3.175 \(\int \frac{\text{csch}^4(x)}{a+b \cosh (x)} \, dx\)
Optimal. Leaf size=110 \[ \frac{\text{csch}^3(x) (b-a \cosh (x))}{3 \left (a^2-b^2\right )}+\frac{\text{csch}(x) \left (a \left (2 a^2-5 b^2\right ) \cosh (x)+3 b^3\right )}{3 \left (a^2-b^2\right )^2}+\frac{2 b^4 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2}} \]
[Out]
(2*b^4*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)) + ((3*b^3 + a*(2*a^2 - 5*b^
2)*Cosh[x])*Csch[x])/(3*(a^2 - b^2)^2) + ((b - a*Cosh[x])*Csch[x]^3)/(3*(a^2 - b^2))
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Rubi [A] time = 0.245871, antiderivative size = 110, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 5, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used =
{2696, 2866, 12, 2659, 208} \[ \frac{\text{csch}^3(x) (b-a \cosh (x))}{3 \left (a^2-b^2\right )}+\frac{\text{csch}(x) \left (a \left (2 a^2-5 b^2\right ) \cosh (x)+3 b^3\right )}{3 \left (a^2-b^2\right )^2}+\frac{2 b^4 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2}} \]
Antiderivative was successfully verified.
[In]
Int[Csch[x]^4/(a + b*Cosh[x]),x]
[Out]
(2*b^4*ArcTanh[(Sqrt[a - b]*Tanh[x/2])/Sqrt[a + b]])/((a - b)^(5/2)*(a + b)^(5/2)) + ((3*b^3 + a*(2*a^2 - 5*b^
2)*Cosh[x])*Csch[x])/(3*(a^2 - b^2)^2) + ((b - a*Cosh[x])*Csch[x]^3)/(3*(a^2 - b^2))
Rule 2696
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[((g*Co
s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*(b - a*Sin[e + f*x]))/(f*g*(a^2 - b^2)*(p + 1)), x] + Dist[1/
(g^2*(a^2 - b^2)*(p + 1)), Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*(a^2*(p + 2) - b^2*(m + p + 2)
+ a*b*(m + p + 3)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] &&
IntegersQ[2*m, 2*p]
Rule 2866
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*sin[(e_.)
+ (f_.)*(x_)]), x_Symbol] :> Simp[((g*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*(b*c - a*d - (a*c -
b*d)*Sin[e + f*x]))/(f*g*(a^2 - b^2)*(p + 1)), x] + Dist[1/(g^2*(a^2 - b^2)*(p + 1)), Int[(g*Cos[e + f*x])^(p
+ 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e
+ f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && LtQ[p, -1] && IntegerQ[2*m]
Rule 12
Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] && !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]
Rule 2659
Int[((a_) + (b_.)*sin[Pi/2 + (c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = FreeFactors[Tan[(c + d*x)/2], x
]}, Dist[(2*e)/d, Subst[Int[1/(a + b + (a - b)*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 208
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]
Rubi steps
\begin{align*} \int \frac{\text{csch}^4(x)}{a+b \cosh (x)} \, dx &=\frac{(b-a \cosh (x)) \text{csch}^3(x)}{3 \left (a^2-b^2\right )}+\frac{\int \frac{\left (-2 a^2+3 b^2-2 a b \cosh (x)\right ) \text{csch}^2(x)}{a+b \cosh (x)} \, dx}{3 \left (a^2-b^2\right )}\\ &=\frac{\left (3 b^3+a \left (2 a^2-5 b^2\right ) \cosh (x)\right ) \text{csch}(x)}{3 \left (a^2-b^2\right )^2}+\frac{(b-a \cosh (x)) \text{csch}^3(x)}{3 \left (a^2-b^2\right )}+\frac{\int \frac{3 b^4}{a+b \cosh (x)} \, dx}{3 \left (a^2-b^2\right )^2}\\ &=\frac{\left (3 b^3+a \left (2 a^2-5 b^2\right ) \cosh (x)\right ) \text{csch}(x)}{3 \left (a^2-b^2\right )^2}+\frac{(b-a \cosh (x)) \text{csch}^3(x)}{3 \left (a^2-b^2\right )}+\frac{b^4 \int \frac{1}{a+b \cosh (x)} \, dx}{\left (a^2-b^2\right )^2}\\ &=\frac{\left (3 b^3+a \left (2 a^2-5 b^2\right ) \cosh (x)\right ) \text{csch}(x)}{3 \left (a^2-b^2\right )^2}+\frac{(b-a \cosh (x)) \text{csch}^3(x)}{3 \left (a^2-b^2\right )}+\frac{\left (2 b^4\right ) \operatorname{Subst}\left (\int \frac{1}{a+b-(a-b) x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{\left (a^2-b^2\right )^2}\\ &=\frac{2 b^4 \tanh ^{-1}\left (\frac{\sqrt{a-b} \tanh \left (\frac{x}{2}\right )}{\sqrt{a+b}}\right )}{(a-b)^{5/2} (a+b)^{5/2}}+\frac{\left (3 b^3+a \left (2 a^2-5 b^2\right ) \cosh (x)\right ) \text{csch}(x)}{3 \left (a^2-b^2\right )^2}+\frac{(b-a \cosh (x)) \text{csch}^3(x)}{3 \left (a^2-b^2\right )}\\ \end{align*}
Mathematica [A] time = 0.534108, size = 141, normalized size = 1.28 \[ \frac{1}{24} \left (-\frac{48 b^4 \tan ^{-1}\left (\frac{(a-b) \tanh \left (\frac{x}{2}\right )}{\sqrt{b^2-a^2}}\right )}{\left (b^2-a^2\right )^{5/2}}-\frac{14 b \tanh \left (\frac{x}{2}\right )}{(a-b)^2}+\frac{8 a \tanh \left (\frac{x}{2}\right )}{(a-b)^2}+\frac{2 (4 a+7 b) \coth \left (\frac{x}{2}\right )}{(a+b)^2}+\frac{8 \sinh ^4\left (\frac{x}{2}\right ) \text{csch}^3(x)}{a-b}-\frac{\sinh (x) \text{csch}^4\left (\frac{x}{2}\right )}{2 (a+b)}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[x]^4/(a + b*Cosh[x]),x]
[Out]
((-48*b^4*ArcTan[((a - b)*Tanh[x/2])/Sqrt[-a^2 + b^2]])/(-a^2 + b^2)^(5/2) + (2*(4*a + 7*b)*Coth[x/2])/(a + b)
^2 + (8*Csch[x]^3*Sinh[x/2]^4)/(a - b) - (Csch[x/2]^4*Sinh[x])/(2*(a + b)) + (8*a*Tanh[x/2])/(a - b)^2 - (14*b
*Tanh[x/2])/(a - b)^2)/24
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Maple [A] time = 0.024, size = 127, normalized size = 1.2 \begin{align*} -{\frac{1}{8\, \left ( a-b \right ) ^{2}} \left ({\frac{a}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-{\frac{b}{3} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{3}}-3\,a\tanh \left ( x/2 \right ) +5\,\tanh \left ( x/2 \right ) b \right ) }+2\,{\frac{{b}^{4}}{ \left ( a-b \right ) ^{2} \left ( a+b \right ) ^{2}\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}{\it Artanh} \left ({\frac{ \left ( a-b \right ) \tanh \left ( x/2 \right ) }{\sqrt{ \left ( a+b \right ) \left ( a-b \right ) }}} \right ) }-{\frac{1}{24\,a+24\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-3}}-{\frac{-3\,a-5\,b}{8\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-1}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(x)^4/(a+b*cosh(x)),x)
[Out]
-1/8/(a-b)^2*(1/3*a*tanh(1/2*x)^3-1/3*tanh(1/2*x)^3*b-3*a*tanh(1/2*x)+5*tanh(1/2*x)*b)+2/(a-b)^2/(a+b)^2*b^4/(
(a+b)*(a-b))^(1/2)*arctanh((a-b)*tanh(1/2*x)/((a+b)*(a-b))^(1/2))-1/24/(a+b)/tanh(1/2*x)^3-1/8/(a+b)^2*(-3*a-5
*b)/tanh(1/2*x)
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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(csch(x)^4/(a+b*cosh(x)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [B] time = 2.15178, size = 5405, normalized size = 49.14 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)^4/(a+b*cosh(x)),x, algorithm="fricas")
[Out]
[1/3*(6*(a^2*b^3 - b^5)*cosh(x)^5 + 6*(a^2*b^3 - b^5)*sinh(x)^5 + 4*a^5 - 14*a^3*b^2 + 10*a*b^4 - 6*(a^3*b^2 -
a*b^4)*cosh(x)^4 - 6*(a^3*b^2 - a*b^4 - 5*(a^2*b^3 - b^5)*cosh(x))*sinh(x)^4 + 4*(2*a^4*b - 7*a^2*b^3 + 5*b^5
