Integrand size = 15, antiderivative size = 34 \[ \int \coth ^2(a+b x) \text {csch}(a+b x) \, dx=-\frac {\text {arctanh}(\cosh (a+b x))}{2 b}-\frac {\coth (a+b x) \text {csch}(a+b x)}{2 b} \] Output:
-1/2*arctanh(cosh(b*x+a))/b-1/2*coth(b*x+a)*csch(b*x+a)/b
Leaf count is larger than twice the leaf count of optimal. \(75\) vs. \(2(34)=68\).
Time = 0.01 (sec) , antiderivative size = 75, normalized size of antiderivative = 2.21 \[ \int \coth ^2(a+b x) \text {csch}(a+b x) \, dx=-\frac {\text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {\log \left (\cosh \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}+\frac {\log \left (\sinh \left (\frac {1}{2} (a+b x)\right )\right )}{2 b}-\frac {\text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{8 b} \] Input:
Integrate[Coth[a + b*x]^2*Csch[a + b*x],x]
Output:
-1/8*Csch[(a + b*x)/2]^2/b - Log[Cosh[(a + b*x)/2]]/(2*b) + Log[Sinh[(a + b*x)/2]]/(2*b) - Sech[(a + b*x)/2]^2/(8*b)
Result contains complex when optimal does not.
Time = 0.27 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.24, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.467, Rules used = {3042, 26, 3091, 26, 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.
\(\displaystyle \int \coth ^2(a+b x) \text {csch}(a+b x) \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int -i \tan \left (i a+i b x-\frac {\pi }{2}\right )^2 \sec \left (i a+i b x-\frac {\pi }{2}\right )dx\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \int \sec \left (\frac {1}{2} (2 i a-\pi )+i b x\right ) \tan \left (\frac {1}{2} (2 i a-\pi )+i b x\right )^2dx\) |
\(\Big \downarrow \) 3091 |
\(\displaystyle -i \left (-\frac {1}{2} \int -i \text {csch}(a+b x)dx-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (\frac {1}{2} i \int \text {csch}(a+b x)dx-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle -i \left (\frac {1}{2} i \int i \csc (i a+i b x)dx-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 26 |
\(\displaystyle -i \left (-\frac {1}{2} \int \csc (i a+i b x)dx-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
\(\Big \downarrow \) 4257 |
\(\displaystyle -i \left (-\frac {i \text {arctanh}(\cosh (a+b x))}{2 b}-\frac {i \coth (a+b x) \text {csch}(a+b x)}{2 b}\right )\) |
Input:
Int[Coth[a + b*x]^2*Csch[a + b*x],x]
Output:
(-I)*(((-1/2*I)*ArcTanh[Cosh[a + b*x]])/b - ((I/2)*Coth[a + b*x]*Csch[a + b*x])/b)
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[((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((b_.)*tan[(e_.) + (f_.)*(x_)])^( n_), x_Symbol] :> Simp[b*(a*Sec[e + f*x])^m*((b*Tan[e + f*x])^(n - 1)/(f*(m + n - 1))), x] - Simp[b^2*((n - 1)/(m + n - 1)) Int[(a*Sec[e + f*x])^m*( b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, e, f, m}, x] && GtQ[n, 1] & & NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.07 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.32
method | result | size |
derivativedivides | \(\frac {-\frac {\cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{2}}+\frac {\coth \left (b x +a \right ) \operatorname {csch}\left (b x +a \right )}{2}-\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}\) | \(45\) |
default | \(\frac {-\frac {\cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{2}}+\frac {\coth \left (b x +a \right ) \operatorname {csch}\left (b x +a \right )}{2}-\operatorname {arctanh}\left ({\mathrm e}^{b x +a}\right )}{b}\) | \(45\) |
risch | \(-\frac {{\mathrm e}^{b x +a} \left ({\mathrm e}^{2 b x +2 a}+1\right )}{b \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}}+\frac {\ln \left ({\mathrm e}^{b x +a}-1\right )}{2 b}-\frac {\ln \left ({\mathrm e}^{b x +a}+1\right )}{2 b}\) | \(65\) |
Input:
int(coth(b*x+a)^2*csch(b*x+a),x,method=_RETURNVERBOSE)
Output:
1/b*(-cosh(b*x+a)/sinh(b*x+a)^2+1/2*coth(b*x+a)*csch(b*x+a)-arctanh(exp(b* x+a)))
Leaf count of result is larger than twice the leaf count of optimal. 387 vs. \(2 (30) = 60\).
