3.243 \(\int \frac{\coth ^4(x)}{(a+b \sinh (x))^2} \, dx\)
Optimal. Leaf size=159 \[ -\frac{2 \sqrt{a^2+b^2} \left (a^2+4 b^2\right ) \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{a^5}-\frac{\left (7 a^2+12 b^2\right ) \coth (x)}{3 a^4}+\frac{b \left (3 a^2+4 b^2\right ) \tanh ^{-1}(\cosh (x))}{a^5}+\frac{\left (a^2+2 b^2\right ) \coth (x) \text{csch}(x)}{a^3 b}-\frac{\left (\frac{4 b^2}{a^2}+3\right ) \coth (x) \text{csch}(x)}{3 b (a+b \sinh (x))}-\frac{\coth (x) \text{csch}^2(x)}{3 a (a+b \sinh (x))} \]
[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]))
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Rubi [A] time = 0.670978, antiderivative size = 159, normalized size of antiderivative = 1.,
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 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 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 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 3770
Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]
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 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 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])
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*}
Mathematica [A] time = 0.818166, size = 214, normalized size = 1.35 \[ \frac{-4 a \left (4 a^2+9 b^2\right ) \tanh \left (\frac{x}{2}\right )-4 a \left (4 a^2+9 b^2\right ) \coth \left (\frac{x}{2}\right )-24 b \left (3 a^2+4 b^2\right ) \log \left (\tanh \left (\frac{x}{2}\right )\right )+\frac{48 \left (5 a^2 b^2+a^4+4 b^4\right ) \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}-\frac{24 a b \left (a^2+b^2\right ) \cosh (x)}{a+b \sinh (x)}+6 a^2 b \text{csch}^2\left (\frac{x}{2}\right )+6 a^2 b \text{sech}^2\left (\frac{x}{2}\right )+8 a^3 \sinh ^4\left (\frac{x}{2}\right ) \text{csch}^3(x)-\frac{1}{2} a^3 \sinh (x) \text{csch}^4\left (\frac{x}{2}\right )}{24 a^5} \]
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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Maple [B] time = 0.056, size = 357, normalized size = 2.3 \begin{align*} -{\frac{1}{24\,{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{b}{4\,{a}^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\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{1}{24\,{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}-{\frac{5}{8\,{a}^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}-{\frac{3\,{b}^{2}}{2\,{a}^{4}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}}+{\frac{b}{4\,{a}^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-3\,{\frac{b\ln \left ( \tanh \left ( x/2 \right ) \right ) }{{a}^{3}}}-4\,{\frac{{b}^{3}\ln \left ( \tanh \left ( x/2 \right ) \right ) }{{a}^{5}}}+2\,{\frac{{b}^{2}\tanh \left ( x/2 \right ) }{{a}^{3} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) }}+2\,{\frac{\tanh \left ( x/2 \right ){b}^{4}}{{a}^{5} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) }}+2\,{\frac{b}{{a}^{2} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) }}+2\,{\frac{{b}^{3}}{{a}^{4} \left ( a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a \right ) }}+2\,{\frac{1}{a\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }+10\,{\frac{{b}^{2}}{{a}^{3}\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) }+8\,{\frac{{b}^{4}}{{a}^{5}\sqrt{{a}^{2}+{b}^{2}}}{\it Artanh} \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) -2\,b}{\sqrt{{a}^{2}+{b}^{2}}}} \right ) } \end{align*}
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)-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))+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*tanh(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
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)^4/(a+b*sinh(x))^2,x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 3.02804, size = 9129, normalized size = 57.42 \begin{align*} \text{result too large to display} \end{align*}
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))
________________________________________________________________________________________
Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(coth(x)**4/(a+b*sinh(x))**2,x)
[Out]
Timed out
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Giac [A] time = 1.20544, size = 327, normalized size = 2.06 \begin{align*} \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}} \end{align*}
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)