3.1.79 \(\int \frac {\text {csch}^4(x)}{a+b \sinh (x)} \, dx\) [79]

Optimal. Leaf size=109 \[ -\frac {b \left (a^2-2 b^2\right ) \tanh ^{-1}(\cosh (x))}{2 a^4}-\frac {2 b^4 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^4 \sqrt {a^2+b^2}}+\frac {\left (2 a^2-3 b^2\right ) \coth (x)}{3 a^3}+\frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {\coth (x) \text {csch}^2(x)}{3 a} \]

[Out]

-1/2*b*(a^2-2*b^2)*arctanh(cosh(x))/a^4+1/3*(2*a^2-3*b^2)*coth(x)/a^3+1/2*b*coth(x)*csch(x)/a^2-1/3*coth(x)*cs
ch(x)^2/a-2*b^4*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/a^4/(a^2+b^2)^(1/2)

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Rubi [A]
time = 0.35, antiderivative size = 109, 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 = {2881, 3134, 3080, 3855, 2739, 632, 212} \begin {gather*} \frac {b \coth (x) \text {csch}(x)}{2 a^2}-\frac {b \left (a^2-2 b^2\right ) \tanh ^{-1}(\cosh (x))}{2 a^4}-\frac {2 b^4 \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^4 \sqrt {a^2+b^2}}+\frac {\left (2 a^2-3 b^2\right ) \coth (x)}{3 a^3}-\frac {\coth (x) \text {csch}^2(x)}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x]^4/(a + b*Sinh[x]),x]

[Out]

-1/2*(b*(a^2 - 2*b^2)*ArcTanh[Cosh[x]])/a^4 - (2*b^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^4*Sqrt[a^2
+ b^2]) + ((2*a^2 - 3*b^2)*Coth[x])/(3*a^3) + (b*Coth[x]*Csch[x])/(2*a^2) - (Coth[x]*Csch[x]^2)/(3*a)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[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 2881

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[(-b^2)*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*(b*c - a*d)*(m + 1) + b^2*d*(m + n + 2) - (b^2*c + b*(b*c - a*d)*(m + 1))*Sin[e + f*x] - b^2*d*(m +
n + 3)*Sin[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
&& NeQ[c^2 - d^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*n] && ((EqQ[a, 0] && IntegerQ[m] &&  !IntegerQ[n]) ||
!(IntegerQ[2*n] && LtQ[n, -1] && ((IntegerQ[n] &&  !IntegerQ[m]) || EqQ[a, 0])))

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 3855

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

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Mathematica [A]
time = 0.66, size = 186, normalized size = 1.71 \begin {gather*} \frac {\frac {48 b^4 \text {ArcTan}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\sqrt {-a^2-b^2}}+4 a \left (2 a^2-3 b^2\right ) \coth \left (\frac {x}{2}\right )+3 a^2 b \text {csch}^2\left (\frac {x}{2}\right )+12 a^2 b \log \left (\tanh \left (\frac {x}{2}\right )\right )-24 b^3 \log \left (\tanh \left (\frac {x}{2}\right )\right )+3 a^2 b \text {sech}^2\left (\frac {x}{2}\right )+8 a^3 \text {csch}^3(x) \sinh ^4\left (\frac {x}{2}\right )-\frac {1}{2} a^3 \text {csch}^4\left (\frac {x}{2}\right ) \sinh (x)+8 a^3 \tanh \left (\frac {x}{2}\right )-12 a b^2 \tanh \left (\frac {x}{2}\right )}{24 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((48*b^4*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + 4*a*(2*a^2 - 3*b^2)*Coth[x/2] + 3*a^2*
b*Csch[x/2]^2 + 12*a^2*b*Log[Tanh[x/2]] - 24*b^3*Log[Tanh[x/2]] + 3*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 + 8*a^3*Tanh[x/2] - 12*a*b^2*Tanh[x/2])/(24*a^4)

