\(\int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx\) [484]

Optimal result

Integrand size = 27, antiderivative size = 111 \[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\left (a^2+2 b^2\right ) \text {arctanh}(\cosh (c+d x))}{2 a^3 d}+\frac {2 b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]

[Out]

-1/2*(a^2+2*b^2)*arctanh(cosh(d*x+c))/a^3/d+b*coth(d*x+c)/a^2/d-1/2*coth(d*x+c)*csch(d*x+c)/a/d+2*b*arctanh((b
-a*tanh(1/2*d*x+1/2*c))/(a^2+b^2)^(1/2))*(a^2+b^2)^(1/2)/a^3/d

Rubi [A] (verified)

Time = 0.38 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.296, Rules used = {2968, 3135, 3134, 3080, 3855, 2739, 632, 210} \[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {b \coth (c+d x)}{a^2 d}-\frac {\left (a^2+2 b^2\right ) \text {arctanh}(\cosh (c+d x))}{2 a^3 d}+\frac {2 b \sqrt {a^2+b^2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2+b^2}}\right )}{a^3 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \]

[In]

Int[(Coth[c + d*x]^2*Csch[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

-1/2*((a^2 + 2*b^2)*ArcTanh[Cosh[c + d*x]])/(a^3*d) + (2*b*Sqrt[a^2 + b^2]*ArcTanh[(b - a*Tanh[(c + d*x)/2])/S
qrt[a^2 + b^2]])/(a^3*d) + (b*Coth[c + d*x])/(a^2*d) - (Coth[c + d*x]*Csch[c + d*x])/(2*a*d)

Rule 210

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])

Rule 632

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 2739

Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x]}, Dis
t[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]

Rule 2968

Int[cos[(e_.) + (f_.)*(x_)]^2*((d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)
, x_Symbol] :> Int[(d*Sin[e + f*x])^n*(a + b*Sin[e + f*x])^m*(1 - Sin[e + f*x]^2), x] /; FreeQ[{a, b, d, e, f,
m, n}, x] && NeQ[a^2 - b^2, 0] && (IGtQ[m, 0] || IntegersQ[2*m, 2*n])

Rule 3080

Int[((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])/(((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)])), x_Symbol] :> Dist[(A*b - a*B)/(b*c - a*d), Int[1/(a + b*Sin[e + f*x]), x], x] + Dist[(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]

Rule 3134

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*s
in[(e_.) + (f_.)*(x_)] + (C_.)*sin[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(-(A*b^2 - a*b*B + a^2*C))*Cos[e
+ f*x]*(a + b*Sin[e + f*x])^(m + 1)*((c + d*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + D
ist[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[(m + 1)*
(b*c - a*d)*(a*A - b*B + a*C) + d*(A*b^2 - a*b*B + a^2*C)*(m + n + 2) - (c*(A*b^2 - a*b*B + a^2*C) + (m + 1)*(
b*c - a*d)*(A*b - a*B + b*C))*Sin[e + f*x] - d*(A*b^2 - a*b*B + a^2*C)*(m + n + 3)*Sin[e + f*x]^2, x], x], x]
/; FreeQ[{a, b, c, d, e, f, A, B, 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])))

