Optimal. Leaf size=122 \[ \frac{a^2 b \tan (d+e x) \sqrt{a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2}}{e \left (a^2 \tan (d+e x)+a b\right )}-\frac{\left (a^2+b^2\right ) \log (\cos (d+e x)) \sqrt{a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2}}{e (a \tan (d+e x)+b)} \]
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Rubi [A] time = 0.100776, antiderivative size = 122, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.073, Rules used = {3710, 3525, 3475} \[ \frac{a^2 b \tan (d+e x) \sqrt{a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2}}{e \left (a^2 \tan (d+e x)+a b\right )}-\frac{\left (a^2+b^2\right ) \log (\cos (d+e x)) \sqrt{a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2}}{e (a \tan (d+e x)+b)} \]
Antiderivative was successfully verified.
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Rule 3710
Rule 3525
Rule 3475
Rubi steps
\begin{align*} \int (a+b \tan (d+e x)) \sqrt{b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)} \, dx &=\frac{\sqrt{b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)} \int \left (2 a b+2 a^2 \tan (d+e x)\right ) (a+b \tan (d+e x)) \, dx}{2 a b+2 a^2 \tan (d+e x)}\\ &=\frac{a^2 b \tan (d+e x) \sqrt{b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}}{e \left (a b+a^2 \tan (d+e x)\right )}+\frac{\left (2 a \left (a^2+b^2\right ) \sqrt{b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}\right ) \int \tan (d+e x) \, dx}{2 a b+2 a^2 \tan (d+e x)}\\ &=-\frac{\left (a^2+b^2\right ) \log (\cos (d+e x)) \sqrt{b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}}{e (b+a \tan (d+e x))}+\frac{a^2 b \tan (d+e x) \sqrt{b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)}}{e \left (a b+a^2 \tan (d+e x)\right )}\\ \end{align*}
Mathematica [A] time = 0.288498, size = 58, normalized size = 0.48 \[ \frac{\sqrt{(a \tan (d+e x)+b)^2} \left (a b \tan (d+e x)-\left (a^2+b^2\right ) \log (\cos (d+e x))\right )}{e (a \tan (d+e x)+b)} \]
Antiderivative was successfully verified.
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Maple [C] time = 0.095, size = 75, normalized size = 0.6 \begin{align*}{\frac{{\it csgn} \left ( b+a\tan \left ( ex+d \right ) \right ) \left ( \ln \left ({a}^{2} \left ( \tan \left ( ex+d \right ) \right ) ^{2}+{a}^{2} \right ){a}^{2}+\ln \left ({a}^{2} \left ( \tan \left ( ex+d \right ) \right ) ^{2}+{a}^{2} \right ){b}^{2}+2\,ab\tan \left ( ex+d \right ) +2\,{b}^{2} \right ) }{2\,e}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.55801, size = 88, normalized size = 0.72 \begin{align*} \frac{{\left (2 \,{\left (e x + d\right )} b + a \log \left (\tan \left (e x + d\right )^{2} + 1\right )\right )} a -{\left (2 \,{\left (e x + d\right )} a - b \log \left (\tan \left (e x + d\right )^{2} + 1\right ) - 2 \, a \tan \left (e x + d\right )\right )} b}{2 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 1.73161, size = 95, normalized size = 0.78 \begin{align*} \frac{2 \, a b \tan \left (e x + d\right ) -{\left (a^{2} + b^{2}\right )} \log \left (\frac{1}{\tan \left (e x + d\right )^{2} + 1}\right )}{2 \, e} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (a + b \tan{\left (d + e x \right )}\right ) \sqrt{\left (a \tan{\left (d + e x \right )} + b\right )^{2}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A] time = 1.25722, size = 100, normalized size = 0.82 \begin{align*} a b e^{\left (-1\right )} \mathrm{sgn}\left (a \tan \left (x e + d\right ) + b\right ) \tan \left (x e + d\right ) + \frac{1}{2} \,{\left (a^{2} \mathrm{sgn}\left (a \tan \left (x e + d\right ) + b\right ) + b^{2} \mathrm{sgn}\left (a \tan \left (x e + d\right ) + b\right )\right )} e^{\left (-1\right )} \log \left (\tan \left (x e + d\right )^{2} + 1\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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