Integrand size = 23, antiderivative size = 85 \[ \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=-\frac {\coth ^2(c+d x)}{2 a d}+\frac {\log (\cosh (c+d x))}{(a+b) d}+\frac {(a-b) \log (\tanh (c+d x))}{a^2 d}+\frac {b^2 \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^2 (a+b) d} \]
-1/2*coth(d*x+c)^2/a/d+ln(cosh(d*x+c))/(a+b)/d+(a-b)*ln(tanh(d*x+c))/a^2/d +1/2*b^2*ln(a+b*tanh(d*x+c)^2)/a^2/(a+b)/d
Time = 0.21 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.71 \[ \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=-\frac {\frac {\coth ^2(c+d x)}{a}-\frac {b^2 \log \left (b+a \coth ^2(c+d x)\right )}{a^2 (a+b)}-\frac {2 \log (\sinh (c+d x))}{a+b}}{2 d} \]
-1/2*(Coth[c + d*x]^2/a - (b^2*Log[b + a*Coth[c + d*x]^2])/(a^2*(a + b)) - (2*Log[Sinh[c + d*x]])/(a + b))/d
Time = 0.33 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.96, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {3042, 26, 4153, 26, 354, 93, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {i}{\tan (i c+i d x)^3 \left (a-b \tan (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -i \int \frac {1}{\tan (i c+i d x)^3 \left (a-b \tan (i c+i d x)^2\right )}dx\) |
| \(\Big \downarrow \) 4153 |
| \(\displaystyle -\frac {i \int \frac {i \coth ^3(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\int \frac {\coth ^3(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )}d\tanh (c+d x)}{d}\) |
| \(\Big \downarrow \) 354 |
| \(\displaystyle \frac {\int \frac {\coth ^2(c+d x)}{\left (1-\tanh ^2(c+d x)\right ) \left (b \tanh ^2(c+d x)+a\right )}d\tanh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 93 |
| \(\displaystyle \frac {\int \left (\frac {b^3}{a^2 (a+b) \left (b \tanh ^2(c+d x)+a\right )}+\frac {\coth ^2(c+d x)}{a}+\frac {(a-b) \coth (c+d x)}{a^2}-\frac {1}{(a+b) \left (\tanh ^2(c+d x)-1\right )}\right )d\tanh ^2(c+d x)}{2 d}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {b^2 \log \left (a+b \tanh ^2(c+d x)\right )}{a^2 (a+b)}+\frac {(a-b) \log \left (\tanh ^2(c+d x)\right )}{a^2}-\frac {\log \left (1-\tanh ^2(c+d x)\right )}{a+b}-\frac {\coth (c+d x)}{a}}{2 d}\) |
(-(Coth[c + d*x]/a) + ((a - b)*Log[Tanh[c + d*x]^2])/a^2 - Log[1 - Tanh[c + d*x]^2]/(a + b) + (b^2*Log[a + b*Tanh[c + d*x]^2])/(a^2*(a + b)))/(2*d)
3.2.78.3.1 Defintions of rubi rules used
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
Int[((e_.) + (f_.)*(x_))^(p_)/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_] :> Int[ExpandIntegrand[(e + f*x)^p/((a + b*x)*(c + d*x)), x], x] /; Fre eQ[{a, b, c, d, e, f}, x] && IntegerQ[p]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S ymbol] :> Simp[1/2 Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x , x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ [(m - 1)/2]
Int[((d_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_.), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[c*(ff/f) Subst[Int[(d*ff*(x/c))^m*((a + b*(ff*x)^n)^p/(c^2 + f f^2*x^2)), x], x, c*(Tan[e + f*x]/ff)], x]] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && (IGtQ[p, 0] || EqQ[n, 2] || EqQ[n, 4] || (IntegerQ[p] && Ratio nalQ[n]))
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 | \(\frac {x}{a +b}-\frac {2 x}{a}-\frac {2 c}{d a}+\frac {2 b x}{a^{2}}+\frac {2 b c}{a^{2} d}-\frac {2 b^{2} x}{a^{2} \left (a +b \right )}-\frac {2 b^{2} c}{d \,a^{2} \left (a +b \right )}-\frac {2 \,{\mathrm e}^{2 d x +2 c}}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{d a}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{d \,a^{2}}+\frac {b^{2} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{2 d \,a^{2} \left (a +b \right )}\) | \(191\) |
1/2*(b^2*ln(a+b*tanh(d*x+c)^2)-2*ln(1-tanh(d*x+c))*a^2+(2*a^2-2*b^2)*ln(ta nh(d*x+c))-coth(d*x+c)^2*a*(a+b)-2*a^2*d*x)/d/a^2/(a+b)
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 )}} \]