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Maple [A]
time = 0.62, size = 151, normalized size = 1.39

method	result	size
default	\(-\frac {\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right ) a^{2}}{3}+a b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-3 a^{2} \tanh \left (\frac {x}{2}\right )+4 b^{2} \tanh \left (\frac {x}{2}\right )}{8 a^{3}}+\frac {2 b^{4} \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{a^{4} \sqrt {a^{2}+b^{2}}}-\frac {1}{24 a \tanh \left (\frac {x}{2}\right )^{3}}-\frac {-3 a^{2}+4 b^{2}}{8 a^{3} \tanh \left (\frac {x}{2}\right )}+\frac {b}{8 a^{2} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {b \left (a^{2}-2 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{2 a^{4}}\)	\(151\)
risch	\(-\frac {-3 a b \,{\mathrm e}^{5 x}+6 b^{2} {\mathrm e}^{4 x}+12 a^{2} {\mathrm e}^{2 x}-12 b^{2} {\mathrm e}^{2 x}+3 b \,{\mathrm e}^{x} a -4 a^{2}+6 b^{2}}{3 a^{3} \left ({\mathrm e}^{2 x}-1\right )^{3}}-\frac {b \ln \left ({\mathrm e}^{x}+1\right )}{2 a^{2}}+\frac {b^{3} \ln \left ({\mathrm e}^{x}+1\right )}{a^{4}}+\frac {b^{4} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}-a^{2}-b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a^{4}}-\frac {b^{4} \ln \left ({\mathrm e}^{x}+\frac {a \sqrt {a^{2}+b^{2}}+a^{2}+b^{2}}{\sqrt {a^{2}+b^{2}}\, b}\right )}{\sqrt {a^{2}+b^{2}}\, a^{4}}+\frac {b \ln \left ({\mathrm e}^{x}-1\right )}{2 a^{2}}-\frac {b^{3} \ln \left ({\mathrm e}^{x}-1\right )}{a^{4}}\)	\(221\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^4/(a+b*sinh(x)),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.50, size = 194, normalized size = 1.78 \begin {gather*} \frac {b^{4} \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^{4}} - \frac {3 \, a b e^{\left (-x\right )} - 6 \, b^{2} e^{\left (-4 \, x\right )} - 3 \, a b e^{\left (-5 \, x\right )} + 4 \, a^{2} - 6 \, b^{2} - 12 \, {\left (a^{2} - b^{2}\right )} e^{\left (-2 \, x\right )}}{3 \, {\left (3 \, a^{3} e^{\left (-2 \, x\right )} - 3 \, a^{3} e^{\left (-4 \, x\right )} + a^{3} e^{\left (-6 \, x\right )} - a^{3}\right )}} - \frac {{\left (a^{2} b - 2 \, b^{3}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{2 \, a^{4}} + \frac {{\left (a^{2} b - 2 \, b^{3}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{2 \, a^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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^4) - 1/3*(3*a*b*
e^(-x) - 6*b^2*e^(-4*x) - 3*a*b*e^(-5*x) + 4*a^2 - 6*b^2 - 12*(a^2 - b^2)*e^(-2*x))/(3*a^3*e^(-2*x) - 3*a^3*e^
(-4*x) + a^3*e^(-6*x) - a^3) - 1/2*(a^2*b - 2*b^3)*log(e^(-x) + 1)/a^4 + 1/2*(a^2*b - 2*b^3)*log(e^(-x) - 1)/a
^4

