3.1.87 \(\int \frac {\text {csch}^4(x)}{(a+b \sinh (x))^2} \, dx\) [87]

Optimal. Leaf size=198 \[ -\frac {b \left (a^2-4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}-\frac {2 b^4 \left (5 a^2+4 b^2\right ) \tanh ^{-1}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{a^5 \left (a^2+b^2\right )^{3/2}}+\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac {b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 \left (a^2+b^2\right )}-\frac {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))} \]

[Out]

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

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Rubi [A]
time = 0.62, antiderivative size = 198, normalized size of antiderivative = 1.00, number of steps used = 9, 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 {\left (a^2+4 b^2\right ) \coth (x) \text {csch}^2(x)}{3 a^2 \left (a^2+b^2\right )}+\frac {b^2 \coth (x) \text {csch}^2(x)}{a \left (a^2+b^2\right ) (a+b \sinh (x))}-\frac {b \left (a^2-4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}-\frac {2 b^4 \left (5 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 \left (a^2+b^2\right )^{3/2}}+\frac {\left (2 a^4-7 a^2 b^2-12 b^4\right ) \coth (x)}{3 a^4 \left (a^2+b^2\right )}+\frac {b \left (a^2+2 b^2\right ) \coth (x) \text {csch}(x)}{a^3 \left (a^2+b^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Csch[x]^4/(a + b*Sinh[x])^2,x]

[Out]

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

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

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Mathematica [A]
time = 0.65, size = 214, normalized size = 1.08 \begin {gather*} \frac {-\frac {48 b^4 \left (5 a^2+4 b^2\right ) \text {ArcTan}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {-a^2-b^2}}\right )}{\left (-a^2-b^2\right )^{3/2}}+4 a \left (2 a^2-9 b^2\right ) \coth \left (\frac {x}{2}\right )+6 a^2 b \text {csch}^2\left (\frac {x}{2}\right )+24 (a-2 b) b (a+2 b) \log \left (\tanh \left (\frac {x}{2}\right )\right )+6 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)-\frac {24 a b^5 \cosh (x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+4 a \left (2 a^2-9 b^2\right ) \tanh \left (\frac {x}{2}\right )}{24 a^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]
time = 0.78, size = 219, normalized size = 1.11

