Integrand size = 16, antiderivative size = 47 \[ \int x \cosh (a+b x) \coth ^2(a+b x) \, dx=-\frac {\text {arctanh}(\cosh (a+b x))}{b^2}-\frac {\cosh (a+b x)}{b^2}-\frac {x \text {csch}(a+b x)}{b}+\frac {x \sinh (a+b x)}{b} \]
Time = 0.32 (sec) , antiderivative size = 79, normalized size of antiderivative = 1.68 \[ \int x \cosh (a+b x) \coth ^2(a+b x) \, dx=\frac {-2 \cosh (a+b x)-b x \coth \left (\frac {1}{2} (a+b x)\right )-2 \log \left (\cosh \left (\frac {1}{2} (a+b x)\right )\right )+2 \log \left (\sinh \left (\frac {1}{2} (a+b x)\right )\right )+2 b x \sinh (a+b x)+b x \tanh \left (\frac {1}{2} (a+b x)\right )}{2 b^2} \]
(-2*Cosh[a + b*x] - b*x*Coth[(a + b*x)/2] - 2*Log[Cosh[(a + b*x)/2]] + 2*L og[Sinh[(a + b*x)/2]] + 2*b*x*Sinh[a + b*x] + b*x*Tanh[(a + b*x)/2])/(2*b^ 2)
Time = 0.44 (sec) , antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.688, Rules used = {5973, 3042, 3777, 26, 3042, 26, 3118, 5942, 3042, 26, 4257}
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.
\(\displaystyle \int x \cosh (a+b x) \coth ^2(a+b x) \, dx\) |
\(\Big \downarrow \) 5973 |
\(\displaystyle \int x \cosh (a+b x)dx+\int x \coth (a+b x) \text {csch}(a+b x)dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int x \coth (a+b x) \text {csch}(a+b x)dx+\int x \sin \left (i a+i b x+\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 3777 |
\(\displaystyle -\frac {i \int -i \sinh (a+b x)dx}{b}+\int x \coth (a+b x) \text {csch}(a+b x)dx+\frac {x \sinh (a+b x)}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -\frac {\int \sinh (a+b x)dx}{b}+\int x \coth (a+b x) \text {csch}(a+b x)dx+\frac {x \sinh (a+b x)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -\frac {\int -i \sin (i a+i b x)dx}{b}+\int x \coth (a+b x) \text {csch}(a+b x)dx+\frac {x \sinh (a+b x)}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \int \sin (i a+i b x)dx}{b}+\int x \coth (a+b x) \text {csch}(a+b x)dx+\frac {x \sinh (a+b x)}{b}\) |
\(\Big \downarrow \) 3118 |
\(\displaystyle \int x \coth (a+b x) \text {csch}(a+b x)dx-\frac {\cosh (a+b x)}{b^2}+\frac {x \sinh (a+b x)}{b}\) |
\(\Big \downarrow \) 5942 |
\(\displaystyle \frac {\int \text {csch}(a+b x)dx}{b}-\frac {\cosh (a+b x)}{b^2}+\frac {x \sinh (a+b x)}{b}-\frac {x \text {csch}(a+b x)}{b}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\int i \csc (i a+i b x)dx}{b}-\frac {\cosh (a+b x)}{b^2}+\frac {x \sinh (a+b x)}{b}-\frac {x \text {csch}(a+b x)}{b}\) |
\(\Big \downarrow \) 26 |
\(\displaystyle \frac {i \int \csc (i a+i b x)dx}{b}-\frac {\cosh (a+b x)}{b^2}+\frac {x \sinh (a+b x)}{b}-\frac {x \text {csch}(a+b x)}{b}\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -\frac {\text {arctanh}(\cosh (a+b x))}{b^2}-\frac {\cosh (a+b x)}{b^2}+\frac {x \sinh (a+b x)}{b}-\frac {x \text {csch}(a+b x)}{b}\) |
-(ArcTanh[Cosh[a + b*x]]/b^2) - Cosh[a + b*x]/b^2 - (x*Csch[a + b*x])/b + (x*Sinh[a + b*x])/b
3.5.41.3.1 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[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + (f_.)*(x_)], x_Symbol] :> Simp[( -(c + d*x)^m)*(Cos[e + f*x]/f), x] + Simp[d*(m/f) Int[(c + d*x)^(m - 1)*C os[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Int[Coth[(a_.) + (b_.)*(x_)^(n_.)]^(q_.)*Csch[(a_.) + (b_.)*(x_)^(n_.)]^(p_ .)*(x_)^(m_.), x_Symbol] :> Simp[(-x^(m - n + 1))*(Csch[a + b*x^n]^p/(b*n*p )), x] + Simp[(m - n + 1)/(b*n*p) Int[x^(m - n)*Csch[a + b*x^n]^p, x], x] /; FreeQ[{a, b, p}, x] && RationalQ[m] && IntegerQ[n] && GeQ[m - n, 0] && EqQ[q, 1]
