3.88 \(\int \frac {\text {csch}^5(x)}{a+b \tanh (x)} \, dx\)

Optimal. Leaf size=255 \[ \frac {b^4 \text {sech}(x)}{a^5}-\frac {b^4 \tanh ^{-1}(\cosh (x))}{a^5}+\frac {3 b^3 \text {csch}(x)}{2 a^4}+\frac {b^3 \tan ^{-1}(\sinh (x))}{a^4}-\frac {b^3 \text {csch}(x) \text {sech}^2(x)}{2 a^4}-\frac {b^3 \tanh (x) \text {sech}(x)}{2 a^4}-\frac {3 b^2 \text {sech}(x)}{2 a^3}+\frac {3 b^2 \tanh ^{-1}(\cosh (x))}{2 a^3}-\frac {b^2 \text {csch}^2(x) \text {sech}(x)}{2 a^3}+\frac {b \text {csch}^3(x)}{3 a^2}-\frac {b \text {csch}(x)}{a^2}-\frac {b \tan ^{-1}(\sinh (x))}{a^2}+\frac {b^2 \left (a^2-b^2\right ) \text {sech}(x)}{a^5}-\frac {b \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{a^5}+\frac {b \left (a^2-b^2\right ) \tan ^{-1}(\sinh (x))}{a^4}-\frac {3 \tanh ^{-1}(\cosh (x))}{8 a}-\frac {\coth (x) \text {csch}^3(x)}{4 a}+\frac {3 \coth (x) \text {csch}(x)}{8 a} \]

[Out]

-b*arctan(sinh(x))/a^2+b^3*arctan(sinh(x))/a^4+b*(a^2-b^2)*arctan(sinh(x))/a^4-b*(a^2-b^2)^(3/2)*arctan((b*cos
h(x)+a*sinh(x))/(a^2-b^2)^(1/2))/a^5-3/8*arctanh(cosh(x))/a+3/2*b^2*arctanh(cosh(x))/a^3-b^4*arctanh(cosh(x))/
a^5-b*csch(x)/a^2+3/2*b^3*csch(x)/a^4+3/8*coth(x)*csch(x)/a+1/3*b*csch(x)^3/a^2-1/4*coth(x)*csch(x)^3/a-3/2*b^
2*sech(x)/a^3+b^4*sech(x)/a^5+b^2*(a^2-b^2)*sech(x)/a^5-1/2*b^2*csch(x)^2*sech(x)/a^3-1/2*b^3*csch(x)*sech(x)^
2/a^4-1/2*b^3*sech(x)*tanh(x)/a^4

________________________________________________________________________________________

Rubi [A]  time = 0.55, antiderivative size = 255, normalized size of antiderivative = 1.00, number of steps used = 29, number of rules used = 13, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.000, Rules used = {3518, 3110, 3768, 3770, 2621, 302, 207, 2622, 288, 321, 3104, 3074, 206} \[ \frac {3 b^3 \text {csch}(x)}{2 a^4}+\frac {b^4 \text {sech}(x)}{a^5}+\frac {b^2 \left (a^2-b^2\right ) \text {sech}(x)}{a^5}-\frac {3 b^2 \text {sech}(x)}{2 a^3}+\frac {b^3 \tan ^{-1}(\sinh (x))}{a^4}+\frac {b \left (a^2-b^2\right ) \tan ^{-1}(\sinh (x))}{a^4}-\frac {b^4 \tanh ^{-1}(\cosh (x))}{a^5}+\frac {3 b^2 \tanh ^{-1}(\cosh (x))}{2 a^3}-\frac {b^2 \text {csch}^2(x) \text {sech}(x)}{2 a^3}-\frac {b^3 \text {csch}(x) \text {sech}^2(x)}{2 a^4}-\frac {b^3 \tanh (x) \text {sech}(x)}{2 a^4}-\frac {b \left (a^2-b^2\right )^{3/2} \tan ^{-1}\left (\frac {a \sinh (x)+b \cosh (x)}{\sqrt {a^2-b^2}}\right )}{a^5}+\frac {b \text {csch}^3(x)}{3 a^2}-\frac {b \text {csch}(x)}{a^2}-\frac {b \tan ^{-1}(\sinh (x))}{a^2}-\frac {3 \tanh ^{-1}(\cosh (x))}{8 a}-\frac {\coth (x) \text {csch}^3(x)}{4 a}+\frac {3 \coth (x) \text {csch}(x)}{8 a} \]

Antiderivative was successfully verified.