method	result	size
default	\(-\frac {\frac {\left (\tanh ^{3}\left (\frac {x}{2}\right )\right ) a^{2}}{3}+2 a b \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-3 a^{2} \tanh \left (\frac {x}{2}\right )+12 b^{2} \tanh \left (\frac {x}{2}\right )}{8 a^{4}}-\frac {1}{24 a^{2} \tanh \left (\frac {x}{2}\right )^{3}}-\frac {-3 a^{2}+12 b^{2}}{8 a^{4} \tanh \left (\frac {x}{2}\right )}+\frac {b}{4 a^{3} \tanh \left (\frac {x}{2}\right )^{2}}+\frac {b \left (a^{2}-4 b^{2}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )\right )}{a^{5}}-\frac {2 b^{4} \left (\frac {-\frac {b^{2} \tanh \left (\frac {x}{2}\right )}{a^{2}+b^{2}}-\frac {a b}{a^{2}+b^{2}}}{a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 b \tanh \left (\frac {x}{2}\right )-a}-\frac {\left (5 a^{2}+4 b^{2}\right ) \arctanh \left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{a^{5}}\)	\(219\)
risch	\(-\frac {2 \left (-3 a^{3} b^{2} {\mathrm e}^{7 x}-6 a \,b^{4} {\mathrm e}^{7 x}-6 a^{4} b \,{\mathrm e}^{6 x}+3 a^{2} b^{3} {\mathrm e}^{6 x}+12 b^{5} {\mathrm e}^{6 x}+21 a^{3} b^{2} {\mathrm e}^{5 x}+30 a \,b^{4} {\mathrm e}^{5 x}+6 a^{4} b \,{\mathrm e}^{4 x}-21 a^{2} b^{3} {\mathrm e}^{4 x}-36 b^{5} {\mathrm e}^{4 x}+12 a^{5} {\mathrm e}^{3 x}-21 a^{3} b^{2} {\mathrm e}^{3 x}-42 a \,b^{4} {\mathrm e}^{3 x}-2 a^{4} b \,{\mathrm e}^{2 x}+25 a^{2} b^{3} {\mathrm e}^{2 x}+36 b^{5} {\mathrm e}^{2 x}-4 a^{5} {\mathrm e}^{x}+11 a^{3} b^{2} {\mathrm e}^{x}+18 b^{4} {\mathrm e}^{x} a +2 a^{4} b -7 a^{2} b^{3}-12 b^{5}\right )}{3 a^{4} \left ({\mathrm e}^{2 x}-1\right )^{3} \left (a^{2}+b^{2}\right ) \left (b \,{\mathrm e}^{2 x}+2 a \,{\mathrm e}^{x}-b \right )}+\frac {5 b^{4} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{3}}+\frac {4 b^{6} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}-a^{4}-2 a^{2} b^{2}-b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{5}}-\frac {5 b^{4} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{3}}-\frac {4 b^{6} \ln \left ({\mathrm e}^{x}+\frac {a \left (a^{2}+b^{2}\right )^{\frac {3}{2}}+a^{4}+2 a^{2} b^{2}+b^{4}}{b \left (a^{2}+b^{2}\right )^{\frac {3}{2}}}\right )}{\left (a^{2}+b^{2}\right )^{\frac {3}{2}} a^{5}}+\frac {b \ln \left ({\mathrm e}^{x}-1\right )}{a^{3}}-\frac {4 b^{3} \ln \left ({\mathrm e}^{x}-1\right )}{a^{5}}-\frac {b \ln \left ({\mathrm e}^{x}+1\right )}{a^{3}}+\frac {4 b^{3} \ln \left ({\mathrm e}^{x}+1\right )}{a^{5}}\)	\(549\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csch(x)^4/(a+b*sinh(x))^2,x,method=_RETURNVERBOSE)

[Out]