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1676 vs. \(2 (97) = 194\).
time = 0.56, size = 1676, normalized size = 15.38 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*(a^4*b + a^2*b^3)*cosh(x)^5 + 6*(a^4*b + a^2*b^3)*sinh(x)^5 + 8*a^5 - 4*a^3*b^2 - 12*a*b^4 - 12*(a^3*b^
2 + a*b^4)*cosh(x)^4 - 6*(2*a^3*b^2 + 2*a*b^4 - 5*(a^4*b + a^2*b^3)*cosh(x))*sinh(x)^4 + 12*(5*(a^4*b + a^2*b^
3)*cosh(x)^2 - 4*(a^3*b^2 + a*b^4)*cosh(x))*sinh(x)^3 - 24*(a^5 - a*b^4)*cosh(x)^2 - 12*(2*a^5 - 2*a*b^4 - 5*(
a^4*b + a^2*b^3)*cosh(x)^3 + 6*(a^3*b^2 + a*b^4)*cosh(x)^2)*sinh(x)^2 + 6*(b^4*cosh(x)^6 + 6*b^4*cosh(x)*sinh(
x)^5 + b^4*sinh(x)^6 - 3*b^4*cosh(x)^4 + 3*b^4*cosh(x)^2 + 3*(5*b^4*cosh(x)^2 - b^4)*sinh(x)^4 - b^4 + 4*(5*b^
4*cosh(x)^3 - 3*b^4*cosh(x))*sinh(x)^3 + 3*(5*b^4*cosh(x)^4 - 6*b^4*cosh(x)^2 + b^4)*sinh(x)^2 + 6*(b^4*cosh(x
)^5 - 2*b^4*cosh(x)^3 + b^4*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*cosh(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)) - 6*(a^4*b + a^2*b^3)*cosh(x) - 3*((a^4*b - a^2
*b^3 - 2*b^5)*cosh(x)^6 + 6*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)*sinh(x)^5 + (a^4*b - a^2*b^3 - 2*b^5)*sinh(x)^6
- a^4*b + a^2*b^3 + 2*b^5 - 3*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^4 - 3*(a^4*b - a^2*b^3 - 2*b^5 - 5*(a^4*b - a^
2*b^3 - 2*b^5)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^3 - 3*(a^4*b - a^2*b^3 - 2*b^5)*c
osh(x))*sinh(x)^3 + 3*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^2 + 3*(a^4*b - a^2*b^3 - 2*b^5 + 5*(a^4*b - a^2*b^3 -
2*b^5)*cosh(x)^4 - 6*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^2)*sinh(x)^2 + 6*((a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^5 -
2*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^3 + (a^4*b - a^2*b^3 - 2*b^5)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1
) + 3*((a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^6 + 6*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)*sinh(x)^5 + (a^4*b - a^2*b^3
- 2*b^5)*sinh(x)^6 - a^4*b + a^2*b^3 + 2*b^5 - 3*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^4 - 3*(a^4*b - a^2*b^3 - 2*
b^5 - 5*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^3 - 3*(a^4*b -
a^2*b^3 - 2*b^5)*cosh(x))*sinh(x)^3 + 3*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^2 + 3*(a^4*b - a^2*b^3 - 2*b^5 + 5*
(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^4 - 6*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^2)*sinh(x)^2 + 6*((a^4*b - a^2*b^3 -
2*b^5)*cosh(x)^5 - 2*(a^4*b - a^2*b^3 - 2*b^5)*cosh(x)^3 + (a^4*b - a^2*b^3 - 2*b^5)*cosh(x))*sinh(x))*log(co
sh(x) + sinh(x) - 1) - 6*(a^4*b + a^2*b^3 - 5*(a^4*b + a^2*b^3)*cosh(x)^4 + 8*(a^3*b^2 + a*b^4)*cosh(x)^3 + 8*
(a^5 - a*b^4)*cosh(x))*sinh(x))/((a^6 + a^4*b^2)*cosh(x)^6 + 6*(a^6 + a^4*b^2)*cosh(x)*sinh(x)^5 + (a^6 + a^4*
b^2)*sinh(x)^6 - a^6 - a^4*b^2 - 3*(a^6 + a^4*b^2)*cosh(x)^4 - 3*(a^6 + a^4*b^2 - 5*(a^6 + a^4*b^2)*cosh(x)^2)
*sinh(x)^4 + 4*(5*(a^6 + a^4*b^2)*cosh(x)^3 - 3*(a^6 + a^4*b^2)*cosh(x))*sinh(x)^3 + 3*(a^6 + a^4*b^2)*cosh(x)
^2 + 3*(a^6 + a^4*b^2 + 5*(a^6 + a^4*b^2)*cosh(x)^4 - 6*(a^6 + a^4*b^2)*cosh(x)^2)*sinh(x)^2 + 6*((a^6 + a^4*b
^2)*cosh(x)^5 - 2*(a^6 + a^4*b^2)*cosh(x)^3 + (a^6 + a^4*b^2)*cosh(x))*sinh(x))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {csch}^{4}{\left (x \right )}}{a + b \sinh {\left (x \right )}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.44, size = 171, normalized size = 1.57 \begin {gather*} \frac {b^{4} \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^{4}} - \frac {{\left (a^{2} b - 2 \, b^{3}\right )} \log \left (e^{x} + 1\right )}{2 \, a^{4}} + \frac {{\left (a^{2} b - 2 \, b^{3}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{2 \, a^{4}} + \frac {3 \, a b e^{\left (5 \, x\right )} - 6 \, b^{2} e^{\left (4 \, x\right )} - 12 \, a^{2} e^{\left (2 \, x\right )} + 12 \, b^{2} e^{\left (2 \, x\right )} - 3 \, a b e^{x} + 4 \, a^{2} - 6 \, b^{2}}{3 \, a^{3} {\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b^4*log(abs(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^4) -
1/2*(a^2*b - 2*b^3)*log(e^x + 1)/a^4 + 1/2*(a^2*b - 2*b^3)*log(abs(e^x - 1))/a^4 + 1/3*(3*a*b*e^(5*x) - 6*b^2
*e^(4*x) - 12*a^2*e^(2*x) + 12*b^2*e^(2*x) - 3*a*b*e^x + 4*a^2 - 6*b^2)/(a^3*(e^(2*x) - 1)^3)

