Optimal. Leaf size=179 \[ \frac{a (A b-a B)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{a^2 A+2 a b B-A b^2}{d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{\left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}+\frac{x \left (3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3\right )}{\left (a^2+b^2\right )^3} \]
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Rubi [A] time = 0.274716, antiderivative size = 179, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138, Rules used = {3591, 3529, 3531, 3530} \[ \frac{a (A b-a B)}{2 b d \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{a^2 A+2 a b B-A b^2}{d \left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{\left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )^3}+\frac{x \left (3 a^2 A b+a^3 (-B)+3 a b^2 B-A b^3\right )}{\left (a^2+b^2\right )^3} \]
Antiderivative was successfully verified.
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Rule 3591
Rule 3529
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{\tan (c+d x) (A+B \tan (c+d x))}{(a+b \tan (c+d x))^3} \, dx &=\frac{a (A b-a B)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{\int \frac{A b-a B+(a A+b B) \tan (c+d x)}{(a+b \tan (c+d x))^2} \, dx}{a^2+b^2}\\ &=\frac{a (A b-a B)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a^2 A-A b^2+2 a b B}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}+\frac{\int \frac{2 a A b-a^2 B+b^2 B+\left (a^2 A-A b^2+2 a b B\right ) \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\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{a (A b-a B)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a^2 A-A b^2+2 a b B}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}-\frac{\left (a^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) \int \frac{b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{\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^3 A-3 a A b^2+3 a^2 b B-b^3 B\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right )^3 d}+\frac{a (A b-a B)}{2 b \left (a^2+b^2\right ) d (a+b \tan (c+d x))^2}+\frac{a^2 A-A b^2+2 a b B}{\left (a^2+b^2\right )^2 d (a+b \tan (c+d x))}\\ \end{align*}
Mathematica [C] time = 3.56999, size = 188, normalized size = 1.05 \[ \frac{\frac{a (A b-a B)}{b \left (a^2+b^2\right ) (a+b \tan (c+d x))^2}+\frac{2 \left (a^2 A+2 a b B-A b^2\right )}{\left (a^2+b^2\right )^2 (a+b \tan (c+d x))}-\frac{2 \left (a^3 A+3 a^2 b B-3 a A b^2-b^3 B\right ) \log (a+b \tan (c+d x))}{\left (a^2+b^2\right )^3}+\frac{(A+i B) \log (-\tan (c+d x)+i)}{(a+i b)^3}+\frac{(A-i B) \log (\tan (c+d x)+i)}{(a-i b)^3}}{2 d} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.045, size = 488, normalized size = 2.7 \begin{align*}{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) A{a}^{3}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{3\,\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) Aa{b}^{2}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{3\,\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) B{a}^{2}b}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) B{b}^{3}}{2\,d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+3\,{\frac{A\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{2}b}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{A\arctan \left ( \tan \left ( dx+c \right ) \right ){b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ){a}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+3\,{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ) a{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{Aa}{2\,d \left ({a}^{2}+{b}^{2} \right ) \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}-{\frac{{a}^{2}B}{2\,d \left ({a}^{2}+{b}^{2} \right ) b \left ( a+b\tan \left ( dx+c \right ) \right ) ^{2}}}+{\frac{{a}^{2}A}{d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-{\frac{A{b}^{2}}{d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) }}+2\,{\frac{Bab}{d \left ({a}^{2}+{b}^{2} \right ) ^{2} \left ( a+b\tan \left ( dx+c \right ) \right ) }}-{\frac{{a}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) A}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+3\,{\frac{a{b}^{2}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) A}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}-3\,{\frac{b{a}^{2}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) B}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}}+{\frac{\ln \left ( a+b\tan \left ( dx+c \right ) \right ) B{b}^{3}}{d \left ({a}^{2}+{b}^{2} \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.55077, size = 446, normalized size = 2.49 \begin{align*} -\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 (A a^{3} + 3 \, B a^{2} b - 3 \, A a b^{2} - B b^{3}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + 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{B a^{4} - 3 \, A a^{3} b - 3 \, B a^{2} b^{2} + A a b^{3} - 2 \,{\left (A a^{2} b^{2} + 2 \, B a b^{3} - A b^{4}\right )} \tan \left (d x + c\right )}{a^{6} b + 2 \, a^{4} b^{3} + a^{2} b^{5} +{\left (a^{4} b^{3} + 2 \, a^{2} b^{5} + b^{7}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{5} b^{2} + 2 \, a^{3} b^{4} + a b^{6}\right )} \tan \left (d x + c\right )}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] time = 1.90427, size = 1058, normalized size = 5.91 \begin{align*} -\frac{3 \, B a^{4} b - 5 \, A a^{3} b^{2} - 3 \, B a^{2} b^{3} + A a b^{4} + 2 \,{\left (B a^{5} - 3 \, A a^{4} b - 3 \, B a^{3} b^{2} + A a^{2} b^{3}\right )} d x -{\left (B a^{4} b - 3 \, A a^{3} b^{2} - 5 \, B a^{2} b^{3} + 3 \, A a b^{4} - 2 \,{\left (B a^{3} b^{2} - 3 \, A a^{2} b^{3} - 3 \, B a b^{4} + A b^{5}\right )} d x\right )} \tan \left (d x + c\right )^{2} +{\left (A a^{5} + 3 \, B a^{4} b - 3 \, A a^{3} b^{2} - B a^{2} b^{3} +{\left (A a^{3} b^{2} + 3 \, B a^{2} b^{3} - 3 \, A a b^{4} - B b^{5}\right )} \tan \left (d x + c\right )^{2} + 2 \,{\left (A a^{4} b + 3 \, B a^{3} b^{2} - 3 \, A a^{2} b^{3} - B a b^{4}\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 ) - 2 \,{\left (B a^{5} - 2 \, A a^{4} b - 3 \, B a^{3} b^{2} + 3 \, A a^{2} b^{3} + 2 \, B a b^{4} - A b^{5} - 2 \,{\left (B a^{4} b - 3 \, A a^{3} b^{2} - 3 \, B a^{2} b^{3} + A a b^{4}\right )} d x\right )} \tan \left (d x + c\right )}{2 \,{\left ({\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} d \tan \left (d x + c\right )^{2} + 2 \,{\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} d \tan \left (d x + c\right ) +{\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} d\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 1.24453, size = 554, normalized size = 3.09 \begin{align*} -\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{{\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{2 \,{\left (A a^{3} b + 3 \, B a^{2} b^{2} - 3 \, A a b^{3} - B b^{4}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac{3 \, A a^{3} b^{3} \tan \left (d x + c\right )^{2} + 9 \, B a^{2} b^{4} \tan \left (d x + c\right )^{2} - 9 \, A a b^{5} \tan \left (d x + c\right )^{2} - 3 \, B b^{6} \tan \left (d x + c\right )^{2} + 8 \, A a^{4} b^{2} \tan \left (d x + c\right ) + 22 \, B a^{3} b^{3} \tan \left (d x + c\right ) - 18 \, A a^{2} b^{4} \tan \left (d x + c\right ) - 2 \, B a b^{5} \tan \left (d x + c\right ) - 2 \, A b^{6} \tan \left (d x + c\right ) - B a^{6} + 6 \, A a^{5} b + 11 \, B a^{4} b^{2} - 7 \, A a^{3} b^{3} - A a b^{5}}{{\left (a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}\right )}{\left (b \tan \left (d x + c\right ) + a\right )}^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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