-1/8/a^4*(1/3*tanh(1/2*x)^3*a^2+2*a*b*tanh(1/2*x)^2-3*a^2*tanh(1/2*x)+12*b^2*tanh(1/2*x))-1/24/a^2/tanh(1/2*x)
^3-1/8/a^4*(-3*a^2+12*b^2)/tanh(1/2*x)+1/4/a^3*b/tanh(1/2*x)^2+1/a^5*b*(a^2-4*b^2)*ln(tanh(1/2*x))-2*b^4/a^5*(
(-b^2/(a^2+b^2)*tanh(1/2*x)-a*b/(a^2+b^2))/(a*tanh(1/2*x)^2-2*b*tanh(1/2*x)-a)-(5*a^2+4*b^2)/(a^2+b^2)^(3/2)*a
rctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/2)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (190) = 380\).
time = 0.50, size = 477, normalized size = 2.41 \begin {gather*} \frac {{\left (5 \, a^{2} b^{4} + 4 \, b^{6}\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 )}{{\left (a^{7} + a^{5} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (2 \, a^{4} b - 7 \, a^{2} b^{3} - 12 \, b^{5} + {\left (4 \, a^{5} - 11 \, a^{3} b^{2} - 18 \, a b^{4}\right )} e^{\left (-x\right )} - {\left (2 \, a^{4} b - 25 \, a^{2} b^{3} - 36 \, b^{5}\right )} e^{\left (-2 \, x\right )} - 3 \, {\left (4 \, a^{5} - 7 \, a^{3} b^{2} - 14 \, a b^{4}\right )} e^{\left (-3 \, x\right )} + 3 \, {\left (2 \, a^{4} b - 7 \, a^{2} b^{3} - 12 \, b^{5}\right )} e^{\left (-4 \, x\right )} - 3 \, {\left (7 \, a^{3} b^{2} + 10 \, a b^{4}\right )} e^{\left (-5 \, x\right )} - 3 \, {\left (2 \, a^{4} b - a^{2} b^{3} - 4 \, b^{5}\right )} e^{\left (-6 \, x\right )} + 3 \, {\left (a^{3} b^{2} + 2 \, a b^{4}\right )} e^{\left (-7 \, x\right )}\right )}}{3 \, {\left (a^{6} b + a^{4} b^{3} + 2 \, {\left (a^{7} + a^{5} b^{2}\right )} e^{\left (-x\right )} - 4 \, {\left (a^{6} b + a^{4} b^{3}\right )} e^{\left (-2 \, x\right )} - 6 \, {\left (a^{7} + a^{5} b^{2}\right )} e^{\left (-3 \, x\right )} + 6 \, {\left (a^{6} b + a^{4} b^{3}\right )} e^{\left (-4 \, x\right )} + 6 \, {\left (a^{7} + a^{5} b^{2}\right )} e^{\left (-5 \, x\right )} - 4 \, {\left (a^{6} b + a^{4} b^{3}\right )} e^{\left (-6 \, x\right )} - 2 \, {\left (a^{7} + a^{5} b^{2}\right )} e^{\left (-7 \, x\right )} + {\left (a^{6} b + a^{4} b^{3}\right )} e^{\left (-8 \, x\right )}\right )}} - \frac {{\left (a^{2} b - 4 \, b^{3}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{a^{5}} + \frac {{\left (a^{2} b - 4 \, b^{3}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{a^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(5*a^2*b^4 + 4*b^6)*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-x) - a + sqrt(a^2 + b^2)))/((a^7 + a^5*b^2)*sq
rt(a^2 + b^2)) + 2/3*(2*a^4*b - 7*a^2*b^3 - 12*b^5 + (4*a^5 - 11*a^3*b^2 - 18*a*b^4)*e^(-x) - (2*a^4*b - 25*a^
2*b^3 - 36*b^5)*e^(-2*x) - 3*(4*a^5 - 7*a^3*b^2 - 14*a*b^4)*e^(-3*x) + 3*(2*a^4*b - 7*a^2*b^3 - 12*b^5)*e^(-4*
x) - 3*(7*a^3*b^2 + 10*a*b^4)*e^(-5*x) - 3*(2*a^4*b - a^2*b^3 - 4*b^5)*e^(-6*x) + 3*(a^3*b^2 + 2*a*b^4)*e^(-7*
x))/(a^6*b + a^4*b^3 + 2*(a^7 + a^5*b^2)*e^(-x) - 4*(a^6*b + a^4*b^3)*e^(-2*x) - 6*(a^7 + a^5*b^2)*e^(-3*x) +
6*(a^6*b + a^4*b^3)*e^(-4*x) + 6*(a^7 + a^5*b^2)*e^(-5*x) - 4*(a^6*b + a^4*b^3)*e^(-6*x) - 2*(a^7 + a^5*b^2)*e
^(-7*x) + (a^6*b + a^4*b^3)*e^(-8*x)) - (a^2*b - 4*b^3)*log(e^(-x) + 1)/a^5 + (a^2*b - 4*b^3)*log(e^(-x) - 1)/
a^5

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 6430 vs. \(2 (190) = 380\).
time = 0.59, size = 6430, normalized size = 32.47 \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))^2,x, algorithm="fricas")

[Out]

