3.243 \(\int \frac {\coth ^4(x)}{(a+b \sinh (x))^2} \, dx\)

Optimal. Leaf size=159 \[ -\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {2 \sqrt {a^2+b^2} \left (a^2+4 b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^5}+\frac {b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))} \]

[Out]

b*(3*a^2+4*b^2)*arctanh(cosh(x))/a^5-1/3*(7*a^2+12*b^2)*coth(x)/a^4+(a^2+2*b^2)*coth(x)*csch(x)/a^3/b-1/3*(3+4
*b^2/a^2)*coth(x)*csch(x)/b/(a+b*sinh(x))-1/3*coth(x)*csch(x)^2/a/(a+b*sinh(x))-2*(a^2+4*b^2)*arctanh((b-a*tan
h(1/2*x))/(a^2+b^2)^(1/2))*(a^2+b^2)^(1/2)/a^5

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Rubi [A]  time = 0.67, antiderivative size = 159, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.538, Rules used = {2724, 3055, 3001, 3770, 2660, 618, 206} \[ -\frac {2 \sqrt {a^2+b^2} \left (a^2+4 b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^5}-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (\frac {4 b^2}{a^2}+3\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^4/(a + b*Sinh[x])^2,x]

[Out]

(b*(3*a^2 + 4*b^2)*ArcTanh[Cosh[x]])/a^5 - (2*Sqrt[a^2 + b^2]*(a^2 + 4*b^2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2
+ b^2]])/a^5 - ((7*a^2 + 12*b^2)*Coth[x])/(3*a^4) + ((a^2 + 2*b^2)*Coth[x]*Csch[x])/(a^3*b) - ((3 + (4*b^2)/a
^2)*Coth[x]*Csch[x])/(3*b*(a + b*Sinh[x])) - (Coth[x]*Csch[x]^2)/(3*a*(a + b*Sinh[x]))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 618

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 2660

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 2724

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)/tan[(e_.) + (f_.)*(x_)]^4, x_Symbol] :> -Simp[(Cos[e + f*x]*(a
+ b*Sin[e + f*x])^(m + 1))/(3*a*f*Sin[e + f*x]^3), x] + (-Dist[1/(3*a^2*b*(m + 1)), Int[((a + b*Sin[e + f*x])
^(m + 1)*Simp[6*a^2 - b^2*(m - 1)*(m - 2) + a*b*(m + 1)*Sin[e + f*x] - (3*a^2 - b^2*m*(m - 2))*Sin[e + f*x]^2,
x])/Sin[e + f*x]^3, x], x] - Simp[((3*a^2 + b^2*(m - 2))*Cos[e + f*x]*(a + b*Sin[e + f*x])^(m + 1))/(3*a^2*b*
f*(m + 1)*Sin[e + f*x]^2), x]) /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]

Rule 3001

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 3055

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] + Dis
t[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] && Lt
Q[m, -1] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||  !(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&
!IntegerQ[m]) || EqQ[a, 0])))

Rule 3770

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

Rubi steps

\begin {align*} \int \frac {\coth ^4(x)}{(a+b \sinh (x))^2} \, dx &=-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}-\frac {\int \frac {\text {csch}^3(x) \left (6 \left (a^2+2 b^2\right )-a b \sinh (x)+\left (3 a^2+8 b^2\right ) \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{3 a^2 b}\\ &=\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}-\frac {i \int \frac {\text {csch}^2(x) \left (2 i b \left (7 a^2+12 b^2\right )-4 i a b^2 \sinh (x)+6 i b \left (a^2+2 b^2\right ) \sinh ^2(x)\right )}{a+b \sinh (x)} \, dx}{6 a^3 b}\\ &=-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}+\frac {\int \frac {\text {csch}(x) \left (-6 b^2 \left (3 a^2+4 b^2\right )+6 a b \left (a^2+2 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)} \, dx}{6 a^4 b}\\ &=-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}+\frac {\left (\left (a^2+b^2\right ) \left (a^2+4 b^2\right )\right ) \int \frac {1}{a+b \sinh (x)} \, dx}{a^5}-\frac {\left (b \left (3 a^2+4 b^2\right )\right ) \int \text {csch}(x) \, dx}{a^5}\\ &=\frac {b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}+\frac {\left (2 \left (a^2+b^2\right ) \left (a^2+4 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{a^5}\\ &=\frac {b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}-\frac {\left (4 \left (a^2+b^2\right ) \left (a^2+4 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{4 \left (a^2+b^2\right )-x^2} \, dx,x,2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{a^5}\\ &=\frac {b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}-\frac {2 \sqrt {a^2+b^2} \left (a^2+4 b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^5}-\frac {\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac {\left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 b}-\frac {\left (3+\frac {4 b^2}{a^2}\right ) \coth (x) \text {csch}(x)}{3 b (a+b \sinh (x))}-\frac {\coth (x) \text {csch}^2(x)}{3 a (a+b \sinh (x))}\\ \end {align*}

