Optimal. Leaf size=101 \[ \frac{a^2 (A b-a B) \log (a+b \tan (c+d x))}{b^2 d \left (a^2+b^2\right )}-\frac{(A b-a B) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac{x (a A+b B)}{a^2+b^2}+\frac{B \tan (c+d x)}{b d} \]
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Rubi [A] time = 0.197371, antiderivative size = 101, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.161, Rules used = {3606, 3626, 3617, 31, 3475} \[ \frac{a^2 (A b-a B) \log (a+b \tan (c+d x))}{b^2 d \left (a^2+b^2\right )}-\frac{(A b-a B) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}-\frac{x (a A+b B)}{a^2+b^2}+\frac{B \tan (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 3606
Rule 3626
Rule 3617
Rule 31
Rule 3475
Rubi steps
\begin{align*} \int \frac{\tan ^2(c+d x) (A+B \tan (c+d x))}{a+b \tan (c+d x)} \, dx &=\frac{B \tan (c+d x)}{b d}+\frac{\int \frac{-a B-b B \tan (c+d x)+(A b-a B) \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b}\\ &=-\frac{(a A+b B) x}{a^2+b^2}+\frac{B \tan (c+d x)}{b d}+\frac{(A b-a B) \int \tan (c+d x) \, dx}{a^2+b^2}+\frac{\left (a^2 (A b-a B)\right ) \int \frac{1+\tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx}{b \left (a^2+b^2\right )}\\ &=-\frac{(a A+b B) x}{a^2+b^2}-\frac{(A b-a B) \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}+\frac{B \tan (c+d x)}{b d}+\frac{\left (a^2 (A b-a B)\right ) \operatorname{Subst}\left (\int \frac{1}{a+x} \, dx,x,b \tan (c+d x)\right )}{b^2 \left (a^2+b^2\right ) d}\\ &=-\frac{(a A+b B) x}{a^2+b^2}-\frac{(A b-a B) \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}+\frac{a^2 (A b-a B) \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right ) d}+\frac{B \tan (c+d x)}{b d}\\ \end{align*}
Mathematica [C] time = 0.555355, size = 118, normalized size = 1.17 \[ \frac{\frac{2 a^2 (A b-a B) \log (a+b \tan (c+d x))}{b^2 \left (a^2+b^2\right )}+\frac{i (A+i B) \log (-\tan (c+d x)+i)}{a+i b}-\frac{(B+i A) \log (\tan (c+d x)+i)}{a-i b}+\frac{2 B \tan (c+d x)}{b}}{2 d} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.035, size = 179, normalized size = 1.8 \begin{align*}{\frac{B\tan \left ( dx+c \right ) }{bd}}+{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) Ab}{2\,d \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) aB}{2\,d \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{A\arctan \left ( \tan \left ( dx+c \right ) \right ) a}{d \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{B\arctan \left ( \tan \left ( dx+c \right ) \right ) b}{d \left ({a}^{2}+{b}^{2} \right ) }}+{\frac{{a}^{2}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) A}{bd \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{B{a}^{3}\ln \left ( a+b\tan \left ( dx+c \right ) \right ) }{ \left ({a}^{2}+{b}^{2} \right ){b}^{2}d}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.49143, size = 147, normalized size = 1.46 \begin{align*} -\frac{\frac{2 \,{\left (A a + B b\right )}{\left (d x + c\right )}}{a^{2} + b^{2}} + \frac{2 \,{\left (B a^{3} - A a^{2} b\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{2} b^{2} + b^{4}} + \frac{{\left (B a - A b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac{2 \, B \tan \left (d x + c\right )}{b}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.97233, size = 333, normalized size = 3.3 \begin{align*} -\frac{2 \,{\left (A a b^{2} + B b^{3}\right )} d x +{\left (B a^{3} - A a^{2} b\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 (B a^{3} - A a^{2} b + B a b^{2} - A b^{3}\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right ) - 2 \,{\left (B a^{2} b + B b^{3}\right )} \tan \left (d x + c\right )}{2 \,{\left (a^{2} b^{2} + b^{4}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 8.95012, size = 1015, normalized size = 10.05 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.44274, size = 149, normalized size = 1.48 \begin{align*} -\frac{\frac{2 \,{\left (A a + B b\right )}{\left (d x + c\right )}}{a^{2} + b^{2}} + \frac{{\left (B a - A b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} + \frac{2 \,{\left (B a^{3} - A a^{2} b\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{2} b^{2} + b^{4}} - \frac{2 \, B \tan \left (d x + c\right )}{b}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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