Integrand size = 16, antiderivative size = 113 \[ \int e^{a+b x} \coth ^4(a+b x) \, dx=\frac {e^{a+b x}}{b}+\frac {8 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {14 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^2}+\frac {5 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {3 \text {arctanh}\left (e^{a+b x}\right )}{b} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {2320, 398, 1272, 1171, 393, 212} \[ \int e^{a+b x} \coth ^4(a+b x) \, dx=-\frac {3 \text {arctanh}\left (e^{a+b x}\right )}{b}+\frac {e^{a+b x}}{b}+\frac {5 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {14 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^2}+\frac {8 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3} \]
[In]
[Out]
Rule 212
Rule 393
Rule 398
Rule 1171
Rule 1272
Rule 2320
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (1+x^2\right )^4}{\left (1-x^2\right )^4} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {\text {Subst}\left (\int \left (1+\frac {8 x^2 \left (1+x^4\right )}{\left (1-x^2\right )^4}\right ) \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {e^{a+b x}}{b}+\frac {8 \text {Subst}\left (\int \frac {x^2 \left (1+x^4\right )}{\left (1-x^2\right )^4} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {e^{a+b x}}{b}+\frac {8 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}+\frac {4 \text {Subst}\left (\int \frac {-2-6 x^2-6 x^4}{\left (1-x^2\right )^3} \, dx,x,e^{a+b x}\right )}{3 b} \\ & = \frac {e^{a+b x}}{b}+\frac {8 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {14 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^2}-\frac {\text {Subst}\left (\int \frac {-6-24 x^2}{\left (1-x^2\right )^2} \, dx,x,e^{a+b x}\right )}{3 b} \\ & = \frac {e^{a+b x}}{b}+\frac {8 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {14 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^2}+\frac {5 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {3 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,e^{a+b x}\right )}{b} \\ & = \frac {e^{a+b x}}{b}+\frac {8 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^3}-\frac {14 e^{a+b x}}{3 b \left (1-e^{2 a+2 b x}\right )^2}+\frac {5 e^{a+b x}}{b \left (1-e^{2 a+2 b x}\right )}-\frac {3 \text {arctanh}\left (e^{a+b x}\right )}{b} \\ \end{align*}
Time = 10.13 (sec) , antiderivative size = 115, normalized size of antiderivative = 1.02 \[ \int e^{a+b x} \coth ^4(a+b x) \, dx=\frac {-24 e^{a+b x}+50 e^{3 (a+b x)}-48 e^{5 (a+b x)}+6 e^{7 (a+b x)}+9 \left (-1+e^{2 (a+b x)}\right )^3 \log \left (1-e^{a+b x}\right )-9 \left (-1+e^{2 (a+b x)}\right )^3 \log \left (1+e^{a+b x}\right )}{6 b \left (-1+e^{2 (a+b x)}\right )^3} \]
[In]
[Out]
Time = 0.36 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.78
method | result | size |
risch | \(\frac {{\mathrm e}^{b x +a}}{b}-\frac {{\mathrm e}^{b x +a} \left (15 \,{\mathrm e}^{4 b x +4 a}-16 \,{\mathrm e}^{2 b x +2 a}+9\right )}{3 b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{3}}-\frac {3 \ln \left ({\mathrm e}^{b x +a}+1\right )}{2 b}+\frac {3 \ln \left ({\mathrm e}^{b x +a}-1\right )}{2 b}\) | \(88\) |
derivativedivides | \(\frac {\frac {\cosh \left (b x +a \right )^{3}}{\sinh \left (b x +a \right )^{2}}-\frac {3 \cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{2}}+\frac {3 \,\operatorname {csch}\left (b x +a \right ) \coth \left (b x +a \right )}{2}-3 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )+\frac {\cosh \left (b x +a \right )^{4}}{\sinh \left (b x +a \right )^{3}}-\frac {4 \cosh \left (b x +a \right )^{2}}{\sinh \left (b x +a \right )^{3}}+\frac {8}{3 \sinh \left (b x +a \right )^{3}}}{b}\) | \(107\) |
default | \(\frac {\frac {\cosh \left (b x +a \right )^{3}}{\sinh \left (b x +a \right )^{2}}-\frac {3 \cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{2}}+\frac {3 \,\operatorname {csch}\left (b x +a \right ) \coth \left (b x +a \right )}{2}-3 \,\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )+\frac {\cosh \left (b x +a \right )^{4}}{\sinh \left (b x +a \right )^{3}}-\frac {4 \cosh \left (b x +a \right )^{2}}{\sinh \left (b x +a \right )^{3}}+\frac {8}{3 \sinh \left (b x +a \right )^{3}}}{b}\) | \(107\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 796 vs. \(2 (95) = 190\).
Time = 0.26 (sec) , antiderivative size = 796, normalized size of antiderivative = 7.04 \[ \int e^{a+b x} \coth ^4(a+b x) \, dx=\text {Too large to display} \]
[In]
[Out]
\[ \int e^{a+b x} \coth ^4(a+b x) \, dx=e^{a} \int e^{b x} \coth ^{4}{\left (a + b x \right )}\, dx \]
[In]
[Out]
none
Time = 0.21 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.97 \[ \int e^{a+b x} \coth ^4(a+b x) \, dx=\frac {e^{\left (b x + a\right )}}{b} - \frac {3 \, \log \left (e^{\left (b x + a\right )} + 1\right )}{2 \, b} + \frac {3 \, \log \left (e^{\left (b x + a\right )} - 1\right )}{2 \, b} - \frac {15 \, e^{\left (5 \, b x + 5 \, a\right )} - 16 \, e^{\left (3 \, b x + 3 \, a\right )} + 9 \, e^{\left (b x + a\right )}}{3 \, b {\left (e^{\left (6 \, b x + 6 \, a\right )} - 3 \, e^{\left (4 \, b x + 4 \, a\right )} + 3 \, e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}} \]
[In]
[Out]
none
Time = 0.29 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.73 \[ \int e^{a+b x} \coth ^4(a+b x) \, dx=-\frac {\frac {2 \, {\left (15 \, e^{\left (5 \, b x + 5 \, a\right )} - 16 \, e^{\left (3 \, b x + 3 \, a\right )} + 9 \, e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{3}} - 6 \, e^{\left (b x + a\right )} + 9 \, \log \left (e^{\left (b x + a\right )} + 1\right ) - 9 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{6 \, b} \]
[In]
[Out]
Time = 0.07 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.42 \[ \int e^{a+b x} \coth ^4(a+b x) \, dx=\frac {{\mathrm {e}}^{a+b\,x}}{b}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {\frac {4\,{\mathrm {e}}^{a+b\,x}}{3\,b}+\frac {4\,{\mathrm {e}}^{5\,a+5\,b\,x}}{3\,b}}{3\,{\mathrm {e}}^{2\,a+2\,b\,x}-3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}-1}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {11\,{\mathrm {e}}^{a+b\,x}}{3\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
[In]
[Out]