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Mathematica [A]  time = 0.82, size = 214, normalized size = 1.35 \[ \frac {-\frac {1}{2} a^3 \sinh (x) \text {csch}^4\left (\frac {x}{2}\right )+8 a^3 \sinh ^4\left (\frac {x}{2}\right ) \text {csch}^3(x)-4 a \left (4 a^2+9 b^2\right ) \tanh \left (\frac {x}{2}\right )-4 a \left (4 a^2+9 b^2\right ) \coth \left (\frac {x}{2}\right )-24 b \left (3 a^2+4 b^2\right ) \log \left (\tanh \left (\frac {x}{2}\right )\right )-\frac {24 a b \left (a^2+b^2\right ) \cosh (x)}{a+b \sinh (x)}+6 a^2 b \text {csch}^2\left (\frac {x}{2}\right )+6 a^2 b \text {sech}^2\left (\frac {x}{2}\right )+\frac {48 \left (a^4+5 a^2 b^2+4 b^4\right ) \tan ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}}{24 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^4/(a + b*Sinh[x])^2,x]

[Out]

((48*(a^4 + 5*a^2*b^2 + 4*b^4)*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] - 4*a*(4*a^2 + 9*b
^2)*Coth[x/2] + 6*a^2*b*Csch[x/2]^2 - 24*b*(3*a^2 + 4*b^2)*Log[Tanh[x/2]] + 6*a^2*b*Sech[x/2]^2 + 8*a^3*Csch[x
]^3*Sinh[x/2]^4 - (a^3*Csch[x/2]^4*Sinh[x])/2 - (24*a*b*(a^2 + b^2)*Cosh[x])/(a + b*Sinh[x]) - 4*a*(4*a^2 + 9*
b^2)*Tanh[x/2])/(24*a^5)

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fricas [B]  time = 1.97, size = 3648, normalized size = 22.94 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^4/(a+b*sinh(x))^2,x, algorithm="fricas")

[Out]