-1/3*(4*a^7*b - 10*a^5*b^3 - 38*a^3*b^5 - 24*a*b^7 - 6*(a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^6)*cosh(x)^7 - 6*(a^6*b^
2 + 3*a^4*b^4 + 2*a^2*b^6)*sinh(x)^7 - 6*(2*a^7*b + a^5*b^3 - 5*a^3*b^5 - 4*a*b^7)*cosh(x)^6 - 6*(2*a^7*b + a^
5*b^3 - 5*a^3*b^5 - 4*a*b^7 + 7*(a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^6)*cosh(x))*sinh(x)^6 + 6*(7*a^6*b^2 + 17*a^4*b
^4 + 10*a^2*b^6)*cosh(x)^5 + 6*(7*a^6*b^2 + 17*a^4*b^4 + 10*a^2*b^6 - 21*(a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^6)*cos
h(x)^2 - 6*(2*a^7*b + a^5*b^3 - 5*a^3*b^5 - 4*a*b^7)*cosh(x))*sinh(x)^5 + 6*(2*a^7*b - 5*a^5*b^3 - 19*a^3*b^5
- 12*a*b^7)*cosh(x)^4 + 6*(2*a^7*b - 5*a^5*b^3 - 19*a^3*b^5 - 12*a*b^7 - 35*(a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^6)*
cosh(x)^3 - 15*(2*a^7*b + a^5*b^3 - 5*a^3*b^5 - 4*a*b^7)*cosh(x)^2 + 5*(7*a^6*b^2 + 17*a^4*b^4 + 10*a^2*b^6)*c
osh(x))*sinh(x)^4 + 6*(4*a^8 - 3*a^6*b^2 - 21*a^4*b^4 - 14*a^2*b^6)*cosh(x)^3 + 6*(4*a^8 - 3*a^6*b^2 - 21*a^4*
b^4 - 14*a^2*b^6 - 35*(a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^6)*cosh(x)^4 - 20*(2*a^7*b + a^5*b^3 - 5*a^3*b^5 - 4*a*b^
7)*cosh(x)^3 + 10*(7*a^6*b^2 + 17*a^4*b^4 + 10*a^2*b^6)*cosh(x)^2 + 4*(2*a^7*b - 5*a^5*b^3 - 19*a^3*b^5 - 12*a
*b^7)*cosh(x))*sinh(x)^3 - 2*(2*a^7*b - 23*a^5*b^3 - 61*a^3*b^5 - 36*a*b^7)*cosh(x)^2 - 2*(2*a^7*b - 23*a^5*b^
3 - 61*a^3*b^5 - 36*a*b^7 + 63*(a^6*b^2 + 3*a^4*b^4 + 2*a^2*b^6)*cosh(x)^5 + 45*(2*a^7*b + a^5*b^3 - 5*a^3*b^5
- 4*a*b^7)*cosh(x)^4 - 30*(7*a^6*b^2 + 17*a^4*b^4 + 10*a^2*b^6)*cosh(x)^3 - 18*(2*a^7*b - 5*a^5*b^3 - 19*a^3*
b^5 - 12*a*b^7)*cosh(x)^2 - 9*(4*a^8 - 3*a^6*b^2 - 21*a^4*b^4 - 14*a^2*b^6)*cosh(x))*sinh(x)^2 - 3*((5*a^2*b^5
+ 4*b^7)*cosh(x)^8 + (5*a^2*b^5 + 4*b^7)*sinh(x)^8 + 2*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^7 + 2*(5*a^3*b^4 + 4*a*b
^6 + 4*(5*a^2*b^5 + 4*b^7)*cosh(x))*sinh(x)^7 + 5*a^2*b^5 + 4*b^7 - 4*(5*a^2*b^5 + 4*b^7)*cosh(x)^6 - 2*(10*a^
2*b^5 + 8*b^7 - 14*(5*a^2*b^5 + 4*b^7)*cosh(x)^2 - 7*(5*a^3*b^4 + 4*a*b^6)*cosh(x))*sinh(x)^6 - 6*(5*a^3*b^4 +
4*a*b^6)*cosh(x)^5 - 2*(15*a^3*b^4 + 12*a*b^6 - 28*(5*a^2*b^5 + 4*b^7)*cosh(x)^3 - 21*(5*a^3*b^4 + 4*a*b^6)*c
osh(x)^2 + 12*(5*a^2*b^5 + 4*b^7)*cosh(x))*sinh(x)^5 + 6*(5*a^2*b^5 + 4*b^7)*cosh(x)^4 + 2*(15*a^2*b^5 + 12*b^
7 + 35*(5*a^2*b^5 + 4*b^7)*cosh(x)^4 + 35*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^3 - 30*(5*a^2*b^5 + 4*b^7)*cosh(x)^2 -
15*(5*a^3*b^4 + 4*a*b^6)*cosh(x))*sinh(x)^4 + 6*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^3 + 2*(15*a^3*b^4 + 12*a*b^6 +
28*(5*a^2*b^5 + 4*b^7)*cosh(x)^5 + 35*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^4 - 40*(5*a^2*b^5 + 4*b^7)*cosh(x)^3 - 30*