[In]

Int[Csch[x]^5/(a + b*Tanh[x]),x]

[Out]

-((b*ArcTan[Sinh[x]])/a^2) + (b^3*ArcTan[Sinh[x]])/a^4 + (b*(a^2 - b^2)*ArcTan[Sinh[x]])/a^4 - (b*(a^2 - b^2)^
(3/2)*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/a^5 - (3*ArcTanh[Cosh[x]])/(8*a) + (3*b^2*ArcTanh[Cosh[
x]])/(2*a^3) - (b^4*ArcTanh[Cosh[x]])/a^5 - (b*Csch[x])/a^2 + (3*b^3*Csch[x])/(2*a^4) + (3*Coth[x]*Csch[x])/(8
*a) + (b*Csch[x]^3)/(3*a^2) - (Coth[x]*Csch[x]^3)/(4*a) - (3*b^2*Sech[x])/(2*a^3) + (b^4*Sech[x])/a^5 + (b^2*(
a^2 - b^2)*Sech[x])/a^5 - (b^2*Csch[x]^2*Sech[x])/(2*a^3) - (b^3*Csch[x]*Sech[x]^2)/(2*a^4) - (b^3*Sech[x]*Tan
h[x])/(2*a^4)

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 207

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

Rule 288

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^
n)^(p + 1))/(b*n*(p + 1)), x] - Dist[(c^n*(m - n + 1))/(b*n*(p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^(p + 1), x
], x] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0] && LtQ[p, -1] && GtQ[m + 1, n] &&  !ILtQ[(m + n*(p + 1) + 1)/n, 0]
&& IntBinomialQ[a, b, c, n, m, p, x]

Rule 302

Int[(x_)^(m_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[PolynomialDivide[x^m, a + b*x^n, x], x] /; FreeQ[{a,
b}, x] && IGtQ[m, 0] && IGtQ[n, 0] && GtQ[m, 2*n - 1]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 2621

Int[(csc[(e_.) + (f_.)*(x_)]*(a_.))^(m_)*sec[(e_.) + (f_.)*(x_)]^(n_.), x_Symbol] :> -Dist[(f*a^n)^(-1), Subst
[Int[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Csc[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && Integer
Q[(n + 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 2622

Int[csc[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sec[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Dist[1/(f*a^n), Subst[Int
[x^(m + n - 1)/(-1 + x^2/a^2)^((n + 1)/2), x], x, a*Sec[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n
+ 1)/2] &&  !(IntegerQ[(m + 1)/2] && LtQ[0, m, n])

Rule 3074

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

Rule 3104

Int[cos[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[Cos[c + d*x]^(m + 1)/(b*d*(m + 1)), x] + (-Dist[a/b^2, Int[Cos[c + d*x]^(m + 1), x], x] + Dist[(a^2 + b
^2)/b^2, Int[Cos[c + d*x]^(m + 2)/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[
a^2 + b^2, 0] && LtQ[m, -1]

Rule 3110

Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(
c_.) + (d_.)*(x_)]), x_Symbol] :> Int[ExpandTrig[(cos[c + d*x]^m*sin[c + d*x]^n)/(a*cos[c + d*x] + b*sin[c + d
*x]), x], x] /; FreeQ[{a, b, c, d, m, n}, x] && IntegersQ[m, n]

Rule 3518

Int[sin[(e_.) + (f_.)*(x_)]^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[(Sin[e + f*x]
^m*(a*Cos[e + f*x] + b*Sin[e + f*x])^n)/Cos[e + f*x]^n, x] /; FreeQ[{a, b, e, f}, x] && IntegerQ[(m - 1)/2] &&
ILtQ[n, 0] && ((LtQ[m, 5] && GtQ[n, -4]) || (EqQ[m, 5] && EqQ[n, -1]))

Rule 3768

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Csc[c + d*x])^(n - 1))/(d*(n -
1)), x] + Dist[(b^2*(n - 2))/(n - 1), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1
] && IntegerQ[2*n]

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

________________________________________________________________________________________

Mathematica [A]  time = 0.89, size = 296, normalized size = 1.16 \[ \frac {3 a^4 \text {sech}^4\left (\frac {x}{2}\right )+18 a^4 \text {sech}^2\left (\frac {x}{2}\right )+72 a^4 \log \left (\tanh \left (\frac {x}{2}\right )\right )+112 a^3 b \tanh \left (\frac {x}{2}\right )+a^3 \text {csch}^4\left (\frac {x}{2}\right ) (4 b \sinh (x)-3 a)+64 a^3 b \sinh ^4\left (\frac {x}{2}\right ) \text {csch}^3(x)-16 a b \left (7 a^2-6 b^2\right ) \coth \left (\frac {x}{2}\right )+6 a^2 \left (3 a^2-4 b^2\right ) \text {csch}^2\left (\frac {x}{2}\right )-24 a^2 b^2 \text {sech}^2\left (\frac {x}{2}\right )-288 a^2 b^2 \log \left (\tanh \left (\frac {x}{2}\right )\right )-384 a^2 b \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a-b} \sqrt {a+b}}\right )-96 a b^3 \tanh \left (\frac {x}{2}\right )+384 b^3 \sqrt {a-b} \sqrt {a+b} \tan ^{-1}\left (\frac {a \tanh \left (\frac {x}{2}\right )+b}{\sqrt {a-b} \sqrt {a+b}}\right )+192 b^4 \log \left (\tanh \left (\frac {x}{2}\right )\right )}{192 a^5} \]

Antiderivative was successfully verified.