1/3*(6*(a^4 + 2*a^2*b^2)*cosh(x)^7 + 6*(a^4 + 2*a^2*b^2)*sinh(x)^7 - 6*(a^3*b + 4*a*b^3)*cosh(x)^6 - 6*(a^3*b
+ 4*a*b^3 - 7*(a^4 + 2*a^2*b^2)*cosh(x))*sinh(x)^6 - 6*(7*a^4 + 10*a^2*b^2)*cosh(x)^5 - 6*(7*a^4 + 10*a^2*b^2
- 21*(a^4 + 2*a^2*b^2)*cosh(x)^2 + 6*(a^3*b + 4*a*b^3)*cosh(x))*sinh(x)^5 + 6*(7*a^3*b + 12*a*b^3)*cosh(x)^4 +
6*(7*a^3*b + 12*a*b^3 + 35*(a^4 + 2*a^2*b^2)*cosh(x)^3 - 15*(a^3*b + 4*a*b^3)*cosh(x)^2 - 5*(7*a^4 + 10*a^2*b
^2)*cosh(x))*sinh(x)^4 + 14*a^3*b + 24*a*b^3 + 42*(a^4 + 2*a^2*b^2)*cosh(x)^3 + 6*(35*(a^4 + 2*a^2*b^2)*cosh(x
)^4 + 7*a^4 + 14*a^2*b^2 - 20*(a^3*b + 4*a*b^3)*cosh(x)^3 - 10*(7*a^4 + 10*a^2*b^2)*cosh(x)^2 + 4*(7*a^3*b + 1
2*a*b^3)*cosh(x))*sinh(x)^3 - 2*(25*a^3*b + 36*a*b^3)*cosh(x)^2 + 2*(63*(a^4 + 2*a^2*b^2)*cosh(x)^5 - 45*(a^3*
b + 4*a*b^3)*cosh(x)^4 - 25*a^3*b - 36*a*b^3 - 30*(7*a^4 + 10*a^2*b^2)*cosh(x)^3 + 18*(7*a^3*b + 12*a*b^3)*cos
h(x)^2 + 63*(a^4 + 2*a^2*b^2)*cosh(x))*sinh(x)^2 + 3*((a^2*b + 4*b^3)*cosh(x)^8 + (a^2*b + 4*b^3)*sinh(x)^8 +
2*(a^3 + 4*a*b^2)*cosh(x)^7 + 2*(a^3 + 4*a*b^2 + 4*(a^2*b + 4*b^3)*cosh(x))*sinh(x)^7 - 4*(a^2*b + 4*b^3)*cosh
(x)^6 - 2*(2*a^2*b + 8*b^3 - 14*(a^2*b + 4*b^3)*cosh(x)^2 - 7*(a^3 + 4*a*b^2)*cosh(x))*sinh(x)^6 - 6*(a^3 + 4*
a*b^2)*cosh(x)^5 + 2*(28*(a^2*b + 4*b^3)*cosh(x)^3 - 3*a^3 - 12*a*b^2 + 21*(a^3 + 4*a*b^2)*cosh(x)^2 - 12*(a^2
*b + 4*b^3)*cosh(x))*sinh(x)^5 + 6*(a^2*b + 4*b^3)*cosh(x)^4 + 2*(35*(a^2*b + 4*b^3)*cosh(x)^4 + 35*(a^3 + 4*a
*b^2)*cosh(x)^3 + 3*a^2*b + 12*b^3 - 30*(a^2*b + 4*b^3)*cosh(x)^2 - 15*(a^3 + 4*a*b^2)*cosh(x))*sinh(x)^4 + 6*
(a^3 + 4*a*b^2)*cosh(x)^3 + 2*(28*(a^2*b + 4*b^3)*cosh(x)^5 + 35*(a^3 + 4*a*b^2)*cosh(x)^4 - 40*(a^2*b + 4*b^3
)*cosh(x)^3 + 3*a^3 + 12*a*b^2 - 30*(a^3 + 4*a*b^2)*cosh(x)^2 + 12*(a^2*b + 4*b^3)*cosh(x))*sinh(x)^3 + a^2*b
+ 4*b^3 - 4*(a^2*b + 4*b^3)*cosh(x)^2 + 2*(14*(a^2*b + 4*b^3)*cosh(x)^6 + 21*(a^3 + 4*a*b^2)*cosh(x)^5 - 30*(a
^2*b + 4*b^3)*cosh(x)^4 - 30*(a^3 + 4*a*b^2)*cosh(x)^3 - 2*a^2*b - 8*b^3 + 18*(a^2*b + 4*b^3)*cosh(x)^2 + 9*(a
^3 + 4*a*b^2)*cosh(x))*sinh(x)^2 - 2*(a^3 + 4*a*b^2)*cosh(x) + 2*(4*(a^2*b + 4*b^3)*cosh(x)^7 + 7*(a^3 + 4*a*b
^2)*cosh(x)^6 - 12*(a^2*b + 4*b^3)*cosh(x)^5 - 15*(a^3 + 4*a*b^2)*cosh(x)^4 + 12*(a^2*b + 4*b^3)*cosh(x)^3 - a
