Optimal. Leaf size=58 \[ \frac {2 b B x}{a^2+b^2}-\frac {B \left (a-\frac {b^2}{a}\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )} \]
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Rubi [A] time = 0.08, antiderivative size = 58, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {3531, 3530} \[ \frac {2 b B x}{a^2+b^2}-\frac {B \left (a-\frac {b^2}{a}\right ) \log (a \cos (c+d x)+b \sin (c+d x))}{d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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Rule 3530
Rule 3531
Rubi steps
\begin {align*} \int \frac {\frac {b B}{a}+B \tan (c+d x)}{a+b \tan (c+d x)} \, dx &=\frac {2 b B x}{a^2+b^2}-\frac {\left (\left (a-\frac {b^2}{a}\right ) B\right ) \int \frac {b-a \tan (c+d x)}{a+b \tan (c+d x)} \, dx}{a^2+b^2}\\ &=\frac {2 b B x}{a^2+b^2}-\frac {\left (a-\frac {b^2}{a}\right ) B \log (a \cos (c+d x)+b \sin (c+d x))}{\left (a^2+b^2\right ) d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 65, normalized size = 1.12 \[ \frac {B \left (\left (a^2-b^2\right ) \left (\log \left (\sec ^2(c+d x)\right )-2 \log (a+b \tan (c+d x))\right )+4 a b \tan ^{-1}(\tan (c+d x))\right )}{2 a d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.59, size = 78, normalized size = 1.34 \[ \frac {4 \, B a b d x - {\left (B a^{2} - B b^{2}\right )} \log \left (\frac {b^{2} \tan \left (d x + c\right )^{2} + 2 \, a b \tan \left (d x + c\right ) + a^{2}}{\tan \left (d x + c\right )^{2} + 1}\right )}{2 \, {\left (a^{3} + a b^{2}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.29, size = 99, normalized size = 1.71 \[ \frac {\frac {4 \, {\left (d x + c\right )} B b}{a^{2} + b^{2}} + \frac {{\left (B a^{2} - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{3} + a b^{2}} - \frac {2 \, {\left (B a^{2} b - B b^{3}\right )} \log \left ({\left | b \tan \left (d x + c\right ) + a \right |}\right )}{a^{3} b + a b^{3}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.26, size = 142, normalized size = 2.45 \[ -\frac {B a \ln \left (a +b \tan \left (d x +c \right )\right )}{d \left (a^{2}+b^{2}\right )}+\frac {b^{2} \ln \left (a +b \tan \left (d x +c \right )\right ) B}{d a \left (a^{2}+b^{2}\right )}+\frac {\ln \left (1+\tan ^{2}\left (d x +c \right )\right ) a B}{2 d \left (a^{2}+b^{2}\right )}-\frac {B \ln \left (1+\tan ^{2}\left (d x +c \right )\right ) b^{2}}{2 d a \left (a^{2}+b^{2}\right )}+\frac {2 B \arctan \left (\tan \left (d x +c \right )\right ) b}{d \left (a^{2}+b^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.50, size = 95, normalized size = 1.64 \[ \frac {\frac {4 \, {\left (d x + c\right )} B b}{a^{2} + b^{2}} - \frac {2 \, {\left (B a^{2} - B b^{2}\right )} \log \left (b \tan \left (d x + c\right ) + a\right )}{a^{3} + a b^{2}} + \frac {{\left (B a^{2} - B b^{2}\right )} \log \left (\tan \left (d x + c\right )^{2} + 1\right )}{a^{3} + a b^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 6.57, size = 112, normalized size = 1.93 \[ -\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}\right )\,\left (B\,b+B\,a\,1{}\mathrm {i}\right )}{2\,d\,\left (a\,b-a^2\,1{}\mathrm {i}\right )}-\frac {\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,\left (B\,a+B\,b\,1{}\mathrm {i}\right )}{2\,d\,\left (-a^2+a\,b\,1{}\mathrm {i}\right )}-\frac {B\,\ln \left (a+b\,\mathrm {tan}\left (c+d\,x\right )\right )\,\left (a^2-b^2\right )}{a\,d\,\left (a^2+b^2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.87, size = 235, normalized size = 4.05 \[ \begin {cases} \text {NaN} & \text {for}\: a = 0 \wedge b = 0 \wedge d = 0 \\- \frac {B}{b d \tan {\left (c + d x \right )} - i b d} & \text {for}\: a = - i b \\- \frac {B}{b d \tan {\left (c + d x \right )} + i b d} & \text {for}\: a = i b \\\frac {x \left (B \tan {\relax (c )} + \frac {B b}{a}\right )}{a + b \tan {\relax (c )}} & \text {for}\: d = 0 \\\frac {B \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a d} & \text {for}\: b = 0 \\- \frac {2 B a^{2} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{3} d + 2 a b^{2} d} + \frac {B a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{3} d + 2 a b^{2} d} + \frac {4 B a b d x}{2 a^{3} d + 2 a b^{2} d} + \frac {2 B b^{2} \log {\left (\frac {a}{b} + \tan {\left (c + d x \right )} \right )}}{2 a^{3} d + 2 a b^{2} d} - \frac {B b^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 a^{3} d + 2 a b^{2} d} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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