[In]

Integrate[Csch[x]^5/(a + b*Tanh[x]),x]

[Out]

(-384*a^2*Sqrt[a - b]*b*Sqrt[a + b]*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])] + 384*Sqrt[a - b]*b^3*
Sqrt[a + b]*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])] - 16*a*b*(7*a^2 - 6*b^2)*Coth[x/2] + 6*a^2*(3*
a^2 - 4*b^2)*Csch[x/2]^2 + 72*a^4*Log[Tanh[x/2]] - 288*a^2*b^2*Log[Tanh[x/2]] + 192*b^4*Log[Tanh[x/2]] + 18*a^
4*Sech[x/2]^2 - 24*a^2*b^2*Sech[x/2]^2 + 3*a^4*Sech[x/2]^4 + 64*a^3*b*Csch[x]^3*Sinh[x/2]^4 + a^3*Csch[x/2]^4*
(-3*a + 4*b*Sinh[x]) + 112*a^3*b*Tanh[x/2] - 96*a*b^3*Tanh[x/2])/(192*a^5)

________________________________________________________________________________________

fricas [B]  time = 0.66, size = 5347, normalized size = 20.97 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^5/(a+b*tanh(x)),x, algorithm="fricas")

[Out]

[1/24*(6*(3*a^4 - 8*a^3*b - 4*a^2*b^2 + 8*a*b^3)*cosh(x)^7 + 42*(3*a^4 - 8*a^3*b - 4*a^2*b^2 + 8*a*b^3)*cosh(x
)*sinh(x)^6 + 6*(3*a^4 - 8*a^3*b - 4*a^2*b^2 + 8*a*b^3)*sinh(x)^7 - 2*(33*a^4 - 104*a^3*b - 12*a^2*b^2 + 72*a*
b^3)*cosh(x)^5 - 2*(33*a^4 - 104*a^3*b - 12*a^2*b^2 + 72*a*b^3 - 63*(3*a^4 - 8*a^3*b - 4*a^2*b^2 + 8*a*b^3)*co
sh(x)^2)*sinh(x)^5 + 10*(21*(3*a^4 - 8*a^3*b - 4*a^2*b^2 + 8*a*b^3)*cosh(x)^3 - (33*a^4 - 104*a^3*b - 12*a^2*b
^2 + 72*a*b^3)*cosh(x))*sinh(x)^4 - 2*(33*a^4 + 104*a^3*b - 12*a^2*b^2 - 72*a*b^3)*cosh(x)^3 + 2*(105*(3*a^4 -
8*a^3*b - 4*a^2*b^2 + 8*a*b^3)*cosh(x)^4 - 33*a^4 - 104*a^3*b + 12*a^2*b^2 + 72*a*b^3 - 10*(33*a^4 - 104*a^3*
b - 12*a^2*b^2 + 72*a*b^3)*cosh(x)^2)*sinh(x)^3 + 2*(63*(3*a^4 - 8*a^3*b - 4*a^2*b^2 + 8*a*b^3)*cosh(x)^5 - 10
*(33*a^4 - 104*a^3*b - 12*a^2*b^2 + 72*a*b^3)*cosh(x)^3 - 3*(33*a^4 + 104*a^3*b - 12*a^2*b^2 - 72*a*b^3)*cosh(
x))*sinh(x)^2 - 24*((a^2*b - b^3)*cosh(x)^8 + 8*(a^2*b - b^3)*cosh(x)*sinh(x)^7 + (a^2*b - b^3)*sinh(x)^8 - 4*
(a^2*b - b^3)*cosh(x)^6 - 4*(a^2*b - b^3 - 7*(a^2*b - b^3)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^2*b - b^3)*cosh(x)^3
- 3*(a^2*b - b^3)*cosh(x))*sinh(x)^5 + 6*(a^2*b - b^3)*cosh(x)^4 + 2*(35*(a^2*b - b^3)*cosh(x)^4 + 3*a^2*b -
3*b^3 - 30*(a^2*b - b^3)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^2*b - b^3)*cosh(x)^5 - 10*(a^2*b - b^3)*cosh(x)^3 + 3*
(a^2*b - b^3)*cosh(x))*sinh(x)^3 + a^2*b - b^3 - 4*(a^2*b - b^3)*cosh(x)^2 + 4*(7*(a^2*b - b^3)*cosh(x)^6 - 15
*(a^2*b - b^3)*cosh(x)^4 - a^2*b + b^3 + 9*(a^2*b - b^3)*cosh(x)^2)*sinh(x)^2 + 8*((a^2*b - b^3)*cosh(x)^7 - 3
*(a^2*b - b^3)*cosh(x)^5 + 3*(a^2*b - b^3)*cosh(x)^3 - (a^2*b - b^3)*cosh(x))*sinh(x))*sqrt(-a^2 + b^2)*log(((
a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + 2*sqrt(-a^2 + b^2)*(cosh(x) + sinh(x)) - a
+ b)/((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + a - b)) + 6*(3*a^4 + 8*a^3*b - 4*a^2
*b^2 - 8*a*b^3)*cosh(x) - 3*((3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^8 + 8*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)*s
inh(x)^7 + (3*a^4 - 12*a^2*b^2 + 8*b^4)*sinh(x)^8 - 4*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^6 - 4*(3*a^4 - 12*a
^2*b^2 + 8*b^4 - 7*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(
x)^3 - 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x))*sinh(x)^5 + 6*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^4 + 2*(35*(3
*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^4 + 9*a^4 - 36*a^2*b^2 + 24*b^4 - 30*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^2
)*sinh(x)^4 + 3*a^4 - 12*a^2*b^2 + 8*b^4 + 8*(7*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^5 - 10*(3*a^4 - 12*a^2*b^
2 + 8*b^4)*cosh(x)^3 + 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x))*sinh(x)^3 - 4*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh
(x)^2 + 4*(7*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^6 - 15*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^4 - 3*a^4 + 12*a