^3 - 4*a*b^2 + 9*(a^3 + 4*a*b^2)*cosh(x)^2 - 4*(a^2*b + 4*b^3)*cosh(x))*sinh(x))*sqrt(a^2 + b^2)*log((b^2*cosh
(x)^2 + b^2*sinh(x)^2 + 2*a*b*cosh(x) + 2*a^2 + b^2 + 2*(b^2*cosh(x) + a*b)*sinh(x) - 2*sqrt(a^2 + b^2)*(b*cos
h(x) + b*sinh(x) + a))/(b*cosh(x)^2 + b*sinh(x)^2 + 2*a*cosh(x) + 2*(b*cosh(x) + a)*sinh(x) - b)) - 2*(11*a^4
+ 18*a^2*b^2)*cosh(x) + 3*((3*a^2*b^2 + 4*b^4)*cosh(x)^8 + (3*a^2*b^2 + 4*b^4)*sinh(x)^8 + 2*(3*a^3*b + 4*a*b^
3)*cosh(x)^7 + 2*(3*a^3*b + 4*a*b^3 + 4*(3*a^2*b^2 + 4*b^4)*cosh(x))*sinh(x)^7 - 4*(3*a^2*b^2 + 4*b^4)*cosh(x)
^6 - 2*(6*a^2*b^2 + 8*b^4 - 14*(3*a^2*b^2 + 4*b^4)*cosh(x)^2 - 7*(3*a^3*b + 4*a*b^3)*cosh(x))*sinh(x)^6 - 6*(3
*a^3*b + 4*a*b^3)*cosh(x)^5 - 2*(9*a^3*b + 12*a*b^3 - 28*(3*a^2*b^2 + 4*b^4)*cosh(x)^3 - 21*(3*a^3*b + 4*a*b^3
)*cosh(x)^2 + 12*(3*a^2*b^2 + 4*b^4)*cosh(x))*sinh(x)^5 + 6*(3*a^2*b^2 + 4*b^4)*cosh(x)^4 + 2*(35*(3*a^2*b^2 +
4*b^4)*cosh(x)^4 + 9*a^2*b^2 + 12*b^4 + 35*(3*a^3*b + 4*a*b^3)*cosh(x)^3 - 30*(3*a^2*b^2 + 4*b^4)*cosh(x)^2 -
15*(3*a^3*b + 4*a*b^3)*cosh(x))*sinh(x)^4 + 3*a^2*b^2 + 4*b^4 + 6*(3*a^3*b + 4*a*b^3)*cosh(x)^3 + 2*(28*(3*a^
2*b^2 + 4*b^4)*cosh(x)^5 + 35*(3*a^3*b + 4*a*b^3)*cosh(x)^4 + 9*a^3*b + 12*a*b^3 - 40*(3*a^2*b^2 + 4*b^4)*cosh
(x)^3 - 30*(3*a^3*b + 4*a*b^3)*cosh(x)^2 + 12*(3*a^2*b^2 + 4*b^4)*cosh(x))*sinh(x)^3 - 4*(3*a^2*b^2 + 4*b^4)*c
osh(x)^2 + 2*(14*(3*a^2*b^2 + 4*b^4)*cosh(x)^6 + 21*(3*a^3*b + 4*a*b^3)*cosh(x)^5 - 30*(3*a^2*b^2 + 4*b^4)*cos
h(x)^4 - 6*a^2*b^2 - 8*b^4 - 30*(3*a^3*b + 4*a*b^3)*cosh(x)^3 + 18*(3*a^2*b^2 + 4*b^4)*cosh(x)^2 + 9*(3*a^3*b
+ 4*a*b^3)*cosh(x))*sinh(x)^2 - 2*(3*a^3*b + 4*a*b^3)*cosh(x) + 2*(4*(3*a^2*b^2 + 4*b^4)*cosh(x)^7 + 7*(3*a^3*
b + 4*a*b^3)*cosh(x)^6 - 12*(3*a^2*b^2 + 4*b^4)*cosh(x)^5 - 15*(3*a^3*b + 4*a*b^3)*cosh(x)^4 - 3*a^3*b - 4*a*b
^3 + 12*(3*a^2*b^2 + 4*b^4)*cosh(x)^3 + 9*(3*a^3*b + 4*a*b^3)*cosh(x)^2 - 4*(3*a^2*b^2 + 4*b^4)*cosh(x))*sinh(
x))*log(cosh(x) + sinh(x) + 1) - 3*((3*a^2*b^2 + 4*b^4)*cosh(x)^8 + (3*a^2*b^2 + 4*b^4)*sinh(x)^8 + 2*(3*a^3*b
+ 4*a*b^3)*cosh(x)^7 + 2*(3*a^3*b + 4*a*b^3 + 4*(3*a^2*b^2 + 4*b^4)*cosh(x))*sinh(x)^7 - 4*(3*a^2*b^2 + 4*b^4
)*cosh(x)^6 - 2*(6*a^2*b^2 + 8*b^4 - 14*(3*a^2*b^2 + 4*b^4)*cosh(x)^2 - 7*(3*a^3*b + 4*a*b^3)*cosh(x))*sinh(x)
^6 - 6*(3*a^3*b + 4*a*b^3)*cosh(x)^5 - 2*(9*a^3*b + 12*a*b^3 - 28*(3*a^2*b^2 + 4*b^4)*cosh(x)^3 - 21*(3*a^3*b
