Integrand size = 21, antiderivative size = 77 \[ \int \frac {\coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b \text {arctanh}(\cosh (c+d x))}{a^2 d}-\frac {2 \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^2 d}-\frac {\coth (c+d x)}{a d} \] Output:
b*arctanh(cosh(d*x+c))/a^2/d-2*(a^2+b^2)^(1/2)*arctanh((b-a*tanh(1/2*d*x+1 /2*c))/(a^2+b^2)^(1/2))/a^2/d-coth(d*x+c)/a/d
Time = 0.54 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.45 \[ \int \frac {\coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {4 \sqrt {-a^2-b^2} \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )+a \coth \left (\frac {1}{2} (c+d x)\right )-2 b \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+2 b \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )+a \tanh \left (\frac {1}{2} (c+d x)\right )}{2 a^2 d} \] Input:
Integrate[Coth[c + d*x]^2/(a + b*Sinh[c + d*x]),x]
Output:
-1/2*(4*Sqrt[-a^2 - b^2]*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2] ] + a*Coth[(c + d*x)/2] - 2*b*Log[Cosh[(c + d*x)/2]] + 2*b*Log[Sinh[(c + d *x)/2]] + a*Tanh[(c + d*x)/2])/(a^2*d)
Result contains complex when optimal does not.
Time = 0.76 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.14, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {3042, 25, 3202, 25, 3042, 25, 3535, 3042, 26, 3480, 26, 3042, 26, 3139, 1083, 217, 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.
| \(\displaystyle \int \frac {\coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1}{\tan (i c+i d x)^2 (a-i b \sin (i c+i d x))}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1}{(a-i b \sin (i c+i d x)) \tan (i c+i d x)^2}dx\) |
| \(\Big \downarrow \) 3202 |
| \(\displaystyle -\int -\frac {\text {csch}^2(c+d x) \left (\sinh ^2(c+d x)+1\right )}{a+b \sinh (c+d x)}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \int \frac {\left (\sinh ^2(c+d x)+1\right ) \text {csch}^2(c+d x)}{a+b \sinh (c+d x)}dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {1-\sin (i c+i d x)^2}{\sin (i c+i d x)^2 (a-i b \sin (i c+i d x))}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {1-\sin (i c+i d x)^2}{\sin (i c+i d x)^2 (a-i b \sin (i c+i d x))}dx\) |
| \(\Big \downarrow \) 3535 |
| \(\displaystyle -\frac {\int \frac {\text {csch}(c+d x) (b-a \sinh (c+d x))}{a+b \sinh (c+d x)}dx}{a}-\frac {\coth (c+d x)}{a d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\coth (c+d x)}{a d}-\frac {\int \frac {i (b+i a \sin (i c+i d x))}{\sin (i c+i d x) (a-i b \sin (i c+i d x))}dx}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\coth (c+d x)}{a d}-\frac {i \int \frac {b+i a \sin (i c+i d x)}{\sin (i c+i d x) (a-i b \sin (i c+i d x))}dx}{a}\) |
| \(\Big \downarrow \) 3480 |
| \(\displaystyle -\frac {\coth (c+d x)}{a d}-\frac {i \left (\frac {i \left (a^2+b^2\right ) \int \frac {1}{a+b \sinh (c+d x)}dx}{a}+\frac {b \int -i \text {csch}(c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\coth (c+d x)}{a d}-\frac {i \left (\frac {i \left (a^2+b^2\right ) \int \frac {1}{a+b \sinh (c+d x)}dx}{a}-\frac {i b \int \text {csch}(c+d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\coth (c+d x)}{a d}-\frac {i \left (\frac {i \left (a^2+b^2\right ) \int \frac {1}{a-i b \sin (i c+i d x)}dx}{a}-\frac {i b \int i \csc (i c+i d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle -\frac {\coth (c+d x)}{a d}-\frac {i \left (\frac {i \left (a^2+b^2\right ) \int \frac {1}{a-i b \sin (i c+i d x)}dx}{a}+\frac {b \int \csc (i c+i d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle -\frac {\coth (c+d x)}{a d}-\frac {i \left (\frac {2 \left (a^2+b^2\right ) \int \frac {1}{-a \tanh ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tanh \left (\frac {1}{2} (c+d x)\right )+a}d\left (i \tanh \left (\frac {1}{2} (c+d x)\right )\right )}{a d}+\frac {b \int \csc (i