(5*a^3*b^4 + 4*a*b^6)*cosh(x)^2 + 12*(5*a^2*b^5 + 4*b^7)*cosh(x))*sinh(x)^3 - 4*(5*a^2*b^5 + 4*b^7)*cosh(x)^2
- 2*(10*a^2*b^5 + 8*b^7 - 14*(5*a^2*b^5 + 4*b^7)*cosh(x)^6 - 21*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^5 + 30*(5*a^2*b^
5 + 4*b^7)*cosh(x)^4 + 30*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^3 - 18*(5*a^2*b^5 + 4*b^7)*cosh(x)^2 - 9*(5*a^3*b^4 +
4*a*b^6)*cosh(x))*sinh(x)^2 - 2*(5*a^3*b^4 + 4*a*b^6)*cosh(x) + 2*(4*(5*a^2*b^5 + 4*b^7)*cosh(x)^7 - 5*a^3*b^4
- 4*a*b^6 + 7*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^6 - 12*(5*a^2*b^5 + 4*b^7)*cosh(x)^5 - 15*(5*a^3*b^4 + 4*a*b^6)*c
osh(x)^4 + 12*(5*a^2*b^5 + 4*b^7)*cosh(x)^3 + 9*(5*a^3*b^4 + 4*a*b^6)*cosh(x)^2 - 4*(5*a^2*b^5 + 4*b^7)*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)) - 2*(4*a^8 - 7*a^6*b^2 - 29*a^4*b^4 - 18*a^2*b^6)*cosh(x) + 3*((a^6*b^2 - 2*a^4*b
^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^8 + (a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*sinh(x)^8 + a^6*b^2 - 2*a^4*b^4
- 7*a^2*b^6 - 4*b^8 + 2*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^7 + 2*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5
- 4*a*b^7 + 4*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x))*sinh(x)^7 - 4*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b
^6 - 4*b^8)*cosh(x)^6 - 2*(2*a^6*b^2 - 4*a^4*b^4 - 14*a^2*b^6 - 8*b^8 - 14*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 -
4*b^8)*cosh(x)^2 - 7*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x))*sinh(x)^6 - 6*(a^7*b - 2*a^5*b^3 - 7*a
^3*b^5 - 4*a*b^7)*cosh(x)^5 - 2*(3*a^7*b - 6*a^5*b^3 - 21*a^3*b^5 - 12*a*b^7 - 28*(a^6*b^2 - 2*a^4*b^4 - 7*a^2
*b^6 - 4*b^8)*cosh(x)^3 - 21*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^2 + 12*(a^6*b^2 - 2*a^4*b^4 - 7
*a^2*b^6 - 4*b^8)*cosh(x))*sinh(x)^5 + 6*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^4 + 2*(3*a^6*b^2 -
6*a^4*b^4 - 21*a^2*b^6 - 12*b^8 + 35*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^4 + 35*(a^7*b - 2*a^5*b
^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^3 - 30*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^2 - 15*(a^7*b - 2*a
^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x))*sinh(x)^4 + 6*(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^3 + 2*(
3*a^7*b - 6*a^5*b^3 - 21*a^3*b^5 - 12*a*b^7 + 28*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^5 + 35*(a^7
*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^4 - 40*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x)^3 - 30*
(a^7*b - 2*a^5*b^3 - 7*a^3*b^5 - 4*a*b^7)*cosh(x)^2 + 12*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - 4*b^8)*cosh(x))*si
nh(x)^3 - 4*(a^6*b^2 - 2*a^4*b^4 - 7*a^2*b^6 - ...