+ 4*a*b^3)*cosh(x)^2 + 12*(3*a^2*b^2 + 4*b^4)*cosh(x))*sinh(x)^5 + 6*(3*a^2*b^2 + 4*b^4)*cosh(x)^4 + 2*(35*(3*
a^2*b^2 + 4*b^4)*cosh(x)^4 + 9*a^2*b^2 + 12*b^4 + 35*(3*a^3*b + 4*a*b^3)*cosh(x)^3 - 30*(3*a^2*b^2 + 4*b^4)*co
sh(x)^2 - 15*(3*a^3*b + 4*a*b^3)*cosh(x))*sinh(x)^4 + 3*a^2*b^2 + 4*b^4 + 6*(3*a^3*b + 4*a*b^3)*cosh(x)^3 + 2*
(28*(3*a^2*b^2 + 4*b^4)*cosh(x)^5 + 35*(3*a^3*b + 4*a*b^3)*cosh(x)^4 + 9*a^3*b + 12*a*b^3 - 40*(3*a^2*b^2 + 4*
b^4)*cosh(x)^3 - 30*(3*a^3*b + 4*a*b^3)*cosh(x)^2 + 12*(3*a^2*b^2 + 4*b^4)*cosh(x))*sinh(x)^3 - 4*(3*a^2*b^2 +
4*b^4)*cosh(x)^2 + 2*(14*(3*a^2*b^2 + 4*b^4)*cosh(x)^6 + 21*(3*a^3*b + 4*a*b^3)*cosh(x)^5 - 30*(3*a^2*b^2 + 4
*b^4)*cosh(x)^4 - 6*a^2*b^2 - 8*b^4 - 30*(3*a^3*b + 4*a*b^3)*cosh(x)^3 + 18*(3*a^2*b^2 + 4*b^4)*cosh(x)^2 + 9*
(3*a^3*b + 4*a*b^3)*cosh(x))*sinh(x)^2 - 2*(3*a^3*b + 4*a*b^3)*cosh(x) + 2*(4*(3*a^2*b^2 + 4*b^4)*cosh(x)^7 +
7*(3*a^3*b + 4*a*b^3)*cosh(x)^6 - 12*(3*a^2*b^2 + 4*b^4)*cosh(x)^5 - 15*(3*a^3*b + 4*a*b^3)*cosh(x)^4 - 3*a^3*
b - 4*a*b^3 + 12*(3*a^2*b^2 + 4*b^4)*cosh(x)^3 + 9*(3*a^3*b + 4*a*b^3)*cosh(x)^2 - 4*(3*a^2*b^2 + 4*b^4)*cosh(
x))*sinh(x))*log(cosh(x) + sinh(x) - 1) + 2*(21*(a^4 + 2*a^2*b^2)*cosh(x)^6 - 18*(a^3*b + 4*a*b^3)*cosh(x)^5 -
15*(7*a^4 + 10*a^2*b^2)*cosh(x)^4 - 11*a^4 - 18*a^2*b^2 + 12*(7*a^3*b + 12*a*b^3)*cosh(x)^3 + 63*(a^4 + 2*a^2
*b^2)*cosh(x)^2 - 2*(25*a^3*b + 36*a*b^3)*cosh(x))*sinh(x))/(a^5*b*cosh(x)^8 + a^5*b*sinh(x)^8 + 2*a^6*cosh(x)
^7 - 4*a^5*b*cosh(x)^6 - 6*a^6*cosh(x)^5 + 6*a^5*b*cosh(x)^4 + 6*a^6*cosh(x)^3 - 4*a^5*b*cosh(x)^2 + 2*(4*a^5*
b*cosh(x) + a^6)*sinh(x)^7 - 2*a^6*cosh(x) + 2*(14*a^5*b*cosh(x)^2 + 7*a^6*cosh(x) - 2*a^5*b)*sinh(x)^6 + a^5*
b + 2*(28*a^5*b*cosh(x)^3 + 21*a^6*cosh(x)^2 - 12*a^5*b*cosh(x) - 3*a^6)*sinh(x)^5 + 2*(35*a^5*b*cosh(x)^4 + 3
5*a^6*cosh(x)^3 - 30*a^5*b*cosh(x)^2 - 15*a^6*cosh(x) + 3*a^5*b)*sinh(x)^4 + 2*(28*a^5*b*cosh(x)^5 + 35*a^6*co
sh(x)^4 - 40*a^5*b*cosh(x)^3 - 30*a^6*cosh(x)^2 + 12*a^5*b*cosh(x) + 3*a^6)*sinh(x)^3 + 2*(14*a^5*b*cosh(x)^6
+ 21*a^6*cosh(x)^5 - 30*a^5*b*cosh(x)^4 - 30*a^6*cosh(x)^3 + 18*a^5*b*cosh(x)^2 + 9*a^6*cosh(x) - 2*a^5*b)*sin
h(x)^2 + 2*(4*a^5*b*cosh(x)^7 + 7*a^6*cosh(x)^6 - 12*a^5*b*cosh(x)^5 - 15*a^6*cosh(x)^4 + 12*a^5*b*cosh(x)^3 +
9*a^6*cosh(x)^2 - 4*a^5*b*cosh(x) - a^6)*sinh(x))

