Optimal. Leaf size=85 \[ -\frac{a (b B-a C) \log (a+b \tan (c+d x))}{b d \left (a^2+b^2\right )}-\frac{(a B+b C) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac{x (b B-a C)}{a^2+b^2} \]
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Rubi [A] time = 0.162711, antiderivative size = 85, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125, Rules used = {1629, 635, 203, 260} \[ -\frac{a (b B-a C) \log (a+b \tan (c+d x))}{b d \left (a^2+b^2\right )}-\frac{(a B+b C) \log (\cos (c+d x))}{d \left (a^2+b^2\right )}+\frac{x (b B-a C)}{a^2+b^2} \]
Antiderivative was successfully verified.
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Rule 1629
Rule 635
Rule 203
Rule 260
Rubi steps
\begin{align*} \int \frac{B \tan (c+d x)+C \tan ^2(c+d x)}{a+b \tan (c+d x)} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{x (B+C x)}{(a+b x) \left (1+x^2\right )} \, dx,x,\tan (c+d x)\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \left (\frac{a (-b B+a C)}{\left (a^2+b^2\right ) (a+b x)}+\frac{b B-a C+(a B+b C) x}{\left (a^2+b^2\right ) \left (1+x^2\right )}\right ) \, dx,x,\tan (c+d x)\right )}{d}\\ &=-\frac{a (b B-a C) \log (a+b \tan (c+d x))}{b \left (a^2+b^2\right ) d}+\frac{\operatorname{Subst}\left (\int \frac{b B-a C+(a B+b C) x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=-\frac{a (b B-a C) \log (a+b \tan (c+d x))}{b \left (a^2+b^2\right ) d}+\frac{(b B-a C) \operatorname{Subst}\left (\int \frac{1}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}+\frac{(a B+b C) \operatorname{Subst}\left (\int \frac{x}{1+x^2} \, dx,x,\tan (c+d x)\right )}{\left (a^2+b^2\right ) d}\\ &=\frac{(b B-a C) x}{a^2+b^2}-\frac{(a B+b C) \log (\cos (c+d x))}{\left (a^2+b^2\right ) d}-\frac{a (b B-a C) \log (a+b \tan (c+d x))}{b \left (a^2+b^2\right ) d}\\ \end{align*}
Mathematica [C] time = 0.169879, size = 98, normalized size = 1.15 \[ \frac{b (a-i b) (B+i C) \log (-\tan (c+d x)+i)+b (a+i b) (B-i C) \log (\tan (c+d x)+i)+2 a (a C-b B) \log (a+b \tan (c+d x))}{2 b d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Maple [A] time = 0.033, size = 159, normalized size = 1.9 \begin{align*}{\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 ) Cb}{2\,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{C\arctan \left ( \tan \left ( dx+c \right ) \right ) a}{d \left ({a}^{2}+{b}^{2} \right ) }}-{\frac{a\ln \left ( a+b\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 ) C}{d \left ({a}^{2}+{b}^{2} \right ) b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 2.41237, size = 127, normalized size = 1.49 \begin{align*} -\frac{\frac{2 \,{\left (C a - B b\right )}{\left (d x + c\right )}}{a^{2} + b^{2}} - \frac{2 \,{\left (C a^{2} - B a b\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{2} b + b^{3}} - \frac{{\left (B a + C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.17741, size = 251, normalized size = 2.95 \begin{align*} -\frac{2 \,{\left (C a b - B b^{2}\right )} d x -{\left (C a^{2} - B a 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 (C a^{2} + C b^{2}\right )} \log \left (\frac{1}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \,{\left (a^{2} b + b^{3}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 7.0537, size = 711, normalized size = 8.36 \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 = 1.61591, size = 128, normalized size = 1.51 \begin{align*} -\frac{\frac{2 \,{\left (C a - B b\right )}{\left (d x + c\right )}}{a^{2} + b^{2}} - \frac{{\left (B a + C b\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{2} + b^{2}} - \frac{2 \,{\left (C a^{2} - B a b\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{2} b + b^{3}}}{2 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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