Rule 3135

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*s
in[(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*Sin[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 - b^2))), x] + Dist[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] && L
tQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\text {csch}^3(c+d x) \left (1+\sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx \\ & = -\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {i \int \frac {\text {csch}^2(c+d x) \left (2 i b-i a \sinh (c+d x)+i b \sinh ^2(c+d x)\right )}{a+b \sinh (c+d x)} \, dx}{2 a} \\ & = \frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\int \frac {\text {csch}(c+d x) \left (-a^2-2 b^2+a b \sinh (c+d x)\right )}{a+b \sinh (c+d x)} \, dx}{2 a^2} \\ & = \frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\left (b \left (a^2+b^2\right )\right ) \int \frac {1}{a+b \sinh (c+d x)} \, dx}{a^3}+\frac {\left (a^2+2 b^2\right ) \int \text {csch}(c+d x) \, dx}{2 a^3} \\ & = -\frac {\left (a^2+2 b^2\right ) \text {arctanh}(\cosh (c+d x))}{2 a^3 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}+\frac {\left (2 i b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a-2 i b x+a x^2} \, dx,x,\tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a^3 d} \\ & = -\frac {\left (a^2+2 b^2\right ) \text {arctanh}(\cosh (c+d x))}{2 a^3 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d}-\frac {\left (4 i b \left (a^2+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-4 \left (a^2+b^2\right )-x^2} \, dx,x,-2 i b+2 a \tan \left (\frac {1}{2} (i c+i d x)\right )\right )}{a^3 d} \\ & = -\frac {\left (a^2+2 b^2\right ) \text {arctanh}(\cosh (c+d x))}{2 a^3 d}+\frac {2 b \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^3 d}+\frac {b \coth (c+d x)}{a^2 d}-\frac {\coth (c+d x) \text {csch}(c+d x)}{2 a d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.09 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.50 \[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {16 b \sqrt {-a^2-b^2} \arctan \left (\frac {b-a \tanh \left (\frac {1}{2} (c+d x)\right )}{\sqrt {-a^2-b^2}}\right )+4 a b \coth \left (\frac {1}{2} (c+d x)\right )-a^2 \text {csch}^2\left (\frac {1}{2} (c+d x)\right )-4 \left (a^2+2 b^2\right ) \log \left (\cosh \left (\frac {1}{2} (c+d x)\right )\right )+4 \left (a^2+2 b^2\right ) \log \left (\sinh \left (\frac {1}{2} (c+d x)\right )\right )-a^2 \text {sech}^2\left (\frac {1}{2} (c+d x)\right )+4 a b \tanh \left (\frac {1}{2} (c+d x)\right )}{8 a^3 d} \]

[In]

Integrate[(Coth[c + d*x]^2*Csch[c + d*x])/(a + b*Sinh[c + d*x]),x]

[Out]

(16*b*Sqrt[-a^2 - b^2]*ArcTan[(b - a*Tanh[(c + d*x)/2])/Sqrt[-a^2 - b^2]] + 4*a*b*Coth[(c + d*x)/2] - a^2*Csch
[(c + d*x)/2]^2 - 4*(a^2 + 2*b^2)*Log[Cosh[(c + d*x)/2]] + 4*(a^2 + 2*b^2)*Log[Sinh[(c + d*x)/2]] - a^2*Sech[(
c + d*x)/2]^2 + 4*a*b*Tanh[(c + d*x)/2])/(8*a^3*d)

Maple [A] (verified)

Time = 1.19 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.26

method	result	size
derivativedivides	\(\frac {\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (2 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{3}}}{d}\)	\(140\)
default	\(\frac {\frac {\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a}{2}+2 b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{4 a^{2}}-\frac {1}{8 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (2 a^{2}+4 b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}+\frac {b}{2 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}-\frac {2 b \sqrt {a^{2}+b^{2}}\, \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{3}}}{d}\)	\(140\)
risch	\(-\frac {{\mathrm e}^{3 d x +3 c} a -2 b \,{\mathrm e}^{2 d x +2 c}+a \,{\mathrm e}^{d x +c}+2 b}{a^{2} d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2}}+\frac {\sqrt {a^{2}+b^{2}}\, b \ln \left ({\mathrm e}^{d x +c}+\frac {a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,a^{3}}-\frac {\sqrt {a^{2}+b^{2}}\, b \ln \left ({\mathrm e}^{d x +c}-\frac {-a +\sqrt {a^{2}+b^{2}}}{b}\right )}{d \,a^{3}}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right )}{2 d a}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{d \,a^{3}}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right )}{2 d a}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{d \,a^{3}}\)	\(215\)