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giac [A]  time = 0.25, size = 242, normalized size = 1.52 \[ \frac {{\left (3 \, a^{2} b + 4 \, b^{3}\right )} \log \left (e^{x} + 1\right )}{a^{5}} - \frac {{\left (3 \, a^{2} b + 4 \, b^{3}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{a^{5}} + \frac {{\left (a^{4} + 5 \, a^{2} b^{2} + 4 \, b^{4}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} a^{5}} + \frac {2 \, {\left (a^{3} e^{x} + a b^{2} e^{x} - a^{2} b - b^{3}\right )}}{{\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )} a^{4}} + \frac {2 \, {\left (3 \, a b e^{\left (5 \, x\right )} - 6 \, a^{2} e^{\left (4 \, x\right )} - 9 \, b^{2} e^{\left (4 \, x\right )} + 6 \, a^{2} e^{\left (2 \, x\right )} + 18 \, b^{2} e^{\left (2 \, x\right )} - 3 \, a b e^{x} - 4 \, a^{2} - 9 \, b^{2}\right )}}{3 \, a^{4} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^4/(a+b*sinh(x))^2,x, algorithm="giac")

[Out]

(3*a^2*b + 4*b^3)*log(e^x + 1)/a^5 - (3*a^2*b + 4*b^3)*log(abs(e^x - 1))/a^5 + (a^4 + 5*a^2*b^2 + 4*b^4)*log(a
bs(2*b*e^x + 2*a - 2*sqrt(a^2 + b^2))/abs(2*b*e^x + 2*a + 2*sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*a^5) + 2*(a^3*e
^x + a*b^2*e^x - a^2*b - b^3)/((b*e^(2*x) + 2*a*e^x - b)*a^4) + 2/3*(3*a*b*e^(5*x) - 6*a^2*e^(4*x) - 9*b^2*e^(
4*x) + 6*a^2*e^(2*x) + 18*b^2*e^(2*x) - 3*a*b*e^x - 4*a^2 - 9*b^2)/(a^4*(e^(2*x) - 1)^3)

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maple [B]  time = 0.08, size = 357, normalized size = 2.25 \[ -\frac {\tanh ^{3}\left (\frac {x}{2}\right )}{24 a^{2}}-\frac {b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )}{4 a^{3}}-\frac {5 \tanh \left (\frac {x}{2}\right )}{8 a^{2}}-\frac {3 b^{2} \tanh \left (\frac {x}{2}\right )}{2 a^{4}}+\frac {2 b^{2} \tanh \left (\frac {x}{2}\right )}{a^{3} \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )}+\frac {2 \tanh \left (\frac {x}{2}\right ) b^{4}}{a^{5} \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )}+\frac {2 b}{a^{2} \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )}+\frac {2 b^{3}}{a^{4} \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )}+\frac {2 \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a \sqrt {a^{2}+b^{2}}}+\frac {10 \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right ) b^{2}}{a^{3} \sqrt {a^{2}+b^{2}}}+\frac {8 \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right ) b^{4}}{a^{5} \sqrt {a^{2}+b^{2}}}-\frac {1}{24 a^{2} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {5}{8 a^{2} \tanh \left (\frac {x}{2}\right )}-\frac {3 b^{2}}{2 a^{4} \tanh \left (\frac {x}{2}\right )}+\frac {b}{4 a^{3} \tanh \left (\frac {x}{2}\right )^{2}}-\frac {3 b \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{3}}-\frac {4 b^{3} \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^4/(a+b*sinh(x))^2,x)

[Out]

-1/24/a^2*tanh(1/2*x)^3-1/4/a^3*b*tanh(1/2*x)^2-5/8/a^2*tanh(1/2*x)-3/2/a^4*b^2*tanh(1/2*x)+2/a^3/(a*tanh(1/2*
x)^2-2*tanh(1/2*x)*b-a)*b^2*tanh(1/2*x)+2/a^5/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)*tanh(1/2*x)*b^4+2/a^2/(a*tan
h(1/2*x)^2-2*tanh(1/2*x)*b-a)*b+2/a^4/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)*b^3+2/a/(a^2+b^2)^(1/2)*arctanh(1/2*
(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/2))+10/a^3/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/2
))*b^2+8/a^5/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/2))*b^4-1/24/a^2/tanh(1/2*x)^3-5/8
/a^2/tanh(1/2*x)-3/2/a^4/tanh(1/2*x)*b^2+1/4/a^3*b/tanh(1/2*x)^2-3/a^3*b*ln(tanh(1/2*x))-4/a^5*b^3*ln(tanh(1/2
*x))