c+i d x)dx}{a}\right )}{a}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle -\frac {\coth (c+d x)}{a d}-\frac {i \left (\frac {b \int \csc (i c+i d x)dx}{a}-\frac {4 \left (a^2+b^2\right ) \int \frac {1}{\tanh ^2\left (\frac {1}{2} (c+d x)\right )-4 \left (a^2+b^2\right )}d\left (2 i a \tanh \left (\frac {1}{2} (c+d x)\right )-2 i b\right )}{a d}\right )}{a}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {\coth (c+d x)}{a d}-\frac {i \left (\frac {b \int \csc (i c+i d x)dx}{a}+\frac {2 i \sqrt {a^2+b^2} \text {arctanh}\left (\frac {\tanh \left (\frac {1}{2} (c+d x)\right )}{2 \sqrt {a^2+b^2}}\right )}{a d}\right )}{a}\) |
| \(\Big \downarrow \) 4257 |
| \(\displaystyle -\frac {\coth (c+d x)}{a d}-\frac {i \left (\frac {2 i \sqrt {a^2+b^2} \text {arctanh}\left (\frac {\tanh \left (\frac {1}{2} (c+d x)\right )}{2 \sqrt {a^2+b^2}}\right )}{a d}+\frac {i b \text {arctanh}(\cosh (c+d x))}{a d}\right )}{a}\) |
Input:
Int[Coth[c + d*x]^2/(a + b*Sinh[c + d*x]),x]
Output:
((-I)*((I*b*ArcTanh[Cosh[c + d*x]])/(a*d) + ((2*I)*Sqrt[a^2 + b^2]*ArcTanh [Tanh[(c + d*x)/2]/(2*Sqrt[a^2 + b^2])])/(a*d)))/a - Coth[c + d*x]/(a*d)
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_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^2, x_Symbol] :> Int[(a + b*Sin[e + f*x])^m*((1 - Sin[e + f*x]^2)/Sin[e + f*x]^ 2), x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2 - b^2, 0]
Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_ .)*(x_)])*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])), x_Symbol] :> Simp[(A*b - a*B)/(b*c - a*d) Int[1/(a + b*Sin[e + f*x]), x], x] + Simp[(B*c - A*d)/ (b*c - a*d) Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f , A, B}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 + a^2*C))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*S in[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 - b^2)) Int[(a + b*Sin[e + f*x])^(m + 1)*(c + d*Sin [e + f*x])^n*Simp[a*(m + 1)*(b*c - a*d)*(A + C) + d*(A*b^2 + a^2*C)*(m + n + 2) - (c*(A*b^2 + a^2*C) + b*(m + 1)*(b*c - a*d)*(A + C))*Sin[e + f*x] - d *(A*b^2 + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] && LtQ[m, -1] && ((EqQ[a, 0] && IntegerQ[m] && !IntegerQ[n]) || !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] && !IntegerQ[m]) || EqQ[a, 0])))
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
Time = 0.49 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.36
| method | result | size |
| derivativedivides | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {\left (-4 a^{2}-4 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 a^{2} \sqrt {a^{2}+b^{2}}}}{d}\) | \(105\) |
| default | \(\frac {-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {1}{2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {b \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a^{2}}-\frac {\left (-4 a^{2}-4 b^{2}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{2 a^{2} \sqrt {a^{2}+b^{2}}}}{d}\) | \(105\) |
| risch | \(-\frac {2}{d a \left ({\mathrm e}^{2 d x +2 c}-1\right )}+\frac {b \ln \left ({\mathrm e}^{d x +c}+1\right )}{d \,a^{2}}-\frac {b \ln \left ({\mathrm e}^{d x +c}-1\right )}{d \,a^{2}}+\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{d x +c}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,a^{2}}-\frac {\sqrt {a^{2}+b^{2}}\, \ln \left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,a^{2}}\) | \(140\) |
Input:
int(coth(d*x+c)^2/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)
Output:
1/d*(-1/2/a*tanh(1/2*d*x+1/2*c)-1/2/a/tanh(1/2*d*x+1/2*c)-1/a^2*b*ln(tanh( 1/2*d*x+1/2*c))-1/2/a^2*(-4*a^2-4*b^2)/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*ta nh(1/2*d*x+1/2*c)-2*b)/(a^2+b^2)^(1/2)))
Leaf count of result is larger than twice the leaf count of optimal. 360 vs. \(2 (74) = 148\).
