Optimal. Leaf size=129 \[ \frac{2 a b}{d \left (a^2-b^2\right )^2 (a+b \coth (c+d x))}+\frac{b}{2 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^2}-\frac{b \left (3 a^2+b^2\right ) \log (a \sinh (c+d x)+b \cosh (c+d x))}{d \left (a^2-b^2\right )^3}+\frac{a x \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.179718, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {3483, 3529, 3531, 3530} \[ \frac{2 a b}{d \left (a^2-b^2\right )^2 (a+b \coth (c+d x))}+\frac{b}{2 d \left (a^2-b^2\right ) (a+b \coth (c+d x))^2}-\frac{b \left (3 a^2+b^2\right ) \log (a \sinh (c+d x)+b \cosh (c+d x))}{d \left (a^2-b^2\right )^3}+\frac{a x \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3483
Rule 3529
Rule 3531
Rule 3530
Rubi steps
\begin{align*} \int \frac{1}{(a+b \coth (c+d x))^3} \, dx &=\frac{b}{2 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^2}+\frac{\int \frac{a-b \coth (c+d x)}{(a+b \coth (c+d x))^2} \, dx}{a^2-b^2}\\ &=\frac{b}{2 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^2}+\frac{2 a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))}+\frac{\int \frac{a^2+b^2-2 a b \coth (c+d x)}{a+b \coth (c+d x)} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac{a \left (a^2+3 b^2\right ) x}{\left (a^2-b^2\right )^3}+\frac{b}{2 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^2}+\frac{2 a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))}-\frac{\left (i b \left (3 a^2+b^2\right )\right ) \int \frac{-i b-i a \coth (c+d x)}{a+b \coth (c+d x)} \, dx}{\left (a^2-b^2\right )^3}\\ &=\frac{a \left (a^2+3 b^2\right ) x}{\left (a^2-b^2\right )^3}+\frac{b}{2 \left (a^2-b^2\right ) d (a+b \coth (c+d x))^2}+\frac{2 a b}{\left (a^2-b^2\right )^2 d (a+b \coth (c+d x))}-\frac{b \left (3 a^2+b^2\right ) \log (b \cosh (c+d x)+a \sinh (c+d x))}{\left (a^2-b^2\right )^3 d}\\ \end{align*}
Mathematica [A] time = 3.43228, size = 134, normalized size = 1.04 \[ -\frac{\frac{b \left (\frac{b \left (b^2-a^2\right ) \left (\left (2 a b^2-6 a^3\right ) \tanh (c+d x)-5 a^2 b+b^3\right )}{a^2 (a \tanh (c+d x)+b)^2}+2 \left (3 a^2+b^2\right ) \log (a \tanh (c+d x)+b)\right )}{\left (a^2-b^2\right )^3}+\frac{\log (1-\tanh (c+d x))}{(a+b)^3}-\frac{\log (\tanh (c+d x)+1)}{(a-b)^3}}{2 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.031, size = 166, normalized size = 1.3 \begin{align*}{\frac{\ln \left ({\rm coth} \left (dx+c\right )+1 \right ) }{2\,d \left ( a-b \right ) ^{3}}}-{\frac{\ln \left ({\rm coth} \left (dx+c\right )-1 \right ) }{2\,d \left ( a+b \right ) ^{3}}}+{\frac{b}{2\,d \left ( a-b \right ) \left ( a+b \right ) \left ( a+b{\rm coth} \left (dx+c\right ) \right ) ^{2}}}+2\,{\frac{ab}{d \left ( a+b \right ) ^{2} \left ( a-b \right ) ^{2} \left ( a+b{\rm coth} \left (dx+c\right ) \right ) }}-3\,{\frac{b\ln \left ( a+b{\rm coth} \left (dx+c\right ) \right ){a}^{2}}{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3}}}-{\frac{{b}^{3}\ln \left ( a+b{\rm coth} \left (dx+c\right ) \right ) }{d \left ( a+b \right ) ^{3} \left ( a-b \right ) ^{3}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] time = 1.18432, size = 435, normalized size = 3.37 \begin{align*} -\frac{{\left (3 \, a^{2} b + b^{3}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a + b\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} d} - \frac{2 \,{\left (3 \, a^{2} b^{2} + 3 \, a b^{3} -{\left (3 \, a^{2} b^{2} - 2 \, a b^{3} - b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )}}{{\left (a^{7} + a^{6} b - 3 \, a^{5} b^{2} - 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} + 3 \, a^{2} b^{5} - a b^{6} - b^{7} - 2 \,{\left (a^{7} - a^{6} b - 3 \, a^{5} b^{2} + 3 \, a^{4} b^{3} + 3 \, a^{3} b^{4} - 3 \, a^{2} b^{5} - a b^{6} + b^{7}\right )} e^{\left (-2 \, d x - 2 \, c\right )} +{\left (a^{7} - 3 \, a^{6} b + a^{5} b^{2} + 5 \, a^{4} b^{3} - 5 \, a^{3} b^{4} - a^{2} b^{5} + 3 \, a b^{6} - b^{7}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} + \frac{d x + c}{{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] time = 3.20442, size = 3182, normalized size = 24.67 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.18017, size = 285, normalized size = 2.21 \begin{align*} -\frac{{\left (3 \, a^{2} b + b^{3}\right )} \log \left ({\left | a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} - a + b \right |}\right )}{a^{6} d - 3 \, a^{4} b^{2} d + 3 \, a^{2} b^{4} d - b^{6} d} + \frac{d x + c}{a^{3} d - 3 \, a^{2} b d + 3 \, a b^{2} d - b^{3} d} - \frac{2 \,{\left ({\left (3 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (2 \, d x + 2 \, c\right )} - \frac{3 \,{\left (a^{3} b^{2} - 2 \, a^{2} b^{3} + a b^{4}\right )}}{a + b}\right )}}{{\left (a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} - a + b\right )}^{2}{\left (a + b\right )}^{2}{\left (a - b\right )}^{3} d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]