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 = 0.10, antiderivative size = 122, normalized size of antiderivative = 1.00, 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 3475
Rule 3525
Rule 3710
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*}
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Mathematica [A] time = 0.30, 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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fricas [A] time = 2.63, size = 38, normalized size = 0.31 \[ \frac {2 \, a b \tan \left (e x + d\right ) - {\left (a^{2} + b^{2}\right )} \log \left (\frac {1}{\tan \left (e x + d\right )^{2} + 1}\right )}{2 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.46, size = 395, normalized size = 3.24 \[ -\frac {a^{2} \log \left (\frac {4 \, {\left (\tan \left (x e\right )^{4} \tan \relax (d)^{2} - 2 \, \tan \left (x e\right )^{3} \tan \relax (d) + \tan \left (x e\right )^{2} \tan \relax (d)^{2} + \tan \left (x e\right )^{2} - 2 \, \tan \left (x e\right ) \tan \relax (d) + 1\right )}}{\tan \relax (d)^{2} + 1}\right ) \mathrm {sgn}\left (a \tan \left (x e + d\right ) + b\right ) \tan \left (x e\right ) \tan \relax (d) + b^{2} \log \left (\frac {4 \, {\left (\tan \left (x e\right )^{4} \tan \relax (d)^{2} - 2 \, \tan \left (x e\right )^{3} \tan \relax (d) + \tan \left (x e\right )^{2} \tan \relax (d)^{2} + \tan \left (x e\right )^{2} - 2 \, \tan \left (x e\right ) \tan \relax (d) + 1\right )}}{\tan \relax (d)^{2} + 1}\right ) \mathrm {sgn}\left (a \tan \left (x e + d\right ) + b\right ) \tan \left (x e\right ) \tan \relax (d) - a^{2} \log \left (\frac {4 \, {\left (\tan \left (x e\right )^{4} \tan \relax (d)^{2} - 2 \, \tan \left (x e\right )^{3} \tan \relax (d) + \tan \left (x e\right )^{2} \tan \relax (d)^{2} + \tan \left (x e\right )^{2} - 2 \, \tan \left (x e\right ) \tan \relax (d) + 1\right )}}{\tan \relax (d)^{2} + 1}\right ) \mathrm {sgn}\left (a \tan \left (x e + d\right ) + b\right ) - b^{2} \log \left (\frac {4 \, {\left (\tan \left (x e\right )^{4} \tan \relax (d)^{2} - 2 \, \tan \left (x e\right )^{3} \tan \relax (d) + \tan \left (x e\right )^{2} \tan \relax (d)^{2} + \tan \left (x e\right )^{2} - 2 \, \tan \left (x e\right ) \tan \relax (d) + 1\right )}}{\tan \relax (d)^{2} + 1}\right ) \mathrm {sgn}\left (a \tan \left (x e + d\right ) + b\right ) + 2 \, a b \mathrm {sgn}\left (a \tan \left (x e + d\right ) + b\right ) \tan \left (x e\right ) + 2 \, a b \mathrm {sgn}\left (a \tan \left (x e + d\right ) + b\right ) \tan \relax (d)}{2 \, {\left (e \tan \left (x e\right ) \tan \relax (d) - e\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.32, size = 75, normalized size = 0.61 \[ \frac {\mathrm {csgn}\left (b +a \tan \left (e x +d \right )\right ) \left (\ln \left (a^{2} \left (\tan ^{2}\left (e x +d \right )\right )+a^{2}\right ) a^{2}+\ln \left (a^{2} \left (\tan ^{2}\left (e x +d \right )\right )+a^{2}\right ) b^{2}+2 a b \tan \left (e x +d \right )+2 b^{2}\right )}{2 e} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.42, size = 65, normalized size = 0.53 \[ \frac {{\left (2 \, {\left (e x + d\right )} b + a \log \left (\tan \left (e x + d\right )^{2} + 1\right )\right )} a - {\left (2 \, {\left (e x + d\right )} a - b \log \left (\tan \left (e x + d\right )^{2} + 1\right ) - 2 \, a \tan \left (e x + d\right )\right )} b}{2 \, e} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \left (a+b\,\mathrm {tan}\left (d+e\,x\right )\right )\,\sqrt {a^2\,{\mathrm {tan}\left (d+e\,x\right )}^2+2\,a\,b\,\mathrm {tan}\left (d+e\,x\right )+b^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tan {\left (d + e x \right )}\right ) \sqrt {\left (a \tan {\left (d + e x \right )} + b\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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