^2*b^2 - 8*b^4 + 9*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^2)*sinh(x)^2 + 8*((3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)
^7 - 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^5 + 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^3 - (3*a^4 - 12*a^2*b^2
+ 8*b^4)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) + 3*((3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^8 + 8*(3*a^4
- 12*a^2*b^2 + 8*b^4)*cosh(x)*sinh(x)^7 + (3*a^4 - 12*a^2*b^2 + 8*b^4)*sinh(x)^8 - 4*(3*a^4 - 12*a^2*b^2 + 8*b
^4)*cosh(x)^6 - 4*(3*a^4 - 12*a^2*b^2 + 8*b^4 - 7*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(3*
a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^3 - 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x))*sinh(x)^5 + 6*(3*a^4 - 12*a^2*b^
2 + 8*b^4)*cosh(x)^4 + 2*(35*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^4 + 9*a^4 - 36*a^2*b^2 + 24*b^4 - 30*(3*a^4
- 12*a^2*b^2 + 8*b^4)*cosh(x)^2)*sinh(x)^4 + 3*a^4 - 12*a^2*b^2 + 8*b^4 + 8*(7*(3*a^4 - 12*a^2*b^2 + 8*b^4)*co
sh(x)^5 - 10*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^3 + 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x))*sinh(x)^3 - 4*(3
*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^2 + 4*(7*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^6 - 15*(3*a^4 - 12*a^2*b^2 +
8*b^4)*cosh(x)^4 - 3*a^4 + 12*a^2*b^2 - 8*b^4 + 9*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^2)*sinh(x)^2 + 8*((3*a^
4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^7 - 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^5 + 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*
cosh(x)^3 - (3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) - 1) + 2*(21*(3*a^4 - 8*a^3*b
- 4*a^2*b^2 + 8*a*b^3)*cosh(x)^6 - 5*(33*a^4 - 104*a^3*b - 12*a^2*b^2 + 72*a*b^3)*cosh(x)^4 + 9*a^4 + 24*a^3*
b - 12*a^2*b^2 - 24*a*b^3 - 3*(33*a^4 + 104*a^3*b - 12*a^2*b^2 - 72*a*b^3)*cosh(x)^2)*sinh(x))/(a^5*cosh(x)^8
+ 8*a^5*cosh(x)*sinh(x)^7 + a^5*sinh(x)^8 - 4*a^5*cosh(x)^6 + 6*a^5*cosh(x)^4 - 4*a^5*cosh(x)^2 + 4*(7*a^5*cos
h(x)^2 - a^5)*sinh(x)^6 + 8*(7*a^5*cosh(x)^3 - 3*a^5*cosh(x))*sinh(x)^5 + a^5 + 2*(35*a^5*cosh(x)^4 - 30*a^5*c
osh(x)^2 + 3*a^5)*sinh(x)^4 + 8*(7*a^5*cosh(x)^5 - 10*a^5*cosh(x)^3 + 3*a^5*cosh(x))*sinh(x)^3 + 4*(7*a^5*cosh
(x)^6 - 15*a^5*cosh(x)^4 + 9*a^5*cosh(x)^2 - a^5)*sinh(x)^2 + 8*(a^5*cosh(x)^7 - 3*a^5*cosh(x)^5 + 3*a^5*cosh(
x)^3 - a^5*cosh(x))*sinh(x)), 1/24*(6*(3*a^4 - 8*a^3*b - 4*a^2*b^2 + 8*a*b^3)*cosh(x)^7 + 42*(3*a^4 - 8*a^3*b
- 4*a^2*b^2 + 8*a*b^3)*cosh(x)*sinh(x)^6 + 6*(3*a^4 - 8*a^3*b - 4*a^2*b^2 + 8*a*b^3)*sinh(x)^7 - 2*(33*a^4 - 1
04*a^3*b - 12*a^2*b^2 + 72*a*b^3)*cosh(x)^5 - 2*(33*a^4 - 104*a^3*b - 12*a^2*b^2 + 72*a*b^3 - 63*(3*a^4 - 8*a^
3*b - 4*a^2*b^2 + 8*a*b^3)*cosh(x)^2)*sinh(x)^5 + 10*(21*(3*a^4 - 8*a^3*b - 4*a^2*b^2 + 8*a*b^3)*cosh(x)^3 - (
33*a^4 - 104*a^3*b - 12*a^2*b^2 + 72*a*b^3)*cosh(x))*sinh(x)^4 - 2*(33*a^4 + 104*a^3*b - 12*a^2*b^2 - 72*a*b^3
)*cosh(x)^3 + 2*(105*(3*a^4 - 8*a^3*b - 4*a^2*b^2 + 8*a*b^3)*cosh(x)^4 - 33*a^4 - 104*a^3*b + 12*a^2*b^2 + 72*
a*b^3 - 10*(33*a^4 - 104*a^3*b - 12*a^2*b^2 + 72*a*b^3)*cosh(x)^2)*sinh(x)^3 + 2*(63*(3*a^4 - 8*a^3*b - 4*a^2*
b^2 + 8*a*b^3)*cosh(x)^5 - 10*(33*a^4 - 104*a^3*b - 12*a^2*b^2 + 72*a*b^3)*cosh(x)^3 - 3*(33*a^4 + 104*a^3*b -
12*a^2*b^2 - 72*a*b^3)*cosh(x))*sinh(x)^2 + 48*((a^2*b - b^3)*cosh(x)^8 + 8*(a^2*b - b^3)*cosh(x)*sinh(x)^7 +
(a^2*b - b^3)*sinh(x)^8 - 4*(a^2*b - b^3)*cosh(x)^6 - 4*(a^2*b - b^3 - 7*(a^2*b - b^3)*cosh(x)^2)*sinh(x)^6 +
8*(7*(a^2*b - b^3)*cosh(x)^3 - 3*(a^2*b - b^3)*cosh(x))*sinh(x)^5 + 6*(a^2*b - b^3)*cosh(x)^4 + 2*(35*(a^2*b
- b^3)*cosh(x)^4 + 3*a^2*b - 3*b^3 - 30*(a^2*b - b^3)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^2*b - b^3)*cosh(x)^5 - 10
*(a^2*b - b^3)*cosh(x)^3 + 3*(a^2*b - b^3)*cosh(x))*sinh(x)^3 + a^2*b - b^3 - 4*(a^2*b - b^3)*cosh(x)^2 + 4*(7