Int[Cosh[(a_.) + (b_.)*(x_)]^(n_.)*Coth[(a_.) + (b_.)*(x_)]^(p_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Int[(c + d*x)^m*Cosh[a + b*x]^n*Coth[a + b* x]^(p - 2), x] + Int[(c + d*x)^m*Cosh[a + b*x]^(n - 2)*Coth[a + b*x]^p, x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[n, 0] && IGtQ[p, 0]
Time = 0.56 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.89
method | result | size |
risch | \(\frac {\left (b x -1\right ) {\mathrm e}^{b x +a}}{2 b^{2}}-\frac {\left (b x +1\right ) {\mathrm e}^{-b x -a}}{2 b^{2}}-\frac {2 x \,{\mathrm e}^{b x +a}}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{b^{2}}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{b^{2}}\) | \(89\) |
1/2*(b*x-1)/b^2*exp(b*x+a)-1/2*(b*x+1)/b^2*exp(-b*x-a)-2/b*x*exp(b*x+a)/(e xp(2*b*x+2*a)-1)+1/b^2*ln(exp(b*x+a)-1)-1/b^2*ln(exp(b*x+a)+1)
Leaf count of result is larger than twice the leaf count of optimal. 367 vs. \(2 (47) = 94\).
Time = 0.26 (sec) , antiderivative size = 367, normalized size of antiderivative = 7.81 \[ \int x \cosh (a+b x) \coth ^2(a+b x) \, dx=\frac {{\left (b x - 1\right )} \cosh \left (b x + a\right )^{4} + 4 \, {\left (b x - 1\right )} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + {\left (b x - 1\right )} \sinh \left (b x + a\right )^{4} - 6 \, b x \cosh \left (b x + a\right )^{2} + 6 \, {\left ({\left (b x - 1\right )} \cosh \left (b x + a\right )^{2} - b x\right )} \sinh \left (b x + a\right )^{2} + b x - 2 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) + 2 \, {\left (\cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + \sinh \left (b x + a\right )^{3} + {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right ) - \cosh \left (b x + a\right )\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 4 \, {\left ({\left (b x - 1\right )} \cosh \left (b x + a\right )^{3} - 3 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1}{2 \, {\left (b^{2} \cosh \left (b x + a\right )^{3} + 3 \, b^{2} \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + b^{2} \sinh \left (b x + a\right )^{3} - b^{2} \cosh \left (b x + a\right ) + {\left (3 \, b^{2} \cosh \left (b x + a\right )^{2} - b^{2}\right )} \sinh \left (b x + a\right )\right )}} \]
1/2*((b*x - 1)*cosh(b*x + a)^4 + 4*(b*x - 1)*cosh(b*x + a)*sinh(b*x + a)^3 + (b*x - 1)*sinh(b*x + a)^4 - 6*b*x*cosh(b*x + a)^2 + 6*((b*x - 1)*cosh(b *x + a)^2 - b*x)*sinh(b*x + a)^2 + b*x - 2*(cosh(b*x + a)^3 + 3*cosh(b*x + a)*sinh(b*x + a)^2 + sinh(b*x + a)^3 + (3*cosh(b*x + a)^2 - 1)*sinh(b*x + a) - cosh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a) + 1) + 2*(cosh(b*x + a)^3 + 3*cosh(b*x + a)*sinh(b*x + a)^2 + sinh(b*x + a)^3 + (3*cosh(b*x + a)^2 - 1)*sinh(b*x + a) - cosh(b*x + a))*log(cosh(b*x + a) + sinh(b*x + a ) - 1) + 4*((b*x - 1)*cosh(b*x + a)^3 - 3*b*x*cosh(b*x + a))*sinh(b*x + a) + 1)/(b^2*cosh(b*x + a)^3 + 3*b^2*cosh(b*x + a)*sinh(b*x + a)^2 + b^2*sin h(b*x + a)^3 - b^2*cosh(b*x + a) + (3*b^2*cosh(b*x + a)^2 - b^2)*sinh(b*x + a))
\[ \int x \cosh (a+b x) \coth ^2(a+b x) \, dx=\int x \cosh ^{3}{\left (a + b x \right )} \operatorname {csch}^{2}{\left (a + b x \right )}\, dx \]
Leaf count of result is larger than twice the leaf count of optimal. 109 vs. \(2 (47) = 94\).