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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 )}}{\left (a + b \sinh {\left (x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.44, size = 236, normalized size = 1.19 \begin {gather*} \frac {{\left (5 \, a^{2} b^{4} + 4 \, b^{6}\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 )}{{\left (a^{7} + a^{5} b^{2}\right )} \sqrt {a^{2} + b^{2}}} + \frac {2 \, {\left (a b^{4} e^{x} - b^{5}\right )}}{{\left (a^{6} + a^{4} b^{2}\right )} {\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} - \frac {{\left (a^{2} b - 4 \, b^{3}\right )} \log \left (e^{x} + 1\right )}{a^{5}} + \frac {{\left (a^{2} b - 4 \, b^{3}\right )} \log \left ({\left | e^{x} - 1 \right |}\right )}{a^{5}} + \frac {2 \, {\left (3 \, a b e^{\left (5 \, 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} + 2 \, a^{2} - 9 \, b^{2}\right )}}{3 \, a^{4} {\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))^2,x, algorithm="giac")

[Out]

(5*a^2*b^4 + 4*b^6)*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)))/((a^7 +
a^5*b^2)*sqrt(a^2 + b^2)) + 2*(a*b^4*e^x - b^5)/((a^6 + a^4*b^2)*(b*e^(2*x) + 2*a*e^x - b)) - (a^2*b - 4*b^3)
*log(e^x + 1)/a^5 + (a^2*b - 4*b^3)*log(abs(e^x - 1))/a^5 + 2/3*(3*a*b*e^(5*x) - 9*b^2*e^(4*x) - 6*a^2*e^(2*x)
+ 18*b^2*e^(2*x) - 3*a*b*e^x + 2*a^2 - 9*b^2)/(a^4*(e^(2*x) - 1)^3)

