Integrand size = 23, antiderivative size = 85 \[ \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=-\frac {\coth ^2(c+d x)}{2 a d}+\frac {\log (\cosh (c+d x))}{(a+b) d}+\frac {(a-b) \log (\tanh (c+d x))}{a^2 d}+\frac {b^2 \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^2 (a+b) d} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3751, 457, 84} \[ \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {b^2 \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^2 d (a+b)}+\frac {(a-b) \log (\tanh (c+d x))}{a^2 d}+\frac {\log (\cosh (c+d x))}{d (a+b)}-\frac {\coth ^2(c+d x)}{2 a d} \]
[In]
[Out]
Rule 84
Rule 457
Rule 3751
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {1}{x^3 \left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \frac {1}{(1-x) x^2 (a+b x)} \, dx,x,\tanh ^2(c+d x)\right )}{2 d} \\ & = \frac {\text {Subst}\left (\int \left (-\frac {1}{(a+b) (-1+x)}+\frac {1}{a x^2}+\frac {a-b}{a^2 x}+\frac {b^3}{a^2 (a+b) (a+b x)}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d} \\ & = -\frac {\coth ^2(c+d x)}{2 a d}+\frac {\log (\cosh (c+d x))}{(a+b) d}+\frac {(a-b) \log (\tanh (c+d x))}{a^2 d}+\frac {b^2 \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^2 (a+b) d} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=-\frac {\frac {\coth ^2(c+d x)}{a}-\frac {b^2 \log \left (b+a \coth ^2(c+d x)\right )}{a^2 (a+b)}-\frac {2 \log (\sinh (c+d x))}{a+b}}{2 d} \]
[In]
[Out]
Time = 0.20 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.04
method | result | size |
parallelrisch | \(\frac {b^{2} \ln \left (a +b \tanh \left (d x +c \right )^{2}\right )-2 \ln \left (1-\tanh \left (d x +c \right )\right ) a^{2}+\left (2 a^{2}-2 b^{2}\right ) \ln \left (\tanh \left (d x +c \right )\right )-\coth \left (d x +c \right )^{2} a \left (a +b \right )-2 a^{2} d x}{2 d \,a^{2} \left (a +b \right )}\) | \(88\) |
derivativedivides | \(-\frac {-\frac {b^{2} \ln \left (a +b \tanh \left (d x +c \right )^{2}\right )}{2 \left (a +b \right ) a^{2}}+\frac {\left (-a +b \right ) \ln \left (\tanh \left (d x +c \right )\right )}{a^{2}}+\frac {1}{2 a \tanh \left (d x +c \right )^{2}}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 a +2 b}+\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 a +2 b}}{d}\) | \(100\) |
default | \(-\frac {-\frac {b^{2} \ln \left (a +b \tanh \left (d x +c \right )^{2}\right )}{2 \left (a +b \right ) a^{2}}+\frac {\left (-a +b \right ) \ln \left (\tanh \left (d x +c \right )\right )}{a^{2}}+\frac {1}{2 a \tanh \left (d x +c \right )^{2}}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2 a +2 b}+\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 a +2 b}}{d}\) | \(100\) |
risch | \(\frac {x}{a +b}-\frac {2 x}{a}-\frac {2 c}{d a}+\frac {2 b x}{a^{2}}+\frac {2 b c}{a^{2} d}-\frac {2 b^{2} x}{a^{2} \left (a +b \right )}-\frac {2 b^{2} c}{d \,a^{2} \left (a +b \right )}-\frac {2 \,{\mathrm e}^{2 d x +2 c}}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d a}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{d \,a^{2}}+\frac {b^{2} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{2 d \,a^{2} \left (a +b \right )}\) | \(191\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 747 vs. \(2 (81) = 162\).
Time = 0.34 (sec) , antiderivative size = 747, normalized size of antiderivative = 8.79 \[ \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=-\frac {2 \, a^{2} d x \cosh \left (d x + c\right )^{4} + 8 \, a^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 2 \, a^{2} d x \sinh \left (d x + c\right )^{4} + 2 \, a^{2} d x - 4 \, {\left (a^{2} d x - a^{2} - a b\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (3 \, a^{2} d x \cosh \left (d x + c\right )^{2} - a^{2} d x + a^{2} + a b\right )} \sinh \left (d x + c\right )^{2} - {\left (b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} - 2 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2} + b^{2} + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} - b^{2} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{2} + {\left (a + b\right )} \sinh \left (d x + c\right )^{2} + a - b\right )}}{\cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}}\right ) - 2 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{2} - a^{2} + b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} - b^{2} + 4 \, {\left ({\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{3} - {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \sinh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 8 \, {\left (a^{2} d x \cosh \left (d x + c\right )^{3} - {\left (a^{2} d x - a^{2} - a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left ({\left (a^{3} + a^{2} b\right )} d \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{3} + a^{2} b\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{3} + a^{2} b\right )} d \sinh \left (d x + c\right )^{4} - 2 \, {\left (a^{3} + a^{2} b\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{3} + a^{2} b\right )} d \cosh \left (d x + c\right )^{2} - {\left (a^{3} + a^{2} b\right )} d\right )} \sinh \left (d x + c\right )^{2} + {\left (a^{3} + a^{2} b\right )} d + 4 \, {\left ({\left (a^{3} + a^{2} b\right )} d \cosh \left (d x + c\right )^{3} - {\left (a^{3} + a^{2} b\right )} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \]
[In]
[Out]
\[ \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {\coth ^{3}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
[In]
[Out]
none
Time = 0.22 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.87 \[ \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {b^{2} \log \left (2 \, {\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{2 \, {\left (a^{3} + a^{2} b\right )} d} + \frac {d x + c}{{\left (a + b\right )} d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (2 \, a e^{\left (-2 \, d x - 2 \, c\right )} - a e^{\left (-4 \, d x - 4 \, c\right )} - a\right )} d} + \frac {{\left (a - b\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} + \frac {{\left (a - b\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} \]
[In]
[Out]
none
Time = 0.35 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.56 \[ \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {\frac {b^{2} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}{a^{3} + a^{2} b} - \frac {2 \, {\left (d x + c\right )}}{a + b} + \frac {2 \, {\left (a - b\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{2}} - \frac {4 \, e^{\left (2 \, d x + 2 \, c\right )}}{a {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \]
[In]
[Out]
Time = 2.24 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.68 \[ \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {b^2\,\ln \left (3\,a\,b^2-2\,a^2\,b-2\,a^3+3\,b^3-4\,a^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-2\,a^3\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}-6\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+3\,b^3\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}+6\,a\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+4\,a^2\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+3\,a\,b^2\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}-2\,a^2\,b\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}\right )}{2\,d\,a^3+2\,b\,d\,a^2}-\frac {x}{a+b}-\frac {2}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {\ln \left (4\,a^4\,b+9\,b^5-12\,a^2\,b^3-9\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-4\,a^4\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+12\,a^2\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (a-b\right )}{a^2\,d}-\frac {2\,\left (a^2+b\,a\right )}{a^2\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )\,\left (a+b\right )} \]
[In]
[Out]