[In]

int(coth(d*x+c)^2*csch(d*x+c)/(a+b*sinh(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(1/4/a^2*(1/2*tanh(1/2*d*x+1/2*c)^2*a+2*b*tanh(1/2*d*x+1/2*c))-1/8/a/tanh(1/2*d*x+1/2*c)^2+1/4/a^3*(2*a^2+
4*b^2)*ln(tanh(1/2*d*x+1/2*c))+1/2*b/a^2/tanh(1/2*d*x+1/2*c)-2*b*(a^2+b^2)^(1/2)/a^3*arctanh(1/2*(2*a*tanh(1/2
*d*x+1/2*c)-2*b)/(a^2+b^2)^(1/2)))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 892 vs. \(2 (104) = 208\).

Time = 0.29 (sec) , antiderivative size = 892, normalized size of antiderivative = 8.04 \[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\text {Too large to display} \]

[In]

integrate(coth(d*x+c)^2*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="fricas")

[Out]

-1/2*(2*a^2*cosh(d*x + c)^3 + 2*a^2*sinh(d*x + c)^3 - 4*a*b*cosh(d*x + c)^2 + 2*a^2*cosh(d*x + c) + 2*(3*a^2*c
osh(d*x + c) - 2*a*b)*sinh(d*x + c)^2 - 2*(b*cosh(d*x + c)^4 + 4*b*cosh(d*x + c)*sinh(d*x + c)^3 + b*sinh(d*x
+ c)^4 - 2*b*cosh(d*x + c)^2 + 2*(3*b*cosh(d*x + c)^2 - b)*sinh(d*x + c)^2 + 4*(b*cosh(d*x + c)^3 - b*cosh(d*x
+ c))*sinh(d*x + c) + b)*sqrt(a^2 + b^2)*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*sqrt(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)) + 4*a*b + ((a^2 + 2*b^2)*cosh(d*x + c)^4 + 4*(a^2 + 2*b^2)*cosh(d*x + c)*sinh(d*x + c)^3 + (a^2 + 2*b^2)*
sinh(d*x + c)^4 - 2*(a^2 + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*b^2)*cosh(d*x + c)^2 - a^2 - 2*b^2)*sinh(d*x
+ c)^2 + a^2 + 2*b^2 + 4*((a^2 + 2*b^2)*cosh(d*x + c)^3 - (a^2 + 2*b^2)*cosh(d*x + c))*sinh(d*x + c))*log(cos
h(d*x + c) + sinh(d*x + c) + 1) - ((a^2 + 2*b^2)*cosh(d*x + c)^4 + 4*(a^2 + 2*b^2)*cosh(d*x + c)*sinh(d*x + c)
^3 + (a^2 + 2*b^2)*sinh(d*x + c)^4 - 2*(a^2 + 2*b^2)*cosh(d*x + c)^2 + 2*(3*(a^2 + 2*b^2)*cosh(d*x + c)^2 - a^
2 - 2*b^2)*sinh(d*x + c)^2 + a^2 + 2*b^2 + 4*((a^2 + 2*b^2)*cosh(d*x + c)^3 - (a^2 + 2*b^2)*cosh(d*x + c))*sin
h(d*x + c))*log(cosh(d*x + c) + sinh(d*x + c) - 1) + 2*(3*a^2*cosh(d*x + c)^2 - 4*a*b*cosh(d*x + c) + a^2)*sin
h(d*x + c))/(a^3*d*cosh(d*x + c)^4 + 4*a^3*d*cosh(d*x + c)*sinh(d*x + c)^3 + a^3*d*sinh(d*x + c)^4 - 2*a^3*d*c
osh(d*x + c)^2 + a^3*d + 2*(3*a^3*d*cosh(d*x + c)^2 - a^3*d)*sinh(d*x + c)^2 + 4*(a^3*d*cosh(d*x + c)^3 - a^3*
d*cosh(d*x + c))*sinh(d*x + c))

Sympy [F]

\[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\int \frac {\coth ^{2}{\left (c + d x \right )} \operatorname {csch}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]

[In]

integrate(coth(d*x+c)**2*csch(d*x+c)/(a+b*sinh(d*x+c)),x)

[Out]

Integral(coth(c + d*x)**2*csch(c + d*x)/(a + b*sinh(c + d*x)), x)

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 217 vs. \(2 (104) = 208\).