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Mupad [B]
time = 3.30, size = 975, normalized size = 4.92 \begin {gather*} \frac {\ln \left ({\mathrm {e}}^x-1\right )\,\left (a^2\,b-4\,b^3\right )}{a^5}-\frac {8}{3\,a^2\,\left (3\,{\mathrm {e}}^{2\,x}-3\,{\mathrm {e}}^{4\,x}+{\mathrm {e}}^{6\,x}-1\right )}-\frac {\frac {4}{a^2}-\frac {4\,b\,{\mathrm {e}}^x}{a^3}}{{\mathrm {e}}^{4\,x}-2\,{\mathrm {e}}^{2\,x}+1}-\frac {\frac {6\,b^2}{a^4}-\frac {2\,b\,{\mathrm {e}}^x}{a^3}}{{\mathrm {e}}^{2\,x}-1}-\frac {\frac {2\,b^8}{a^4\,\left (a^2\,b^3+b^5\right )}-\frac {2\,b^7\,{\mathrm {e}}^x}{a^3\,\left (a^2\,b^3+b^5\right )}}{2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}}-\frac {\ln \left ({\mathrm {e}}^x+1\right )\,\left (a^2\,b-4\,b^3\right )}{a^5}-\frac {b^4\,\ln \left (\frac {32\,b\,\left (-5\,a^4+16\,a^2\,b^2+16\,b^4\right )\,\left (-4\,{\mathrm {e}}^x\,a^5+2\,a^4\,b+11\,{\mathrm {e}}^x\,a^3\,b^2-6\,a^2\,b^3+14\,{\mathrm {e}}^x\,a\,b^4-8\,b^5\right )}{a^{12}\,{\left (a^2+b^2\right )}^2}-\frac {32\,b\,\left (5\,a^2+4\,b^2\right )\,\left (5\,a^5\,b^9-32\,b^{11}\,\sqrt {{\left (a^2+b^2\right )}^3}-2\,a^{13}\,b+20\,a^7\,b^7+24\,a^9\,b^5+7\,a^{11}\,b^3+4\,a^{14}\,{\mathrm {e}}^x-80\,a^2\,b^9\,\sqrt {{\left (a^2+b^2\right )}^3}-50\,a^4\,b^7\,\sqrt {{\left (a^2+b^2\right )}^3}-15\,a^6\,b^8\,{\mathrm {e}}^x-50\,a^8\,b^6\,{\mathrm {e}}^x-52\,a^{10}\,b^4\,{\mathrm {e}}^x-13\,a^{12}\,b^2\,{\mathrm {e}}^x+127\,a^3\,b^8\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+79\,a^5\,b^6\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+5\,a\,b^4\,{\mathrm {e}}^x\,{\left ({\left (a^2+b^2\right )}^3\right )}^{3/2}+51\,a\,b^{10}\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}\right )}{a^{12}\,\sqrt {{\left (a^2+b^2\right )}^3}\,{\left (a^2+b^2\right )}^4}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}\,\left (5\,a^2+4\,b^2\right )}{a^{11}+3\,a^9\,b^2+3\,a^7\,b^4+a^5\,b^6}+\frac {b^4\,\ln \left (\frac {32\,b\,\left (-5\,a^4+16\,a^2\,b^2+16\,b^4\right )\,\left (-4\,{\mathrm {e}}^x\,a^5+2\,a^4\,b+11\,{\mathrm {e}}^x\,a^3\,b^2-6\,a^2\,b^3+14\,{\mathrm {e}}^x\,a\,b^4-8\,b^5\right )}{a^{12}\,{\left (a^2+b^2\right )}^2}-\frac {32\,b\,\left (5\,a^2+4\,b^2\right )\,\left (2\,a^{13}\,b-32\,b^{11}\,\sqrt {{\left (a^2+b^2\right )}^3}-5\,a^5\,b^9-20\,a^7\,b^7-24\,a^9\,b^5-7\,a^{11}\,b^3-4\,a^{14}\,{\mathrm {e}}^x-80\,a^2\,b^9\,\sqrt {{\left (a^2+b^2\right )}^3}-50\,a^4\,b^7\,\sqrt {{\left (a^2+b^2\right )}^3}+15\,a^6\,b^8\,{\mathrm {e}}^x+50\,a^8\,b^6\,{\mathrm {e}}^x+52\,a^{10}\,b^4\,{\mathrm {e}}^x+13\,a^{12}\,b^2\,{\mathrm {e}}^x+127\,a^3\,b^8\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+79\,a^5\,b^6\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}+5\,a\,b^4\,{\mathrm {e}}^x\,{\left ({\left (a^2+b^2\right )}^3\right )}^{3/2}+51\,a\,b^{10}\,{\mathrm {e}}^x\,\sqrt {{\left (a^2+b^2\right )}^3}\right )}{a^{12}\,\sqrt {{\left (a^2+b^2\right )}^3}\,{\left (a^2+b^2\right )}^4}\right )\,\sqrt {{\left (a^2+b^2\right )}^3}\,\left (5\,a^2+4\,b^2\right )}{a^{11}+3\,a^9\,b^2+3\,a^7\,b^4+a^5\,b^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sinh(x)^4*(a + b*sinh(x))^2),x)

[Out]

