Optimal. Leaf size=316 \[ -\frac{\left (a^2-b^2\right ) (a \tan (d+e x)+b)}{2 e \left (a^2+b^2\right ) \left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}-\frac{4 b x \left (a^2-b^2\right ) \left (a^2 \tan (d+e x)+a b\right )^3}{a^2 \left (a^2+b^2\right )^3 \left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}-\frac{b \left (3 a^2-b^2\right ) \left (a^2 \tan (d+e x)+a b\right )^3}{e \left (a^2+b^2\right )^2 \left (a^3 b+a^4 \tan (d+e x)\right ) \left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) (a \tan (d+e x)+b)^3 \log (a \sin (d+e x)+b \cos (d+e x))}{e \left (a^2+b^2\right )^3 \left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}} \]
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Rubi [A] time = 0.401571, antiderivative size = 316, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 41, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.098, Rules used = {3710, 3529, 3531, 3530} \[ -\frac{\left (a^2-b^2\right ) (a \tan (d+e x)+b)}{2 e \left (a^2+b^2\right ) \left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}-\frac{4 b x \left (a^2-b^2\right ) \left (a^2 \tan (d+e x)+a b\right )^3}{a^2 \left (a^2+b^2\right )^3 \left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}-\frac{b \left (3 a^2-b^2\right ) \left (a^2 \tan (d+e x)+a b\right )^3}{e \left (a^2+b^2\right )^2 \left (a^3 b+a^4 \tan (d+e x)\right ) \left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}}-\frac{\left (-6 a^2 b^2+a^4+b^4\right ) (a \tan (d+e x)+b)^3 \log (a \sin (d+e x)+b \cos (d+e x))}{e \left (a^2+b^2\right )^3 \left (a^2 \tan ^2(d+e x)+2 a b \tan (d+e x)+b^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 3710
Rule 3529
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{a+b \tan (d+e x)}{\left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}} \, dx &=\frac{\left (2 a b+2 a^2 \tan (d+e x)\right )^3 \int \frac{a+b \tan (d+e x)}{\left (2 a b+2 a^2 \tan (d+e x)\right )^3} \, dx}{\left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}\\ &=-\frac{\left (a^2-b^2\right ) (b+a \tan (d+e x))}{2 \left (a^2+b^2\right ) e \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}+\frac{\left (2 a b+2 a^2 \tan (d+e x)\right )^3 \int \frac{4 a^2 b-2 a \left (a^2-b^2\right ) \tan (d+e x)}{\left (2 a b+2 a^2 \tan (d+e x)\right )^2} \, dx}{4 a^2 \left (a^2+b^2\right ) \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}\\ &=-\frac{\left (a^2-b^2\right ) (b+a \tan (d+e x))}{2 \left (a^2+b^2\right ) e \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}-\frac{b \left (3 a^2-b^2\right ) (b+a \tan (d+e x))^2}{\left (a^2+b^2\right )^2 e \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}+\frac{\left (2 a b+2 a^2 \tan (d+e x)\right )^3 \int \frac{-4 a^3 \left (a^2-3 b^2\right )-4 a^2 b \left (3 a^2-b^2\right ) \tan (d+e x)}{2 a b+2 a^2 \tan (d+e x)} \, dx}{16 a^4 \left (a^2+b^2\right )^2 \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}\\ &=-\frac{\left (a^2-b^2\right ) (b+a \tan (d+e x))}{2 \left (a^2+b^2\right ) e \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}-\frac{b \left (3 a^2-b^2\right ) (b+a \tan (d+e x))^2}{\left (a^2+b^2\right )^2 e \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}-\frac{4 b \left (a^2-b^2\right ) x \left (a b+a^2 \tan (d+e x)\right )^3}{a^2 \left (a^2+b^2\right )^3 \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}-\frac{\left (\left (a^4-6 a^2 b^2+b^4\right ) \left (2 a b+2 a^2 \tan (d+e x)\right )^3\right ) \int \frac{2 a^2-2 a b \tan (d+e x)}{2 a b+2 a^2 \tan (d+e x)} \, dx}{8 a^3 \left (a^2+b^2\right )^3 \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}\\ &=-\frac{\left (a^2-b^2\right ) (b+a \tan (d+e x))}{2 \left (a^2+b^2\right ) e \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}-\frac{b \left (3 a^2-b^2\right ) (b+a \tan (d+e x))^2}{\left (a^2+b^2\right )^2 e \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}-\frac{\left (a^4-6 a^2 b^2+b^4\right ) \log (b \cos (d+e x)+a \sin (d+e x)) (b+a \tan (d+e x))^3}{\left (a^2+b^2\right )^3 e \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}-\frac{4 b \left (a^2-b^2\right ) x \left (a b+a^2 \tan (d+e x)\right )^3}{a^2 \left (a^2+b^2\right )^3 \left (b^2+2 a b \tan (d+e x)+a^2 \tan ^2(d+e x)\right )^{3/2}}\\ \end{align*}