*(a^2*b - b^3)*cosh(x)^6 - 15*(a^2*b - b^3)*cosh(x)^4 - a^2*b + b^3 + 9*(a^2*b - b^3)*cosh(x)^2)*sinh(x)^2 + 8
*((a^2*b - b^3)*cosh(x)^7 - 3*(a^2*b - b^3)*cosh(x)^5 + 3*(a^2*b - b^3)*cosh(x)^3 - (a^2*b - b^3)*cosh(x))*sin
h(x))*sqrt(a^2 - b^2)*arctan(sqrt(a^2 - b^2)/((a + b)*cosh(x) + (a + b)*sinh(x))) + 6*(3*a^4 + 8*a^3*b - 4*a^2
*b^2 - 8*a*b^3)*cosh(x) - 3*((3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^8 + 8*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)*s
inh(x)^7 + (3*a^4 - 12*a^2*b^2 + 8*b^4)*sinh(x)^8 - 4*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^6 - 4*(3*a^4 - 12*a
^2*b^2 + 8*b^4 - 7*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(
x)^3 - 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x))*sinh(x)^5 + 6*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^4 + 2*(35*(3
*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^4 + 9*a^4 - 36*a^2*b^2 + 24*b^4 - 30*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^2
)*sinh(x)^4 + 3*a^4 - 12*a^2*b^2 + 8*b^4 + 8*(7*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^5 - 10*(3*a^4 - 12*a^2*b^
2 + 8*b^4)*cosh(x)^3 + 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x))*sinh(x)^3 - 4*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh
(x)^2 + 4*(7*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^6 - 15*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^4 - 3*a^4 + 12*a
^2*b^2 - 8*b^4 + 9*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^2)*sinh(x)^2 + 8*((3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)
^7 - 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^5 + 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^3 - (3*a^4 - 12*a^2*b^2
+ 8*b^4)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) + 3*((3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^8 + 8*(3*a^4
- 12*a^2*b^2 + 8*b^4)*cosh(x)*sinh(x)^7 + (3*a^4 - 12*a^2*b^2 + 8*b^4)*sinh(x)^8 - 4*(3*a^4 - 12*a^2*b^2 + 8*b
^4)*cosh(x)^6 - 4*(3*a^4 - 12*a^2*b^2 + 8*b^4 - 7*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(3*
a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^3 - 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x))*sinh(x)^5 + 6*(3*a^4 - 12*a^2*b^
2 + 8*b^4)*cosh(x)^4 + 2*(35*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^4 + 9*a^4 - 36*a^2*b^2 + 24*b^4 - 30*(3*a^4
- 12*a^2*b^2 + 8*b^4)*cosh(x)^2)*sinh(x)^4 + 3*a^4 - 12*a^2*b^2 + 8*b^4 + 8*(7*(3*a^4 - 12*a^2*b^2 + 8*b^4)*co
sh(x)^5 - 10*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^3 + 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x))*sinh(x)^3 - 4*(3
*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^2 + 4*(7*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^6 - 15*(3*a^4 - 12*a^2*b^2 +
8*b^4)*cosh(x)^4 - 3*a^4 + 12*a^2*b^2 - 8*b^4 + 9*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^2)*sinh(x)^2 + 8*((3*a^
4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^7 - 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x)^5 + 3*(3*a^4 - 12*a^2*b^2 + 8*b^4)*
cosh(x)^3 - (3*a^4 - 12*a^2*b^2 + 8*b^4)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) - 1) + 2*(21*(3*a^4 - 8*a^3*b
- 4*a^2*b^2 + 8*a*b^3)*cosh(x)^6 - 5*(33*a^4 - 104*a^3*b - 12*a^2*b^2 + 72*a*b^3)*cosh(x)^4 + 9*a^4 + 24*a^3*
b - 12*a^2*b^2 - 24*a*b^3 - 3*(33*a^4 + 104*a^3*b - 12*a^2*b^2 - 72*a*b^3)*cosh(x)^2)*sinh(x))/(a^5*cosh(x)^8
+ 8*a^5*cosh(x)*sinh(x)^7 + a^5*sinh(x)^8 - 4*a^5*cosh(x)^6 + 6*a^5*cosh(x)^4 - 4*a^5*cosh(x)^2 + 4*(7*a^5*cos
h(x)^2 - a^5)*sinh(x)^6 + 8*(7*a^5*cosh(x)^3 - 3*a^5*cosh(x))*sinh(x)^5 + a^5 + 2*(35*a^5*cosh(x)^4 - 30*a^5*c
osh(x)^2 + 3*a^5)*sinh(x)^4 + 8*(7*a^5*cosh(x)^5 - 10*a^5*cosh(x)^3 + 3*a^5*cosh(x))*sinh(x)^3 + 4*(7*a^5*cosh
(x)^6 - 15*a^5*cosh(x)^4 + 9*a^5*cosh(x)^2 - a^5)*sinh(x)^2 + 8*(a^5*cosh(x)^7 - 3*a^5*cosh(x)^5 + 3*a^5*cosh(
x)^3 - a^5*cosh(x))*sinh(x))]

