B-6 a^5 B-a b^4 B+3 A b^5\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 d \left (a^2+b^2\right )^3}-\frac {x \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right )}{\left (a^2+b^2\right )^3}-\frac {(3 A b-a B) \log (\sin (c+d x))}{a^4 d}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^2} \]
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 ^2(c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx &=-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {\int \frac {\cot (c+d x) \left (3 A b-a B+a A \tan (c+d x)+3 A b \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^3} \, dx}{a}\\ &=-\frac {b \left (2 a^2 A+3 A b^2-a b B\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {\int \frac {\cot (c+d x) \left (2 \left (a^2+b^2\right ) (3 A b-a B)+2 a^2 (a A+b B) \tan (c+d x)+2 b \left (2 a^2 A+3 A b^2-a b B\right ) \tan ^2(c+d x)\right )}{(a+b \tan (c+d x))^2} \, dx}{2 a^2 \left (a^2+b^2\right )}\\ &=-\frac {b \left (2 a^2 A+3 A b^2-a b B\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4 A+6 a^2 A b^2+3 A b^4-3 a^3 b B-a b^3 B\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {\int \frac {\cot (c+d x) \left (2 \left (a^2+b^2\right )^2 (3 A b-a B)+2 a^3 \left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)+2 b \left (a^4 A+6 a^2 A b^2+3 A b^4-3 a^3 b B-a b^3 B\right ) \tan ^2(c+d x)\right )}{a+b \tan (c+d x)} \, dx}{2 a^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 {b \left (2 a^2 A+3 A b^2-a b B\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4 A+6 a^2 A b^2+3 A b^4-3 a^3 b B-a b^3 B\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac {(3 A b-a B) \int \cot (c+d x) \, dx}{a^4}+\frac {\left (b^2 \left (10 a^4 A b+9 a^2 A b^3+3 A b^5-6 a^5 B-3 a^3 b^2 B-a b^4 B\right )\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^4 \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 {(3 A b-a B) \log (\sin (c+d x))}{a^4 d}+\frac {b^2 \left (10 a^4 A b+9 a^2 A b^3+3 A b^5-6 a^5 B-3 a^3 b^2 B-a b^4 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{a^4 \left (a^2+b^2\right )^3 d}-\frac {b \left (2 a^2 A+3 A b^2-a b B\right )}{2 a^2 \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}-\frac {A \cot (c+d x)}{a d (a+b \tan (c+d x))^2}-\frac {b \left (a^4 A+6 a^2 A b^2+3 A b^4-3 a^3 b B-a b^3 B\right )}{a^3 \left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end {align*}
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Mathematica [C] time = 6.43, size = 288, normalized size = 1.00 \[ -\frac {(3 A b-a B) \log (\tan (c+d x))}{a^4 d}-\frac {A \cot (c+d x)}{a^3 d}-\frac {b^2 (A b-a B)}{2 a^2 d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}-\frac {b^2 \left (-3 a^3 B+4 a^2 A b-a b^2 B+2 A b^3\right )}{a^3 d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}+\frac {b^2 \left (-6 a^5 B+10 a^4 A b-3 a^3 b^2 B+9 a^2 A b^3-a b^4 B+3 A b^5\right ) \log (a+b \tan (c+d x))}{a^4 d \left (a^2+b^2\right )^3}+\frac {(A+i B) \log (-\tan (c+d x)+i)}{2 d (-b+i a)^3}-\frac {(B+i A) \log (\tan (c+d x)+i)}{2 d (a-i b)^3} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.09, size = 917, normalized size = 3.20 \[ -\frac {2 \, A a^{9} + 6 \, A a^{7} b^{2} + 6 \, A a^{5} b^{4} + 2 \, A a^{3} b^{6} + {\left (7 \, B a^{5} b^{4} - 9 \, A a^{4} b^{5} + B a^{3} b^{6} - 3 \, A a^{2} b^{7} + 2 \, {\left (A a^{7} b^{2} + 3 \, B a^{6} b^{3} - 3 \, A a^{5} b^{4} - B a^{4} b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (A a^{7} b^{2} + 4 \, B a^{6} b^{3} - 2 \, A a^{5} b^{4} - 3 \, B a^{4} b^{5} + 6 \, A a^{3} b^{6} - B a^{2} b^{7} + 3 \, A a b^{8} + 2 \, {\left (A a^{8} b + 3 \, B a^{7} b^{2} - 3 \, A a^{6} b^{3} - B a^{5} b^{4}\right )} d x\right )} \tan \left (d x + c\right )^{2} - {\left ({\left (B a^{7} b^{2} - 3 \, A a^{6} b^{3} + 3 \, B a^{5} b^{4} - 9 \, A a^{4} b^{5} + 3 \, B a^{3} b^{6} - 9 \, A a^{2} b^{7} + B a b^{8} - 3 \, A b^{9}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (B a^{8} b - 3 \, A a^{7} b^{2} + 3 \, B a^{6} b^{3} - 9 \, A a^{5} b^{4} + 3 \, B a^{4} b^{5} - 9 \, A a^{3} b^{6} + B a^{2} b^{7} - 3 \, A a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (B a^{9} - 3 \, A a^{8} b + 3 \, B a^{7} b^{2} - 9 \, A a^{6} b^{3} + 3 \, B a^{5} b^{4} - 9 \, A a^{4} b^{5} + B a^{3} b^{6} - 3 \, A a^{2} b^{7}\right )} \tan \left (d x + c\right )\right )} \log \left (\frac {\tan \left (d x + c\right )^{2}}{\tan \left (d x + c\right )^{2} + 1}\right ) + {\left ({\left (6 \, B a^{5} b^{4} - 10 \, A a^{4} b^{5} + 3 \, B a^{3} b^{6} - 9 \, A a^{2} b^{7} + B a b^{8} - 3 \, A b^{9}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (6 \, B a^{6} b^{3} - 10 \, A a^{5} b^{4} + 3 \, B a^{4} b^{5} - 9 \, A a^{3} b^{6} + B a^{2} b^{7} - 3 \, A a b^{8}\right )} \tan \left (d x + c\right )^{2} + {\left (6 \, B a^{7} b^{2} - 10 \, A a^{6} b^{3} + 3 \, B a^{5} b^{4} - 9 \, A a^{4} b^{5} + B a^{3} b^{6} - 3 \, A a^{2} b^{7}\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 (4 \, A a^{8} b + 12 \, A a^{6} b^{3} - 9 \, B a^{5} b^{4} + 23 \, A a^{4} b^{5} - 3 \, B a^{3} b^{6} + 9 \, A a^{2} b^{7} + 2 \, {\left (A a^{9} + 3 \, B a^{8} b - 3 \, A a^{7} b^{2} - B a^{6} b^{3}\right )} d x\right )} \tan \left (d x + c\right )}{2 \, {\left ({\left (a^{10} b^{2} + 3 \, a^{8} b^{4} + 3 \, a^{6} b^{6} + a^{4} b^{8}\right )} d \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{11} b + 3 \, a^{9} b^{3} + 3 \, a^{7} b^{5} + a^{5} b^{7}\right )} d \tan \left (d x + c\right )^{2} + {\left (a^{12} + 3 \, a^{10} b^{2} + 3 \, a^{8} b^{4} + a^{6} b^{6}\right )} d \tan \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.20, size = 560, normalized size = 1.95 \[ -\frac {\frac {2 \, {\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (6 \, B a^{5} b^{3} - 10 \, A a^{4} b^{4} + 3 \, B a^{3} b^{5} - 9 \, A a^{2} b^{6} + B a b^{7} - 3 \, A b^{8}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{10} b + 3 \, a^{8} b^{3} + 3 \, a^{6} b^{5} + a^{4} b^{7}} - \frac {18 \, B a^{5} b^{4} \tan \left (d x + c\right )^{2} - 30 \, A a^{4} b^{5} \tan \left (d x + c\right )^{2} + 9 \, B a^{3} b^{6} \tan \left (d x + c\right )^{2} - 27 \, A a^{2} b^{7} \tan \left (d x + c\right )^{2} + 3 \, B a b^{8} \tan \left (d x + c\right )^{2} - 9 \, A b^{9} \tan \left (d x + c\right )^{2} + 42 \, B a^{6} b^{3} \tan \left (d x + c\right ) - 68 \, A a^{5} b^{4} \tan \left (d x + c\right ) + 26 \, B a^{4} b^{5} \tan \left (d x + c\right ) - 66 \, A a^{3} b^{6} \tan \left (d x + c\right ) + 8 \, B a^{2} b^{7} \tan \left (d x + c\right ) - 22 \, A a b^{8} \tan \left (d x + c\right ) + 25 \, B a^{7} b^{2} - 39 \, A a^{6} b^{3} + 19 \, B a^{5} b^{4} - 41 \, A a^{4} b^{5} + 6 \, B a^{3} b^{6} - 14 \, A a^{2} b^{7}}{{\left (a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}\right )} {\left (b \tan \left (d x + c\right ) + a\right )}^{2}} - \frac {2 \, {\left (B a - 3 \, A b\right )} \log \left ({\left | \tan \left (d x + c\right ) \right |}\right )}{a^{4}} + \frac {2 \, {\left (B a \tan \left (d x + c\right ) - 3 \, A b \tan \left (d x + c\right ) + A a\right )}}{a^{4} \tan \left (d x + c\right )}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.69, size = 651, normalized size = 2.27 \[ -\frac {4 b^{3} A}{d a \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}-\frac {2 b^{5} A}{d \,a^{3} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {3 b^{2} B}{d \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {b^{4} B}{d \,a^{2} \left (a^{2}+b^{2}\right )^{2} \left (a +b \tan \left (d x +c \right )\right )}+\frac {10 \ln \left (a +b \tan \left (d x +c \right )\right ) A \,b^{3}}{d \left (a^{2}+b^{2}\right )^{3}}+\frac {9 b^{5} \ln \left (a +b \tan \left (d x +c \right )\right ) A}{d \,a^{2} \left (a^{2}+b^{2}\right )^{3}}+\frac {3 b^{7} \ln \left (a +b \tan \left (d x +c \right )\right ) A}{d \,a^{4} \left (a^{2}+b^{2}\right )^{3}}-\frac {6 \ln \left (a +b \tan \left (d x +c \right )\right ) B a \,b^{2}}{d \left (a^{2}+b^{2}\right )^{3}}-\frac {3 b^{4} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d a \left (a^{2}+b^{2}\right )^{3}}-\frac {b^{6} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d \,a^{3} \left (a^{2}+b^{2}\right )^{3}}-\frac {b^{3} A}{2 d \,a^{2} \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}+\frac {b^{2} B}{2 d a \left (a^{2}+b^{2}\right ) \left (a +b \tan \left (d x +c \right )\right )^{2}}-\frac {A}{d \,a^{3} \tan \left (d x +c \right )}-\frac {3 \ln \left (\tan \left (d x +c \right )\right ) A b}{d \,a^{4}}+\frac {\ln \left (\tan \left (d x +c \right )\right ) B}{d \,a^{3}}+\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 = 0.84, size = 454, normalized size = 1.58 \[ -\frac {\frac {2 \, {\left (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} {\left (d x + c\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (6 \, B a^{5} b^{2} - 10 \, A a^{4} b^{3} + 3 \, B a^{3} b^{4} - 9 \, A a^{2} b^{5} + B a b^{6} - 3 \, A b^{7}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{10} + 3 \, a^{8} b^{2} + 3 \, a^{6} b^{4} + a^{4} b^{6}} + \frac {{\left (B a^{3} - 3 \, A a^{2} b - 3 \, B a b^{2} + A b^{3}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, A a^{6} + 4 \, A a^{4} b^{2} + 2 \, A a^{2} b^{4} + 2 \, {\left (A a^{4} b^{2} - 3 \, B a^{3} b^{3} + 6 \, A a^{2} b^{4} - B a b^{5} + 3 \, A b^{6}\right )} \tan \left (d x + c\right )^{2} + {\left (4 \, A a^{5} b - 7 \, B a^{4} b^{2} + 17 \, A a^{3} b^{3} - 3 \, B a^{2} b^{4} + 9 \, A a b^{5}\right )} \tan \left (d x + c\right )}{{\left (a^{7} b^{2} + 2 \, a^{5} b^{4} + a^{3} b^{6}\right )} \tan \left (d x + c\right )^{3} + 2 \, {\left (a^{8} b + 2 \, a^{6} b^{3} + a^{4} b^{5}\right )} \tan \left (d x + c\right )^{2} + {\left (a^{9} + 2 \, a^{7} b^{2} + a^{5} b^{4}\right )} \tan \left (d x + c\right )} - \frac {2 \, {\left (B a - 3 \, A b\right )} \log \left (\tan \left (d x + c\right )\right )}{a^{4}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 11.23, size = 380, normalized size = 1.32 \[ \frac {b^2\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (-6\,B\,a^5+10\,A\,a^4\,b-3\,B\,a^3\,b^2+9\,A\,a^2\,b^3-B\,a\,b^4+3\,A\,b^5\right )}{a^4\,d\,{\left (a^2+b^2\right )}^3}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (-B+A\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3-a^2\,b\,3{}\mathrm {i}+3\,a\,b^2+b^3\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )\right )\,\left (3\,A\,b-B\,a\right )}{a^4\,d}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (A-B\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^3\,1{}\mathrm {i}-3\,a^2\,b+a\,b^2\,3{}\mathrm {i}+b^3\right )}-\frac {\frac {A}{a}+\frac {{\mathrm {tan}\left (c+d\,x\right )}^2\,\left (A\,a^4\,b^2-3\,B\,a^3\,b^3+6\,A\,a^2\,b^4-B\,a\,b^5+3\,A\,b^6\right )}{a^3\,\left (a^4+2\,a^2\,b^2+b^4\right )}+\frac {\mathrm {tan}\left (c+d\,x\right )\,\left (4\,A\,a^4\,b-7\,B\,a^3\,b^2+17\,A\,a^2\,b^3-3\,B\,a\,b^4+9\,A\,b^5\right )}{2\,a^2\,\left (a^4+2\,a^2\,b^2+b^4\right )}}{d\,\left (a^2\,\mathrm {tan}\left (c+d\,x\right )+2\,a\,b\,{\mathrm {tan}\left (c+d\,x\right )}^2+b^2\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )} \]
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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