Mathematica [C] time = 3.18623, size = 268, normalized size = 0.85 \[ \frac{(a \tan (d+e x)+b)^3 \left (b \left (\frac{2 a \left (2 b \log (a \tan (d+e x)+b)-\frac{a^2+b^2}{a \tan (d+e x)+b}\right )}{\left (a^2+b^2\right )^2}+\frac{i \log (-\tan (d+e x)+i)}{(a-i b)^2}-\frac{i \log (\tan (d+e x)+i)}{(a+i b)^2}\right )+(a-b) (a+b) \left (\frac{a \left (-\frac{\left (a^2+b^2\right ) \left (a^2+4 a b \tan (d+e x)+5 b^2\right )}{(a \tan (d+e x)+b)^2}-2 \left (a^2-3 b^2\right ) \log (a \tan (d+e x)+b)\right )}{\left (a^2+b^2\right )^3}+\frac{\log (-\tan (d+e x)+i)}{(a-i b)^3}+\frac{\log (\tan (d+e x)+i)}{(a+i b)^3}\right )\right )}{2 a e \left ((a \tan (d+e x)+b)^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Maple [B] time = 0.094, size = 622, normalized size = 2. \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.54713, size = 672, normalized size = 2.13 \begin{align*} -\frac{{\left (\frac{2 \,{\left (3 \, a^{2} b - b^{3}\right )}{\left (e x + d\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{2 \,{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (a \tan \left (e x + d\right ) + b\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (a^{3} - 3 \, a b^{2}\right )} \log \left (\tan \left (e x + d\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{4 \, a^{2} b \tan \left (e x + d\right ) + a^{3} + 5 \, a b^{2}}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6} +{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \tan \left (e x + d\right )^{2} + 2 \,{\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (e x + d\right )}\right )} a +{\left (\frac{2 \,{\left (a^{3} - 3 \, a b^{2}\right )}{\left (e x + d\right )}}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{2 \,{\left (3 \, a^{2} b - b^{3}\right )} \log \left (a \tan \left (e x + d\right ) + b\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{{\left (3 \, a^{2} b - b^{3}\right )} \log \left (\tan \left (e x + d\right )^{2} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac{a^{2} b - 3 \, b^{3} + 2 \,{\left (a^{3} - a b^{2}\right )} \tan \left (e x + d\right )}{a^{4} b^{2} + 2 \, a^{2} b^{4} + b^{6} +{\left (a^{6} + 2 \, a^{4} b^{2} + a^{2} b^{4}\right )} \tan \left (e x + d\right )^{2} + 2 \,{\left (a^{5} b + 2 \, a^{3} b^{3} + a b^{5}\right )} \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 = 1.73784, size = 764, normalized size = 2.42 \begin{align*} -\frac{a^{6} + 8 \, a^{4} b^{2} - 5 \, a^{2} b^{4} + 8 \,{\left (a^{3} b^{3} - a b^{5}\right )} e x +{\left (a^{6} - 8 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + 8 \,{\left (a^{5} b - a^{3} b^{3}\right )} e x\right )} \tan \left (e x + d\right )^{2} +{\left (a^{4} b^{2} - 6 \, a^{2} b^{4} + b^{6} +{\left (a^{6} - 6 \, a^{4} b^{2} + a^{2} b^{4}\right )} \tan \left (e x + d\right )^{2} + 2 \,{\left (a^{5} b - 6 \, a^{3} b^{3} + a b^{5}\right )} \tan \left (e x + d\right )\right )} \log \left (\frac{a^{2} \tan \left (e x + d\right )^{2} + 2 \, a b \tan \left (e x + d\right ) + b^{2}}{\tan \left (e x + d\right )^{2} + 1}\right ) + 4 \,{\left (2 \, a^{5} b - 3 \, a^{3} b^{3} + a b^{5} + 4 \,{\left (a^{4} b^{2} - a^{2} b^{4}\right )} e x\right )} \tan \left (e x + d\right )}{2 \,{\left ({\left (a^{8} + 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} + a^{2} b^{6}\right )} e \tan \left (e x + d\right )^{2} + 2 \,{\left (a^{7} b + 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} + a b^{7}\right )} e \tan \left (e x + d\right ) +{\left (a^{6} b^{2} + 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} + b^{8}\right )} e\right )}} \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 \frac{a + b \tan{\left (d + e x \right )}}{\left (\left (a \tan{\left (d + e x \right )} + b\right )^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] time = 3.00569, size = 2121, normalized size = 6.71 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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