Time = 0.09 (sec) , antiderivative size = 387, normalized size of antiderivative = 11.38 \[ \int \coth ^2(a+b x) \text {csch}(a+b x) \, dx=-\frac {2 \, \cosh \left (b x + a\right )^{3} + 6 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{2} + 2 \, \sinh \left (b x + a\right )^{3} + {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) + 1\right ) - {\left (\cosh \left (b x + a\right )^{4} + 4 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + \sinh \left (b x + a\right )^{4} + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} - 1\right )} \sinh \left (b x + a\right )^{2} - 2 \, \cosh \left (b x + a\right )^{2} + 4 \, {\left (\cosh \left (b x + a\right )^{3} - \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + 1\right )} \log \left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right ) - 1\right ) + 2 \, {\left (3 \, \cosh \left (b x + a\right )^{2} + 1\right )} \sinh \left (b x + a\right ) + 2 \, \cosh \left (b x + a\right )}{2 \, {\left (b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{3} + b \sinh \left (b x + a\right )^{4} - 2 \, b \cosh \left (b x + a\right )^{2} + 2 \, {\left (3 \, b \cosh \left (b x + a\right )^{2} - b\right )} \sinh \left (b x + a\right )^{2} + 4 \, {\left (b \cosh \left (b x + a\right )^{3} - b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right ) + b\right )}} \] Input:
integrate(coth(b*x+a)^2*csch(b*x+a),x, algorithm="fricas")
Output:
-1/2*(2*cosh(b*x + a)^3 + 6*cosh(b*x + a)*sinh(b*x + a)^2 + 2*sinh(b*x + a )^3 + (cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b*x + a)^4 + 2*(3*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - 2*cosh(b*x + a)^2 + 4*(cosh (b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) + 1)*log(cosh(b*x + a) + sinh(b *x + a) + 1) - (cosh(b*x + a)^4 + 4*cosh(b*x + a)*sinh(b*x + a)^3 + sinh(b *x + a)^4 + 2*(3*cosh(b*x + a)^2 - 1)*sinh(b*x + a)^2 - 2*cosh(b*x + a)^2 + 4*(cosh(b*x + a)^3 - cosh(b*x + a))*sinh(b*x + a) + 1)*log(cosh(b*x + a) + sinh(b*x + a) - 1) + 2*(3*cosh(b*x + a)^2 + 1)*sinh(b*x + a) + 2*cosh(b *x + a))/(b*cosh(b*x + a)^4 + 4*b*cosh(b*x + a)*sinh(b*x + a)^3 + b*sinh(b *x + a)^4 - 2*b*cosh(b*x + a)^2 + 2*(3*b*cosh(b*x + a)^2 - b)*sinh(b*x + a )^2 + 4*(b*cosh(b*x + a)^3 - b*cosh(b*x + a))*sinh(b*x + a) + b)
\[ \int \coth ^2(a+b x) \text {csch}(a+b x) \, dx=\int \coth ^{2}{\left (a + b x \right )} \operatorname {csch}{\left (a + b x \right )}\, dx \] Input:
integrate(coth(b*x+a)**2*csch(b*x+a),x)
Output:
Integral(coth(a + b*x)**2*csch(a + b*x), x)
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (30) = 60\).
Time = 0.04 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.47 \[ \int \coth ^2(a+b x) \text {csch}(a+b x) \, dx=-\frac {\log \left (e^{\left (-b x - a\right )} + 1\right )}{2 \, b} + \frac {\log \left (e^{\left (-b x - a\right )} - 1\right )}{2 \, b} + \frac {e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}}{b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - e^{\left (-4 \, b x - 4 \, a\right )} - 1\right )}} \] Input:
integrate(coth(b*x+a)^2*csch(b*x+a),x, algorithm="maxima")
Output:
-1/2*log(e^(-b*x - a) + 1)/b + 1/2*log(e^(-b*x - a) - 1)/b + (e^(-b*x - a) + e^(-3*b*x - 3*a))/(b*(2*e^(-2*b*x - 2*a) - e^(-4*b*x - 4*a) - 1))
Leaf count of result is larger than twice the leaf count of optimal. 84 vs. \(2 (30) = 60\).
Time = 0.13 (sec) , antiderivative size = 84, normalized size of antiderivative = 2.47 \[ \int \coth ^2(a+b x) \text {csch}(a+b x) \, dx=-\frac {\frac {4 \, {\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}}{{\left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )}\right )}^{2} - 4} + \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} + 2\right ) - \log \left (e^{\left (b x + a\right )} + e^{\left (-b x - a\right )} - 2\right )}{4 \, b} \] Input:
integrate(coth(b*x+a)^2*csch(b*x+a),x, algorithm="giac")
Output:
-1/4*(4*(e^(b*x + a) + e^(-b*x - a))/((e^(b*x + a) + e^(-b*x - a))^2 - 4) + log(e^(b*x + a) + e^(-b*x - a) + 2) - log(e^(b*x + a) + e^(-b*x - a) - 2 ))/b
Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 2.56 \[ \int \coth ^2(a+b x) \text {csch}(a+b x) \, dx=-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \] Input:
int(coth(a + b*x)^2/sinh(a + b*x),x)
Output:
- atan((exp(b*x)*exp(a)*(-b^2)^(1/2))/b)/(-b^2)^(1/2) - (2*exp(a + b*x))/( b*(exp(4*a + 4*b*x) - 2*exp(2*a + 2*b*x) + 1)) - exp(a + b*x)/(b*(exp(2*a + 2*b*x) - 1))
Time = 0.23 (sec) , antiderivative size = 162, normalized size of antiderivative = 4.76 \[ \int \coth ^2(a+b x) \text {csch}(a+b x) \, dx=\frac {e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}-1\right )-e^{4 b x +4 a} \mathrm {log}\left (e^{b x +a}+1\right )-2 e^{3 b x +3 a}-2 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}-1\right )+2 e^{2 b x +2 a} \mathrm {log}\left (e^{b x +a}+1\right )-2 e^{b x +a}+\mathrm {log}\left (e^{b x +a}-1\right )-\mathrm {log}\left (e^{b x +a}+1\right )}{2 b \left (e^{4 b x +4 a}-2 e^{2 b x +2 a}+1\right )} \] Input:
int(coth(b*x+a)^2*csch(b*x+a),x)
Output:
(e**(4*a + 4*b*x)*log(e**(a + b*x) - 1) - e**(4*a + 4*b*x)*log(e**(a + b*x ) + 1) - 2*e**(3*a + 3*b*x) - 2*e**(2*a + 2*b*x)*log(e**(a + b*x) - 1) + 2 *e**(2*a + 2*b*x)*log(e**(a + b*x) + 1) - 2*e**(a + b*x) + log(e**(a + b*x ) - 1) - log(e**(a + b*x) + 1))/(2*b*(e**(4*a + 4*b*x) - 2*e**(2*a + 2*b*x ) + 1))