Time = 0.28 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.95 \[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {a e^{\left (-d x - c\right )} + 2 \, b e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-3 \, d x - 3 \, c\right )} - 2 \, b}{{\left (2 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )} - a^{2} e^{\left (-4 \, d x - 4 \, c\right )} - a^{2}\right )} d} - \frac {{\left (a^{2} + 2 \, b^{2}\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{2 \, a^{3} d} + \frac {{\left (a^{2} + 2 \, b^{2}\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{2 \, a^{3} d} - \frac {{\left (a^{2} b + b^{3}\right )} \log \left (\frac {b e^{\left (-d x - c\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-d x - c\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{3} d} \]

[In]

integrate(coth(d*x+c)^2*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="maxima")

[Out]

(a*e^(-d*x - c) + 2*b*e^(-2*d*x - 2*c) + a*e^(-3*d*x - 3*c) - 2*b)/((2*a^2*e^(-2*d*x - 2*c) - a^2*e^(-4*d*x -
4*c) - a^2)*d) - 1/2*(a^2 + 2*b^2)*log(e^(-d*x - c) + 1)/(a^3*d) + 1/2*(a^2 + 2*b^2)*log(e^(-d*x - c) - 1)/(a^
3*d) - (a^2*b + b^3)*log((b*e^(-d*x - c) - a - sqrt(a^2 + b^2))/(b*e^(-d*x - c) - a + sqrt(a^2 + b^2)))/(sqrt(
a^2 + b^2)*a^3*d)

Giac [A] (verification not implemented)

none

Time = 0.32 (sec) , antiderivative size = 182, normalized size of antiderivative = 1.64 \[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=-\frac {\frac {{\left (a^{2} + 2 \, b^{2}\right )} \log \left (e^{\left (d x + c\right )} + 1\right )}{a^{3}} - \frac {{\left (a^{2} + 2 \, b^{2}\right )} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right )}{a^{3}} + \frac {2 \, {\left (a^{2} b + b^{3}\right )} \log \left (\frac {{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{\left (d x + c\right )} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{3}} + \frac {2 \, {\left (a e^{\left (3 \, d x + 3 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a e^{\left (d x + c\right )} + 2 \, b\right )}}{a^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \]

[In]

integrate(coth(d*x+c)^2*csch(d*x+c)/(a+b*sinh(d*x+c)),x, algorithm="giac")

[Out]

-1/2*((a^2 + 2*b^2)*log(e^(d*x + c) + 1)/a^3 - (a^2 + 2*b^2)*log(abs(e^(d*x + c) - 1))/a^3 + 2*(a^2*b + b^3)*l
og(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)))/(sqrt(a^2 +
b^2)*a^3) + 2*(a*e^(3*d*x + 3*c) - 2*b*e^(2*d*x + 2*c) + a*e^(d*x + c) + 2*b)/(a^2*(e^(2*d*x + 2*c) - 1)^2))/d

Mupad [B] (verification not implemented)