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Mupad [B]
time = 0.88, size = 694, normalized size = 6.37 \begin {gather*} \frac {8}{3\,\left (a-3\,a\,{\mathrm {e}}^{2\,x}+3\,a\,{\mathrm {e}}^{4\,x}-a\,{\mathrm {e}}^{6\,x}\right )}-\frac {4}{a-2\,a\,{\mathrm {e}}^{2\,x}+a\,{\mathrm {e}}^{4\,x}}-\frac {2\,b^2}{a^3\,{\mathrm {e}}^{2\,x}-a^3}+\frac {b\,\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^x-24\,b^4\,{\mathrm {e}}^x+20\,a^2\,b^2\,{\mathrm {e}}^x\right )}{2\,a^2}-\frac {b\,\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2+4\,a^4\,{\mathrm {e}}^x+24\,b^4\,{\mathrm {e}}^x-20\,a^2\,b^2\,{\mathrm {e}}^x\right )}{2\,a^2}-\frac {b^3\,\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2-4\,a^4\,{\mathrm {e}}^x-24\,b^4\,{\mathrm {e}}^x+20\,a^2\,b^2\,{\mathrm {e}}^x\right )}{a^4}+\frac {b^3\,\ln \left (4\,a^4+24\,b^4-20\,a^2\,b^2+4\,a^4\,{\mathrm {e}}^x+24\,b^4\,{\mathrm {e}}^x-20\,a^2\,b^2\,{\mathrm {e}}^x\right )}{a^4}+\frac {2\,b\,{\mathrm {e}}^x}{a^2\,{\mathrm {e}}^{4\,x}-2\,a^2\,{\mathrm {e}}^{2\,x}+a^2}+\frac {b\,{\mathrm {e}}^x}{a^2\,{\mathrm {e}}^{2\,x}-a^2}+\frac {b^4\,\ln \left (16\,a^5\,b^2-48\,a\,b^6-32\,a^3\,b^4-24\,b^6\,\sqrt {a^2+b^2}+24\,b^7\,{\mathrm {e}}^x-40\,a^2\,b^4\,\sqrt {a^2+b^2}+16\,a^4\,b^2\,\sqrt {a^2+b^2}-32\,a^6\,b\,{\mathrm {e}}^x+112\,a^2\,b^5\,{\mathrm {e}}^x+56\,a^4\,b^3\,{\mathrm {e}}^x+72\,a\,b^5\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}-32\,a^5\,b\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+72\,a^3\,b^3\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^6+a^4\,b^2}-\frac {b^4\,\ln \left (24\,b^6\,\sqrt {a^2+b^2}-48\,a\,b^6-32\,a^3\,b^4+16\,a^5\,b^2+24\,b^7\,{\mathrm {e}}^x+40\,a^2\,b^4\,\sqrt {a^2+b^2}-16\,a^4\,b^2\,\sqrt {a^2+b^2}-32\,a^6\,b\,{\mathrm {e}}^x+112\,a^2\,b^5\,{\mathrm {e}}^x+56\,a^4\,b^3\,{\mathrm {e}}^x-72\,a\,b^5\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}+32\,a^5\,b\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}-72\,a^3\,b^3\,{\mathrm {e}}^x\,\sqrt {a^2+b^2}\right )\,\sqrt {a^2+b^2}}{a^6+a^4\,b^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^4*(a + b*sinh(x))),x)