________________________________________________________________________________________

giac [A]  time = 0.13, size = 273, normalized size = 1.07 \[ -\frac {{\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left (e^{x} + 1\right )}{8 \, a^{5}} + \frac {{\left (3 \, a^{4} - 12 \, a^{2} b^{2} + 8 \, b^{4}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{8 \, a^{5}} - \frac {2 \, {\left (a^{4} b - 2 \, a^{2} b^{3} + b^{5}\right )} \arctan \left (\frac {a e^{x} + b e^{x}}{\sqrt {a^{2} - b^{2}}}\right )}{\sqrt {a^{2} - b^{2}} a^{5}} + \frac {9 \, a^{3} e^{\left (7 \, x\right )} - 24 \, a^{2} b e^{\left (7 \, x\right )} - 12 \, a b^{2} e^{\left (7 \, x\right )} + 24 \, b^{3} e^{\left (7 \, x\right )} - 33 \, a^{3} e^{\left (5 \, x\right )} + 104 \, a^{2} b e^{\left (5 \, x\right )} + 12 \, a b^{2} e^{\left (5 \, x\right )} - 72 \, b^{3} e^{\left (5 \, x\right )} - 33 \, a^{3} e^{\left (3 \, x\right )} - 104 \, a^{2} b e^{\left (3 \, x\right )} + 12 \, a b^{2} e^{\left (3 \, x\right )} + 72 \, b^{3} e^{\left (3 \, x\right )} + 9 \, a^{3} e^{x} + 24 \, a^{2} b e^{x} - 12 \, a b^{2} e^{x} - 24 \, b^{3} e^{x}}{12 \, a^{4} {\left (e^{\left (2 \, x\right )} - 1\right )}^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^5/(a+b*tanh(x)),x, algorithm="giac")

[Out]