Time = 0.11 (sec) , antiderivative size = 360, normalized size of antiderivative = 4.68 \[ \int \frac {\coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\sqrt {a^{2} + b^{2}} {\left (\cosh \left (d x + c\right )^{2} + 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2} - 1\right )} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{2} + b^{2} \sinh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) + 2 \, a^{2} + b^{2} + 2 \, {\left (b^{2} \cosh \left (d x + c\right ) + a b\right )} \sinh \left (d x + c\right ) - 2 \, \sqrt {a^{2} + b^{2}} {\left (b \cosh \left (d x + c\right ) + b \sinh \left (d x + c\right ) + a\right )}}{b \cosh \left (d x + c\right )^{2} + b \sinh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) + 2 \, {\left (b \cosh \left (d x + c\right ) + a\right )} \sinh \left (d x + c\right ) - b}\right ) + {\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) + 1\right ) - {\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} - b\right )} \log \left (\cosh \left (d x + c\right ) + \sinh \left (d x + c\right ) - 1\right ) - 2 \, a}{a^{2} d \cosh \left (d x + c\right )^{2} + 2 \, a^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a^{2} d \sinh \left (d x + c\right )^{2} - a^{2} d} \] Input:
integrate(coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="fricas")
Output:
(sqrt(a^2 + b^2)*(cosh(d*x + c)^2 + 2*cosh(d*x + c)*sinh(d*x + c) + sinh(d *x + c)^2 - 1)*log((b^2*cosh(d*x + c)^2 + b^2*sinh(d*x + c)^2 + 2*a*b*cosh (d*x + c) + 2*a^2 + b^2 + 2*(b^2*cosh(d*x + c) + a*b)*sinh(d*x + c) - 2*sq rt(a^2 + b^2)*(b*cosh(d*x + c) + b*sinh(d*x + c) + a))/(b*cosh(d*x + c)^2 + b*sinh(d*x + c)^2 + 2*a*cosh(d*x + c) + 2*(b*cosh(d*x + c) + a)*sinh(d*x + c) - b)) + (b*cosh(d*x + c)^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sin h(d*x + c)^2 - b)*log(cosh(d*x + c) + sinh(d*x + c) + 1) - (b*cosh(d*x + c )^2 + 2*b*cosh(d*x + c)*sinh(d*x + c) + b*sinh(d*x + c)^2 - b)*log(cosh(d* x + c) + sinh(d*x + c) - 1) - 2*a)/(a^2*d*cosh(d*x + c)^2 + 2*a^2*d*cosh(d *x + c)*sinh(d*x + c) + a^2*d*sinh(d*x + c)^2 - a^2*d)
\[ \int \frac {\coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\coth ^{2}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \] Input:
integrate(coth(d*x+c)**2/(a+b*sinh(d*x+c)),x)
Output:
Integral(coth(c + d*x)**2/(a + b*sinh(c + d*x)), x)
Time = 0.12 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.74 \[ \int \frac {\coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{2} d} - \frac {b \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{2} d} + \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{a^{2} d} + \frac {2}{{\left (a e^{\left (-2 \, d x - 2 \, c\right )} - a\right )} d} \] Input:
integrate(coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="maxima")
Output:
b*log(e^(-d*x - c) + 1)/(a^2*d) - b*log(e^(-d*x - c) - 1)/(a^2*d) + sqrt(a ^2 + b^2)*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(a^2*d) + 2/((a*e^(-2*d*x - 2*c) - a)*d)
Time = 0.16 (sec) , antiderivative size = 120, normalized size of antiderivative = 1.56 \[ \int \frac {\coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {\frac {b \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{2}} - \frac {b \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{2}} + \frac {\sqrt {a^{2} + b^{2}} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{a^{2}} - \frac {2}{a {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}}}{d} \] Input:
integrate(coth(d*x+c)^2/(a+b*sinh(d*x+c)),x, algorithm="giac")