(log(exp(x) - 1)*(a^2*b - 4*b^3))/a^5 - 8/(3*a^2*(3*exp(2*x) - 3*exp(4*x) + exp(6*x) - 1)) - (4/a^2 - (4*b*exp
(x))/a^3)/(exp(4*x) - 2*exp(2*x) + 1) - ((6*b^2)/a^4 - (2*b*exp(x))/a^3)/(exp(2*x) - 1) - ((2*b^8)/(a^4*(b^5 +
a^2*b^3)) - (2*b^7*exp(x))/(a^3*(b^5 + a^2*b^3)))/(2*a*exp(x) - b + b*exp(2*x)) - (log(exp(x) + 1)*(a^2*b - 4
*b^3))/a^5 - (b^4*log((32*b*(16*b^4 - 5*a^4 + 16*a^2*b^2)*(2*a^4*b - 8*b^5 - 6*a^2*b^3 - 4*a^5*exp(x) + 14*a*b
^4*exp(x) + 11*a^3*b^2*exp(x)))/(a^12*(a^2 + b^2)^2) - (32*b*(5*a^2 + 4*b^2)*(5*a^5*b^9 - 32*b^11*((a^2 + b^2)
^3)^(1/2) - 2*a^13*b + 20*a^7*b^7 + 24*a^9*b^5 + 7*a^11*b^3 + 4*a^14*exp(x) - 80*a^2*b^9*((a^2 + b^2)^3)^(1/2)
- 50*a^4*b^7*((a^2 + b^2)^3)^(1/2) - 15*a^6*b^8*exp(x) - 50*a^8*b^6*exp(x) - 52*a^10*b^4*exp(x) - 13*a^12*b^2
*exp(x) + 127*a^3*b^8*exp(x)*((a^2 + b^2)^3)^(1/2) + 79*a^5*b^6*exp(x)*((a^2 + b^2)^3)^(1/2) + 5*a*b^4*exp(x)*
((a^2 + b^2)^3)^(3/2) + 51*a*b^10*exp(x)*((a^2 + b^2)^3)^(1/2)))/(a^12*((a^2 + b^2)^3)^(1/2)*(a^2 + b^2)^4))*(
(a^2 + b^2)^3)^(1/2)*(5*a^2 + 4*b^2))/(a^11 + a^5*b^6 + 3*a^7*b^4 + 3*a^9*b^2) + (b^4*log((32*b*(16*b^4 - 5*a^
4 + 16*a^2*b^2)*(2*a^4*b - 8*b^5 - 6*a^2*b^3 - 4*a^5*exp(x) + 14*a*b^4*exp(x) + 11*a^3*b^2*exp(x)))/(a^12*(a^2
+ b^2)^2) - (32*b*(5*a^2 + 4*b^2)*(2*a^13*b - 32*b^11*((a^2 + b^2)^3)^(1/2) - 5*a^5*b^9 - 20*a^7*b^7 - 24*a^9
*b^5 - 7*a^11*b^3 - 4*a^14*exp(x) - 80*a^2*b^9*((a^2 + b^2)^3)^(1/2) - 50*a^4*b^7*((a^2 + b^2)^3)^(1/2) + 15*a
^6*b^8*exp(x) + 50*a^8*b^6*exp(x) + 52*a^10*b^4*exp(x) + 13*a^12*b^2*exp(x) + 127*a^3*b^8*exp(x)*((a^2 + b^2)^
3)^(1/2) + 79*a^5*b^6*exp(x)*((a^2 + b^2)^3)^(1/2) + 5*a*b^4*exp(x)*((a^2 + b^2)^3)^(3/2) + 51*a*b^10*exp(x)*(
(a^2 + b^2)^3)^(1/2)))/(a^12*((a^2 + b^2)^3)^(1/2)*(a^2 + b^2)^4))*((a^2 + b^2)^3)^(1/2)*(5*a^2 + 4*b^2))/(a^1
1 + a^5*b^6 + 3*a^7*b^4 + 3*a^9*b^2)

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