-1/8*(3*a^4 - 12*a^2*b^2 + 8*b^4)*log(e^x + 1)/a^5 + 1/8*(3*a^4 - 12*a^2*b^2 + 8*b^4)*log(abs(e^x - 1))/a^5 -
2*(a^4*b - 2*a^2*b^3 + b^5)*arctan((a*e^x + b*e^x)/sqrt(a^2 - b^2))/(sqrt(a^2 - b^2)*a^5) + 1/12*(9*a^3*e^(7*x
) - 24*a^2*b*e^(7*x) - 12*a*b^2*e^(7*x) + 24*b^3*e^(7*x) - 33*a^3*e^(5*x) + 104*a^2*b*e^(5*x) + 12*a*b^2*e^(5*
x) - 72*b^3*e^(5*x) - 33*a^3*e^(3*x) - 104*a^2*b*e^(3*x) + 12*a*b^2*e^(3*x) + 72*b^3*e^(3*x) + 9*a^3*e^x + 24*
a^2*b*e^x - 12*a*b^2*e^x - 24*b^3*e^x)/(a^4*(e^(2*x) - 1)^4)

________________________________________________________________________________________

maple [A]  time = 0.14, size = 311, normalized size = 1.22 \[ \frac {\tanh ^{4}\left (\frac {x}{2}\right )}{64 a}-\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right ) b}{24 a^{2}}-\frac {\tanh ^{2}\left (\frac {x}{2}\right )}{8 a}+\frac {\left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b^{2}}{8 a^{3}}+\frac {5 \tanh \left (\frac {x}{2}\right ) b}{8 a^{2}}-\frac {\tanh \left (\frac {x}{2}\right ) b^{3}}{2 a^{4}}-\frac {2 b \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a \sqrt {a^{2}-b^{2}}}+\frac {4 b^{3} \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{3} \sqrt {a^{2}-b^{2}}}-\frac {2 b^{5} \arctan \left (\frac {2 a \tanh \left (\frac {x}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{a^{5} \sqrt {a^{2}-b^{2}}}-\frac {1}{64 a \tanh \left (\frac {x}{2}\right )^{4}}+\frac {1}{8 a \tanh \left (\frac {x}{2}\right )^{2}}-\frac {b^{2}}{8 a^{3} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{8 a}-\frac {3 \ln \left (\tanh \left (\frac {x}{2}\right )\right ) b^{2}}{2 a^{3}}+\frac {\ln \left (\tanh \left (\frac {x}{2}\right )\right ) b^{4}}{a^{5}}+\frac {b}{24 a^{2} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {5 b}{8 a^{2} \tanh \left (\frac {x}{2}\right )}+\frac {b^{3}}{2 a^{4} \tanh \left (\frac {x}{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^5/(a+b*tanh(x)),x)

[Out]

1/64/a*tanh(1/2*x)^4-1/24/a^2*tanh(1/2*x)^3*b-1/8/a*tanh(1/2*x)^2+1/8/a^3*tanh(1/2*x)^2*b^2+5/8/a^2*tanh(1/2*x
)*b-1/2/a^4*tanh(1/2*x)*b^3-2/a*b/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^2)^(1/2))+4*b^3/a^3/
(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+2*b)/(a^2-b^2)^(1/2))-2*b^5/a^5/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*ta
nh(1/2*x)+2*b)/(a^2-b^2)^(1/2))-1/64/a/tanh(1/2*x)^4+1/8/a/tanh(1/2*x)^2-1/8/a^3/tanh(1/2*x)^2*b^2+3/8/a*ln(ta
nh(1/2*x))-3/2/a^3*ln(tanh(1/2*x))*b^2+1/a^5*ln(tanh(1/2*x))*b^4+1/24/a^2*b/tanh(1/2*x)^3-5/8/a^2*b/tanh(1/2*x
)+1/2*b^3/a^4/tanh(1/2*x)

________________________________________________________________________________________

maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)^5/(a+b*tanh(x)),x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?`
for more details)Is 4*b^2-4*a^2 positive or negative?