Output:
(b*log(e^(d*x + c) + 1)/a^2 - b*log(abs(e^(d*x + c) - 1))/a^2 + sqrt(a^2 + b^2)*log(abs(2*b*e^(d*x + c) + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^(d*x + c) + 2*a + 2*sqrt(a^2 + b^2)))/a^2 - 2/(a*(e^(2*d*x + 2*c) - 1)))/d
Time = 1.68 (sec) , antiderivative size = 380, normalized size of antiderivative = 4.94 \[ \int \frac {\coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2}{a\,d-a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}}-\frac {b\,\ln \left (32\,a^2+32\,b^2-32\,a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-32\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^2\,d}+\frac {b\,\ln \left (32\,a^2+32\,b^2+32\,a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+32\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^2\,d}+\frac {\ln \left (128\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-64\,a\,b^3-64\,a^3\,b-32\,b^3\,\sqrt {a^2+b^2}+32\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-64\,a^2\,b\,\sqrt {a^2+b^2}+160\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+128\,a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}+96\,a\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^2\,d}-\frac {\ln \left (32\,b^3\,\sqrt {a^2+b^2}-64\,a\,b^3-64\,a^3\,b+128\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+32\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+64\,a^2\,b\,\sqrt {a^2+b^2}+160\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-128\,a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}-96\,a\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^2\,d} \] Input:
int(coth(c + d*x)^2/(a + b*sinh(c + d*x)),x)
Output:
2/(a*d - a*d*exp(2*c + 2*d*x)) - (b*log(32*a^2 + 32*b^2 - 32*a^2*exp(d*x)* exp(c) - 32*b^2*exp(d*x)*exp(c)))/(a^2*d) + (b*log(32*a^2 + 32*b^2 + 32*a^ 2*exp(d*x)*exp(c) + 32*b^2*exp(d*x)*exp(c)))/(a^2*d) + (log(128*a^4*exp(d* x)*exp(c) - 64*a*b^3 - 64*a^3*b - 32*b^3*(a^2 + b^2)^(1/2) + 32*b^4*exp(d* x)*exp(c) - 64*a^2*b*(a^2 + b^2)^(1/2) + 160*a^2*b^2*exp(d*x)*exp(c) + 128 *a^3*exp(d*x)*exp(c)*(a^2 + b^2)^(1/2) + 96*a*b^2*exp(d*x)*exp(c)*(a^2 + b ^2)^(1/2))*(a^2 + b^2)^(1/2))/(a^2*d) - (log(32*b^3*(a^2 + b^2)^(1/2) - 64 *a*b^3 - 64*a^3*b + 128*a^4*exp(d*x)*exp(c) + 32*b^4*exp(d*x)*exp(c) + 64* a^2*b*(a^2 + b^2)^(1/2) + 160*a^2*b^2*exp(d*x)*exp(c) - 128*a^3*exp(d*x)*e xp(c)*(a^2 + b^2)^(1/2) - 96*a*b^2*exp(d*x)*exp(c)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/(a^2*d)
Time = 0.21 (sec) , antiderivative size = 189, normalized size of antiderivative = 2.45 \[ \int \frac {\coth ^2(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {2 e^{2 d x +2 c} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) i -2 \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{d x +c} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) i -e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}-1\right ) b +e^{2 d x +2 c} \mathrm {log}\left (e^{d x +c}+1\right ) b -2 e^{2 d x +2 c} a +\mathrm {log}\left (e^{d x +c}-1\right ) b -\mathrm {log}\left (e^{d x +c}+1\right ) b}{a^{2} d \left (e^{2 d x +2 c}-1\right )} \] Input:
int(coth(d*x+c)^2/(a+b*sinh(d*x+c)),x)
Output:
(2*e**(2*c + 2*d*x)*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a **2 + b**2))*i - 2*sqrt(a**2 + b**2)*atan((e**(c + d*x)*b*i + a*i)/sqrt(a* *2 + b**2))*i - e**(2*c + 2*d*x)*log(e**(c + d*x) - 1)*b + e**(2*c + 2*d*x )*log(e**(c + d*x) + 1)*b - 2*e**(2*c + 2*d*x)*a + log(e**(c + d*x) - 1)*b - log(e**(c + d*x) + 1)*b)/(a**2*d*(e**(2*c + 2*d*x) - 1))