[Out]

8/(3*(a - 3*a*exp(2*x) + 3*a*exp(4*x) - a*exp(6*x))) - 4/(a - 2*a*exp(2*x) + a*exp(4*x)) - (2*b^2)/(a^3*exp(2*
x) - a^3) + (b*log(4*a^4 + 24*b^4 - 20*a^2*b^2 - 4*a^4*exp(x) - 24*b^4*exp(x) + 20*a^2*b^2*exp(x)))/(2*a^2) -
(b*log(4*a^4 + 24*b^4 - 20*a^2*b^2 + 4*a^4*exp(x) + 24*b^4*exp(x) - 20*a^2*b^2*exp(x)))/(2*a^2) - (b^3*log(4*a
^4 + 24*b^4 - 20*a^2*b^2 - 4*a^4*exp(x) - 24*b^4*exp(x) + 20*a^2*b^2*exp(x)))/a^4 + (b^3*log(4*a^4 + 24*b^4 -
20*a^2*b^2 + 4*a^4*exp(x) + 24*b^4*exp(x) - 20*a^2*b^2*exp(x)))/a^4 + (2*b*exp(x))/(a^2*exp(4*x) - 2*a^2*exp(2
*x) + a^2) + (b*exp(x))/(a^2*exp(2*x) - a^2) + (b^4*log(16*a^5*b^2 - 48*a*b^6 - 32*a^3*b^4 - 24*b^6*(a^2 + b^2
)^(1/2) + 24*b^7*exp(x) - 40*a^2*b^4*(a^2 + b^2)^(1/2) + 16*a^4*b^2*(a^2 + b^2)^(1/2) - 32*a^6*b*exp(x) + 112*
a^2*b^5*exp(x) + 56*a^4*b^3*exp(x) + 72*a*b^5*exp(x)*(a^2 + b^2)^(1/2) - 32*a^5*b*exp(x)*(a^2 + b^2)^(1/2) + 7
2*a^3*b^3*exp(x)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/(a^6 + a^4*b^2) - (b^4*log(24*b^6*(a^2 + b^2)^(1/2) - 4
8*a*b^6 - 32*a^3*b^4 + 16*a^5*b^2 + 24*b^7*exp(x) + 40*a^2*b^4*(a^2 + b^2)^(1/2) - 16*a^4*b^2*(a^2 + b^2)^(1/2
) - 32*a^6*b*exp(x) + 112*a^2*b^5*exp(x) + 56*a^4*b^3*exp(x) - 72*a*b^5*exp(x)*(a^2 + b^2)^(1/2) + 32*a^5*b*ex
p(x)*(a^2 + b^2)^(1/2) - 72*a^3*b^3*exp(x)*(a^2 + b^2)^(1/2))*(a^2 + b^2)^(1/2))/(a^6 + a^4*b^2)

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