Time = 1.41 (sec) , antiderivative size = 628, normalized size of antiderivative = 5.66 \[ \int \frac {\coth ^2(c+d x) \text {csch}(c+d x)}{a+b \sinh (c+d x)} \, dx=\frac {{\mathrm {e}}^{c+d\,x}}{a\,d-a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}}-\frac {2\,{\mathrm {e}}^{c+d\,x}}{a\,d-2\,a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}+a\,d\,{\mathrm {e}}^{4\,c+4\,d\,x}}-\frac {2\,b}{a^2\,d-a^2\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}}+\frac {\ln \left (4\,a^4+8\,b^4+12\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-12\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{2\,a\,d}-\frac {\ln \left (4\,a^4+8\,b^4+12\,a^2\,b^2+4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+12\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{2\,a\,d}+\frac {b^2\,\ln \left (4\,a^4+8\,b^4+12\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-12\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^3\,d}-\frac {b^2\,\ln \left (4\,a^4+8\,b^4+12\,a^2\,b^2+4\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+12\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{a^3\,d}-\frac {b\,\ln \left (32\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,a\,b^3-16\,a^3\,b-8\,b^3\,\sqrt {a^2+b^2}+8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,a^2\,b\,\sqrt {a^2+b^2}+40\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+32\,a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}+24\,a\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^3\,d}+\frac {b\,\ln \left (8\,b^3\,\sqrt {a^2+b^2}-16\,a\,b^3-16\,a^3\,b+32\,a^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+8\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+16\,a^2\,b\,\sqrt {a^2+b^2}+40\,a^2\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-32\,a^3\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}-24\,a\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^3\,d} \]

[In]

int(coth(c + d*x)^2/(sinh(c + d*x)*(a + b*sinh(c + d*x))),x)

[Out]

exp(c + d*x)/(a*d - a*d*exp(2*c + 2*d*x)) - (2*exp(c + d*x))/(a*d - 2*a*d*exp(2*c + 2*d*x) + a*d*exp(4*c + 4*d
*x)) - (2*b)/(a^2*d - a^2*d*exp(2*c + 2*d*x)) + log(4*a^4 + 8*b^4 + 12*a^2*b^2 - 4*a^4*exp(d*x)*exp(c) - 8*b^4
*exp(d*x)*exp(c) - 12*a^2*b^2*exp(d*x)*exp(c))/(2*a*d) - log(4*a^4 + 8*b^4 + 12*a^2*b^2 + 4*a^4*exp(d*x)*exp(c
) + 8*b^4*exp(d*x)*exp(c) + 12*a^2*b^2*exp(d*x)*exp(c))/(2*a*d) + (b^2*log(4*a^4 + 8*b^4 + 12*a^2*b^2 - 4*a^4*
exp(d*x)*exp(c) - 8*b^4*exp(d*x)*exp(c) - 12*a^2*b^2*exp(d*x)*exp(c)))/(a^3*d) - (b^2*log(4*a^4 + 8*b^4 + 12*a
^2*b^2 + 4*a^4*exp(d*x)*exp(c) + 8*b^4*exp(d*x)*exp(c) + 12*a^2*b^2*exp(d*x)*exp(c)))/(a^3*d) - (b*log(32*a^4*
exp(d*x)*exp(c) - 16*a*b^3 - 16*a^3*b - 8*b^3*(a^2 + b^2)^(1/2) + 8*b^4*exp(d*x)*exp(c) - 16*a^2*b*(a^2 + b^2)
^(1/2) + 40*a^2*b^2*exp(d*x)*exp(c) + 32*a^3*exp(d*x)*exp(c)*(a^2 + b^2)^(1/2) + 24*a*b^2*exp(d*x)*exp(c)*(a^2
+ b^2)^(1/2))*(a^2 + b^2)^(1/2))/(a^3*d) + (b*log(8*b^3*(a^2 + b^2)^(1/2) - 16*a*b^3 - 16*a^3*b + 32*a^4*exp(
d*x)*exp(c) + 8*b^4*exp(d*x)*exp(c) + 16*a^2*b*(a^2 + b^2)^(1/2) + 40*a^2*b^2*exp(d*x)*exp(c) - 32*a^3*exp(d*x
)*exp(c)*(a^2 + b^2)^(1/2) - 24*a*b^2*exp(d*x)*exp(c)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/(a^3*d)