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maxima [B]  time = 0.45, size = 339, normalized size = 2.13 \[ -\frac {2 \, {\left (7 \, a^{2} b + 12 \, b^{3} + {\left (11 \, a^{3} + 18 \, a b^{2}\right )} e^{\left (-x\right )} - {\left (25 \, a^{2} b + 36 \, b^{3}\right )} e^{\left (-2 \, x\right )} - 21 \, {\left (a^{3} + 2 \, a b^{2}\right )} e^{\left (-3 \, x\right )} + 3 \, {\left (7 \, a^{2} b + 12 \, b^{3}\right )} e^{\left (-4 \, x\right )} + 3 \, {\left (7 \, a^{3} + 10 \, a b^{2}\right )} e^{\left (-5 \, x\right )} - 3 \, {\left (a^{2} b + 4 \, b^{3}\right )} e^{\left (-6 \, x\right )} - 3 \, {\left (a^{3} + 2 \, a b^{2}\right )} e^{\left (-7 \, x\right )}\right )}}{3 \, {\left (2 \, a^{5} e^{\left (-x\right )} - 4 \, a^{4} b e^{\left (-2 \, x\right )} - 6 \, a^{5} e^{\left (-3 \, x\right )} + 6 \, a^{4} b e^{\left (-4 \, x\right )} + 6 \, a^{5} e^{\left (-5 \, x\right )} - 4 \, a^{4} b e^{\left (-6 \, x\right )} - 2 \, a^{5} e^{\left (-7 \, x\right )} + a^{4} b e^{\left (-8 \, x\right )} + a^{4} b\right )}} + \frac {{\left (3 \, a^{2} b + 4 \, b^{3}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{5}} - \frac {{\left (3 \, a^{2} b + 4 \, b^{3}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{5}} + \frac {{\left (a^{4} + 5 \, a^{2} b^{2} + 4 \, b^{4}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} a^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)^4/(a+b*sinh(x))^2,x, algorithm="maxima")

[Out]

-2/3*(7*a^2*b + 12*b^3 + (11*a^3 + 18*a*b^2)*e^(-x) - (25*a^2*b + 36*b^3)*e^(-2*x) - 21*(a^3 + 2*a*b^2)*e^(-3*
x) + 3*(7*a^2*b + 12*b^3)*e^(-4*x) + 3*(7*a^3 + 10*a*b^2)*e^(-5*x) - 3*(a^2*b + 4*b^3)*e^(-6*x) - 3*(a^3 + 2*a
*b^2)*e^(-7*x))/(2*a^5*e^(-x) - 4*a^4*b*e^(-2*x) - 6*a^5*e^(-3*x) + 6*a^4*b*e^(-4*x) + 6*a^5*e^(-5*x) - 4*a^4*
b*e^(-6*x) - 2*a^5*e^(-7*x) + a^4*b*e^(-8*x) + a^4*b) + (3*a^2*b + 4*b^3)*log(e^(-x) + 1)/a^5 - (3*a^2*b + 4*b
^3)*log(e^(-x) - 1)/a^5 + (a^4 + 5*a^2*b^2 + 4*b^4)*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(
a^2 + b^2)))/(sqrt(a^2 + b^2)*a^5)

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mupad [B]  time = 1.53, size = 1450, normalized size = 9.12 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^4/(a + b*sinh(x))^2,x)

[Out]