Time = 0.27 (sec) , antiderivative size = 109, normalized size of antiderivative = 2.32 \[ \int x \cosh (a+b x) \coth ^2(a+b x) \, dx=-\frac {6 \, b x e^{\left (b x + 2 \, a\right )} - {\left (b x e^{\left (4 \, a\right )} - e^{\left (4 \, a\right )}\right )} e^{\left (3 \, b x\right )} - {\left (b x + 1\right )} e^{\left (-b x\right )}}{2 \, {\left (b^{2} e^{\left (2 \, b x + 3 \, a\right )} - b^{2} e^{a}\right )}} - \frac {\log \left ({\left (e^{\left (b x + a\right )} + 1\right )} e^{\left (-a\right )}\right )}{b^{2}} + \frac {\log \left ({\left (e^{\left (b x + a\right )} - 1\right )} e^{\left (-a\right )}\right )}{b^{2}} \]
-1/2*(6*b*x*e^(b*x + 2*a) - (b*x*e^(4*a) - e^(4*a))*e^(3*b*x) - (b*x + 1)* e^(-b*x))/(b^2*e^(2*b*x + 3*a) - b^2*e^a) - log((e^(b*x + a) + 1)*e^(-a))/ b^2 + log((e^(b*x + a) - 1)*e^(-a))/b^2
Leaf count of result is larger than twice the leaf count of optimal. 144 vs. \(2 (47) = 94\).
Time = 0.30 (sec) , antiderivative size = 144, normalized size of antiderivative = 3.06 \[ \int x \cosh (a+b x) \coth ^2(a+b x) \, dx=\frac {b x e^{\left (4 \, b x + 4 \, a\right )} - 6 \, b x e^{\left (2 \, b x + 2 \, a\right )} + b x - 2 \, e^{\left (3 \, b x + 3 \, a\right )} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, e^{\left (b x + a\right )} \log \left (e^{\left (b x + a\right )} + 1\right ) + 2 \, e^{\left (3 \, b x + 3 \, a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - 2 \, e^{\left (b x + a\right )} \log \left (e^{\left (b x + a\right )} - 1\right ) - e^{\left (4 \, b x + 4 \, a\right )} + 1}{2 \, {\left (b^{2} e^{\left (3 \, b x + 3 \, a\right )} - b^{2} e^{\left (b x + a\right )}\right )}} \]
1/2*(b*x*e^(4*b*x + 4*a) - 6*b*x*e^(2*b*x + 2*a) + b*x - 2*e^(3*b*x + 3*a) *log(e^(b*x + a) + 1) + 2*e^(b*x + a)*log(e^(b*x + a) + 1) + 2*e^(3*b*x + 3*a)*log(e^(b*x + a) - 1) - 2*e^(b*x + a)*log(e^(b*x + a) - 1) - e^(4*b*x + 4*a) + 1)/(b^2*e^(3*b*x + 3*a) - b^2*e^(b*x + a))
Time = 0.10 (sec) , antiderivative size = 95, normalized size of antiderivative = 2.02 \[ \int x \cosh (a+b x) \coth ^2(a+b x) \, dx={\mathrm {e}}^{a+b\,x}\,\left (\frac {x}{2\,b}-\frac {1}{2\,b^2}\right )-\frac {2\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^4}}{b^2}\right )}{\sqrt {-b^4}}-{\mathrm {e}}^{-a-b\,x}\,\left (\frac {x}{2\,b}+\frac {1}{2\,b^2}\right )-\frac {2\,x\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]