-1/2*(2*a^2*d*x*cosh(d*x + c)^4 + 8*a^2*d*x*cosh(d*x + c)*sinh(d*x + c)^3 + 2*a^2*d*x*sinh(d*x + c)^4 + 2*a^2*d*x - 4*(a^2*d*x - a^2 - a*b)*cosh(d*x + c)^2 + 4*(3*a^2*d*x*cosh(d*x + c)^2 - a^2*d*x + a^2 + a*b)*sinh(d*x + c )^2 - (b^2*cosh(d*x + c)^4 + 4*b^2*cosh(d*x + c)*sinh(d*x + c)^3 + b^2*sin h(d*x + c)^4 - 2*b^2*cosh(d*x + c)^2 + 2*(3*b^2*cosh(d*x + c)^2 - b^2)*sin h(d*x + c)^2 + b^2 + 4*(b^2*cosh(d*x + c)^3 - b^2*cosh(d*x + c))*sinh(d*x + c))*log(2*((a + b)*cosh(d*x + c)^2 + (a + b)*sinh(d*x + c)^2 + a - b)/(c osh(d*x + c)^2 - 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d*x + c)^2)) - 2*((a ^2 - b^2)*cosh(d*x + c)^4 + 4*(a^2 - b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 - b^2)*sinh(d*x + c)^4 - 2*(a^2 - b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 - b^2)*cosh(d*x + c)^2 - a^2 + b^2)*sinh(d*x + c)^2 + a^2 - b^2 + 4*((a^2 - b^2)*cosh(d*x + c)^3 - (a^2 - b^2)*cosh(d*x + c))*sinh(d*x + c))*log(2*sin h(d*x + c)/(cosh(d*x + c) - sinh(d*x + c))) + 8*(a^2*d*x*cosh(d*x + c)^3 - (a^2*d*x - a^2 - a*b)*cosh(d*x + c))*sinh(d*x + c))/((a^3 + a^2*b)*d*cosh (d*x + c)^4 + 4*(a^3 + a^2*b)*d*cosh(d*x + c)*sinh(d*x + c)^3 + (a^3 + a^2 *b)*d*sinh(d*x + c)^4 - 2*(a^3 + a^2*b)*d*cosh(d*x + c)^2 + 2*(3*(a^3 + a^ 2*b)*d*cosh(d*x + c)^2 - (a^3 + a^2*b)*d)*sinh(d*x + c)^2 + (a^3 + a^2*b)* d + 4*((a^3 + a^2*b)*d*cosh(d*x + c)^3 - (a^3 + a^2*b)*d*cosh(d*x + c))*si nh(d*x + c))
\[ \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {\coth ^{3}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
Time = 0.22 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.87 \[ \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {b^{2} \log \left (2 \, {\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{2 \, {\left (a^{3} + a^{2} b\right )} d} + \frac {d x + c}{{\left (a + b\right )} d} + \frac {2 \, e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (2 \, a e^{\left (-2 \, d x - 2 \, c\right )} - a e^{\left (-4 \, d x - 4 \, c\right )} - a\right )} d} + \frac {{\left (a - b\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} + \frac {{\left (a - b\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} \]
1/2*b^2*log(2*(a - b)*e^(-2*d*x - 2*c) + (a + b)*e^(-4*d*x - 4*c) + a + b) /((a^3 + a^2*b)*d) + (d*x + c)/((a + b)*d) + 2*e^(-2*d*x - 2*c)/((2*a*e^(- 2*d*x - 2*c) - a*e^(-4*d*x - 4*c) - a)*d) + (a - b)*log(e^(-d*x - c) + 1)/ (a^2*d) + (a - b)*log(e^(-d*x - c) - 1)/(a^2*d)
Time = 0.35 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.56 \[ \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {\frac {b^{2} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}{a^{3} + a^{2} b} - \frac {2 \, {\left (d x + c\right )}}{a + b} + \frac {2 \, {\left (a - b\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{2}} - \frac {4 \, e^{\left (2 \, d x + 2 \, c\right )}}{a {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \]
1/2*(b^2*log(a*e^(4*d*x + 4*c) + b*e^(4*d*x + 4*c) + 2*a*e^(2*d*x + 2*c) - 2*b*e^(2*d*x + 2*c) + a + b)/(a^3 + a^2*b) - 2*(d*x + c)/(a + b) + 2*(a - b)*log(abs(e^(2*d*x + 2*c) - 1))/a^2 - 4*e^(2*d*x + 2*c)/(a*(e^(2*d*x + 2 *c) - 1)^2))/d
Time = 2.24 (sec) , antiderivative size = 313, normalized size of antiderivative = 3.68 \[ \int \frac {\coth ^3(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {b^2\,\ln \left (3\,a\,b^2-2\,a^2\,b-2\,a^3+3\,b^3-4\,a^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-2\,a^3\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}-6\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+3\,b^3\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}+6\,a\,b^2\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+4\,a^2\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+3\,a\,b^2\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}-2\,a^2\,b\,{\mathrm {e}}^{4\,c}\,{\mathrm {e}}^{4\,d\,x}\right )}{2\,d\,a^3+2\,b\,d\,a^2}-\frac {x}{a+b}-\frac {2}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {\ln \left (4\,a^4\,b+9\,b^5-12\,a^2\,b^3-9\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}-4\,a^4\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+12\,a^2\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\right )\,\left (a-b\right )}{a^2\,d}-\frac {2\,\left (a^2+b\,a\right )}{a^2\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )\,\left (a+b\right )} \]
(b^2*log(3*a*b^2 - 2*a^2*b - 2*a^3 + 3*b^3 - 4*a^3*exp(2*c)*exp(2*d*x) - 2 *a^3*exp(4*c)*exp(4*d*x) - 6*b^3*exp(2*c)*exp(2*d*x) + 3*b^3*exp(4*c)*exp( 4*d*x) + 6*a*b^2*exp(2*c)*exp(2*d*x) + 4*a^2*b*exp(2*c)*exp(2*d*x) + 3*a*b ^2*exp(4*c)*exp(4*d*x) - 2*a^2*b*exp(4*c)*exp(4*d*x)))/(2*a^3*d + 2*a^2*b* d) - x/(a + b) - 2/(a*d*(exp(4*c + 4*d*x) - 2*exp(2*c + 2*d*x) + 1)) + (lo g(4*a^4*b + 9*b^5 - 12*a^2*b^3 - 9*b^5*exp(2*c)*exp(2*d*x) - 4*a^4*b*exp(2 *c)*exp(2*d*x) + 12*a^2*b^3*exp(2*c)*exp(2*d*x))*(a - b))/(a^2*d) - (2*(a* b + a^2))/(a^2*d*(exp(2*c + 2*d*x) - 1)*(a + b))