(3*b*log(96*a^4 + 128*b^4 + 224*a^2*b^2 + 96*a^4*exp(x) + 128*b^4*exp(x) + 224*a^2*b^2*exp(x)))/a^3 - 4/(a^2*e
xp(2*x) - a^2) - (6*b^2)/(a^4*exp(2*x) - a^4) - 8/(3*(3*a^2*exp(2*x) - 3*a^2*exp(4*x) + a^2*exp(6*x) - a^2)) -
(4*a^3*b^7)/(a^5*b^7*exp(2*x) - a^7*b^5 - a^5*b^7 + a^7*b^5*exp(2*x) + 2*a^6*b^6*exp(x) + 2*a^8*b^4*exp(x)) -
(2*a^5*b^5)/(a^5*b^7*exp(2*x) - a^7*b^5 - a^5*b^7 + a^7*b^5*exp(2*x) + 2*a^6*b^6*exp(x) + 2*a^8*b^4*exp(x)) -
(3*b*log(96*a^4 + 128*b^4 + 224*a^2*b^2 - 96*a^4*exp(x) - 128*b^4*exp(x) - 224*a^2*b^2*exp(x)))/a^3 - 4/(a^2*
exp(4*x) - 2*a^2*exp(2*x) + a^2) - (4*b^3*log(96*a^4 + 128*b^4 + 224*a^2*b^2 - 96*a^4*exp(x) - 128*b^4*exp(x)
- 224*a^2*b^2*exp(x)))/a^5 + (4*b^3*log(96*a^4 + 128*b^4 + 224*a^2*b^2 + 96*a^4*exp(x) + 128*b^4*exp(x) + 224*
a^2*b^2*exp(x)))/a^5 + (log(128*a^6*exp(x) - 256*a*b^5 - 64*a^5*b - 320*a^3*b^3 - 128*b^5*(a^2 + b^2)^(1/2) +
128*b^6*exp(x) - 288*a^2*b^3*(a^2 + b^2)^(1/2) + 128*a^5*exp(x)*(a^2 + b^2)^(1/2) + 672*a^2*b^4*exp(x) + 672*a
^4*b^2*exp(x) - 64*a^4*b*(a^2 + b^2)^(1/2) + 384*a*b^4*exp(x)*(a^2 + b^2)^(1/2) + 608*a^3*b^2*exp(x)*(a^2 + b^
2)^(1/2))*(a^2 + b^2)^(1/2))/a^3 - (log(128*b^5*(a^2 + b^2)^(1/2) - 256*a*b^5 - 64*a^5*b - 320*a^3*b^3 + 128*a
^6*exp(x) + 128*b^6*exp(x) + 288*a^2*b^3*(a^2 + b^2)^(1/2) - 128*a^5*exp(x)*(a^2 + b^2)^(1/2) + 672*a^2*b^4*ex
p(x) + 672*a^4*b^2*exp(x) + 64*a^4*b*(a^2 + b^2)^(1/2) - 384*a*b^4*exp(x)*(a^2 + b^2)^(1/2) - 608*a^3*b^2*exp(
x)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/a^3 - (2*a*b^9)/(a^5*b^7*exp(2*x) - a^7*b^5 - a^5*b^7 + a^7*b^5*exp(2
*x) + 2*a^6*b^6*exp(x) + 2*a^8*b^4*exp(x)) + (4*b*exp(x))/(a^3*exp(4*x) - 2*a^3*exp(2*x) + a^3) + (2*b*exp(x))
/(a^3*exp(2*x) - a^3) + (4*b^2*log(128*a^6*exp(x) - 256*a*b^5 - 64*a^5*b - 320*a^3*b^3 - 128*b^5*(a^2 + b^2)^(
1/2) + 128*b^6*exp(x) - 288*a^2*b^3*(a^2 + b^2)^(1/2) + 128*a^5*exp(x)*(a^2 + b^2)^(1/2) + 672*a^2*b^4*exp(x)
+ 672*a^4*b^2*exp(x) - 64*a^4*b*(a^2 + b^2)^(1/2) + 384*a*b^4*exp(x)*(a^2 + b^2)^(1/2) + 608*a^3*b^2*exp(x)*(a
^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/a^5 - (4*b^2*log(128*b^5*(a^2 + b^2)^(1/2) - 256*a*b^5 - 64*a^5*b - 320*a^
3*b^3 + 128*a^6*exp(x) + 128*b^6*exp(x) + 288*a^2*b^3*(a^2 + b^2)^(1/2) - 128*a^5*exp(x)*(a^2 + b^2)^(1/2) + 6
72*a^2*b^4*exp(x) + 672*a^4*b^2*exp(x) + 64*a^4*b*(a^2 + b^2)^(1/2) - 384*a*b^4*exp(x)*(a^2 + b^2)^(1/2) - 608
*a^3*b^2*exp(x)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/a^5 + (2*a^2*b^9*exp(x))/(a^5*b^8*exp(2*x) - a^7*b^6 - a
^5*b^8 + a^7*b^6*exp(2*x) + 2*a^6*b^7*exp(x) + 2*a^8*b^5*exp(x)) + (4*a^4*b^7*exp(x))/(a^5*b^8*exp(2*x) - a^7*
b^6 - a^5*b^8 + a^7*b^6*exp(2*x) + 2*a^6*b^7*exp(x) + 2*a^8*b^5*exp(x)) + (2*a^6*b^5*exp(x))/(a^5*b^8*exp(2*x)
- a^7*b^6 - a^5*b^8 + a^7*b^6*exp(2*x) + 2*a^6*b^7*exp(x) + 2*a^8*b^5*exp(x))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\coth ^{4}{\relax (x )}}{\left (a + b \sinh {\relax (x )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**4/(a+b*sinh(x))**2,x)

[Out]

Integral(coth(x)**4/(a + b*sinh(x))**2, x)

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