________________________________________________________________________________________

mupad [B]  time = 3.31, size = 753, normalized size = 2.95 \[ \frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (3\,a^4-12\,a^2\,b^2+8\,b^4\right )}{8\,a^5}-\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (3\,a^4-12\,a^2\,b^2+8\,b^4\right )}{8\,a^5}-\frac {4\,{\mathrm {e}}^x}{a\,\left (6\,{\mathrm {e}}^{4\,x}-4\,{\mathrm {e}}^{2\,x}-4\,{\mathrm {e}}^{6\,x}+{\mathrm {e}}^{8\,x}+1\right )}-\frac {2\,{\mathrm {e}}^x\,\left (9\,a-4\,b\right )}{3\,a^2\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {{\mathrm {e}}^x\,\left (3\,a^2-16\,a\,b+12\,b^2\right )}{6\,a^3\,\left ({\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1\right )}-\frac {{\mathrm {e}}^x\,\left (-3\,a^3+8\,a^2\,b+4\,a\,b^2-8\,b^3\right )}{4\,a^4\,\left ({\mathrm {e}}^{2\,x}-1\right )}+\frac {b\,\ln \left (\frac {b\,{\mathrm {e}}^x\,{\left (a-b\right )}^2\,\left (-9\,a^7-24\,a^6\,b+144\,a^5\,b^2+24\,a^4\,b^3-456\,a^3\,b^4+224\,a^2\,b^5+288\,a\,b^6-192\,b^7\right )}{2\,a^{12}\,\left (a+b\right )}-\frac {b\,\left (a-b\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (8\,a^5\,b^3-9\,a^8-9\,a^7\,b+8\,a^6\,b^2+192\,b^5\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}-224\,a^2\,b^3\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}-88\,a^3\,b^2\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}+96\,a\,b^4\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}+24\,a^4\,b\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}\right )}{2\,a^{12}\,{\left (a+b\right )}^4}\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^5}-\frac {b\,\ln \left (\frac {b\,{\mathrm {e}}^x\,{\left (a-b\right )}^2\,\left (-9\,a^7-24\,a^6\,b+144\,a^5\,b^2+24\,a^4\,b^3-456\,a^3\,b^4+224\,a^2\,b^5+288\,a\,b^6-192\,b^7\right )}{2\,a^{12}\,\left (a+b\right )}-\frac {b\,\left (a-b\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}\,\left (9\,a^7\,b+9\,a^8-8\,a^5\,b^3-8\,a^6\,b^2+192\,b^5\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}-224\,a^2\,b^3\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}-88\,a^3\,b^2\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}+96\,a\,b^4\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}+24\,a^4\,b\,{\mathrm {e}}^x\,\sqrt {-{\left (a^2-b^2\right )}^3}\right )}{2\,a^{12}\,{\left (a+b\right )}^4}\right )\,\sqrt {-{\left (a+b\right )}^3\,{\left (a-b\right )}^3}}{a^5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^5*(a + b*tanh(x))),x)

[Out]

(log(exp(x) - 1)*(3*a^4 + 8*b^4 - 12*a^2*b^2))/(8*a^5) - (log(exp(x) + 1)*(3*a^4 + 8*b^4 - 12*a^2*b^2))/(8*a^5
) - (4*exp(x))/(a*(6*exp(4*x) - 4*exp(2*x) - 4*exp(6*x) + exp(8*x) + 1)) - (2*exp(x)*(9*a - 4*b))/(3*a^2*(3*ex
p(2*x) - 3*exp(4*x) + exp(6*x) - 1)) - (exp(x)*(3*a^2 - 16*a*b + 12*b^2))/(6*a^3*(exp(4*x) - 2*exp(2*x) + 1))
- (exp(x)*(4*a*b^2 + 8*a^2*b - 3*a^3 - 8*b^3))/(4*a^4*(exp(2*x) - 1)) + (b*log((b*exp(x)*(a - b)^2*(288*a*b^6
- 24*a^6*b - 9*a^7 - 192*b^7 + 224*a^2*b^5 - 456*a^3*b^4 + 24*a^4*b^3 + 144*a^5*b^2))/(2*a^12*(a + b)) - (b*(a
- b)*(-(a + b)^3*(a - b)^3)^(1/2)*(8*a^5*b^3 - 9*a^8 - 9*a^7*b + 8*a^6*b^2 + 192*b^5*exp(x)*(-(a^2 - b^2)^3)^
(1/2) - 224*a^2*b^3*exp(x)*(-(a^2 - b^2)^3)^(1/2) - 88*a^3*b^2*exp(x)*(-(a^2 - b^2)^3)^(1/2) + 96*a*b^4*exp(x)
*(-(a^2 - b^2)^3)^(1/2) + 24*a^4*b*exp(x)*(-(a^2 - b^2)^3)^(1/2)))/(2*a^12*(a + b)^4))*(-(a + b)^3*(a - b)^3)^
(1/2))/a^5 - (b*log((b*exp(x)*(a - b)^2*(288*a*b^6 - 24*a^6*b - 9*a^7 - 192*b^7 + 224*a^2*b^5 - 456*a^3*b^4 +
24*a^4*b^3 + 144*a^5*b^2))/(2*a^12*(a + b)) - (b*(a - b)*(-(a + b)^3*(a - b)^3)^(1/2)*(9*a^7*b + 9*a^8 - 8*a^5
*b^3 - 8*a^6*b^2 + 192*b^5*exp(x)*(-(a^2 - b^2)^3)^(1/2) - 224*a^2*b^3*exp(x)*(-(a^2 - b^2)^3)^(1/2) - 88*a^3*
b^2*exp(x)*(-(a^2 - b^2)^3)^(1/2) + 96*a*b^4*exp(x)*(-(a^2 - b^2)^3)^(1/2) + 24*a^4*b*exp(x)*(-(a^2 - b^2)^3)^
(1/2)))/(2*a^12*(a + b)^4))*(-(a + b)^3*(a - b)^3)^(1/2))/a^5

________________________________________________________________________________________

sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {csch}^{5}{\relax (x )}}{a + b \tanh {\relax (x )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csch(x)**5/(a+b*tanh(x)),x)

[Out]

Integral(csch(x)**5/(a + b*tanh(x)), x)

________________________________________________________________________________________