)*cosh(x)^3 + 4*(2*a^4*b - 7*a^2*b^3 + 5*b^5 + 15*(a^2*b^3 - b^5)*cosh(x)^2 - 6*(a^3*b^2 - a*b^4)*cosh(x))*sin
h(x)^3 - 12*(a^5 - 3*a^3*b^2 + 2*a*b^4)*cosh(x)^2 - 12*(a^5 - 3*a^3*b^2 + 2*a*b^4 - 5*(a^2*b^3 - b^5)*cosh(x)^
3 + 3*(a^3*b^2 - a*b^4)*cosh(x)^2 - (2*a^4*b - 7*a^2*b^3 + 5*b^5)*cosh(x))*sinh(x)^2 + 3*(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^2*b^3 - b^5)*cosh(x) + 6*
(a^2*b^3 - b^5 + 5*(a^2*b^3 - b^5)*cosh(x)^4 - 4*(a^3*b^2 - a*b^4)*cosh(x)^3 + 2*(2*a^4*b - 7*a^2*b^3 + 5*b^5)
*cosh(x)^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*cosh(x))*sinh(x))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^6 +
6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)*sinh(x)^5 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sinh(x)^6 - a^6
+ 3*a^4*b^2 - 3*a^2*b^4 + b^6 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b
^4 - b^6 - 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*sinh(x)^4 + 4*(5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^
6)*cosh(x)^3 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x))*sinh(x)^3 + 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6
)*cosh(x)^2 + 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 + 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 - 6*(a^6
- 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*sinh(x)^2 + 6*((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^5 - 2*(a^
6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x))*sinh(x)), 2/3*(3*(a^
2*b^3 - b^5)*cosh(x)^5 + 3*(a^2*b^3 - b^5)*sinh(x)^5 + 2*a^5 - 7*a^3*b^2 + 5*a*b^4 - 3*(a^3*b^2 - a*b^4)*cosh(
x)^4 - 3*(a^3*b^2 - a*b^4 - 5*(a^2*b^3 - b^5)*cosh(x))*sinh(x)^4 + 2*(2*a^4*b - 7*a^2*b^3 + 5*b^5)*cosh(x)^3 +
2*(2*a^4*b - 7*a^2*b^3 + 5*b^5 + 15*(a^2*b^3 - b^5)*cosh(x)^2 - 6*(a^3*b^2 - a*b^4)*cosh(x))*sinh(x)^3 - 6*(a
^5 - 3*a^3*b^2 + 2*a*b^4)*cosh(x)^2 - 6*(a^5 - 3*a^3*b^2 + 2*a*b^4 - 5*(a^2*b^3 - b^5)*cosh(x)^3 + 3*(a^3*b^2
- a*b^4)*cosh(x)^2 - (2*a^4*b - 7*a^2*b^3 + 5*b^5)*cosh(x))*sinh(x)^2 - 3*(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)*arctan(-sqrt(-a^2 + b^2)*(b*cosh(x) + b*sinh(x)
+ a)/(a^2 - b^2)) + 3*(a^2*b^3 - b^5)*cosh(x) + 3*(a^2*b^3 - b^5 + 5*(a^2*b^3 - b^5)*cosh(x)^4 - 4*(a^3*b^2 -
a*b^4)*cosh(x)^3 + 2*(2*a^4*b - 7*a^2*b^3 + 5*b^5)*cosh(x)^2 - 4*(a^5 - 3*a^3*b^2 + 2*a*b^4)*cosh(x))*sinh(x)
)/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^6 + 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)*sinh(x)^5 + (
a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sinh(x)^6 - a^6 + 3*a^4*b^2 - 3*a^2*b^4 + b^6 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*
b^4 - b^6)*cosh(x)^4 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)
*sinh(x)^4 + 4*(5*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x
))*sinh(x)^3 + 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2 + 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 + 5*(a^6
- 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 - 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*sinh(x)^2 + 6*((a
^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^5 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^3 + (a^6 - 3*a^4*b
^2 + 3*a^2*b^4 - b^6)*cosh(x))*sinh(x))]
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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(csch(x)**4/(a+b*cosh(x)),x)
[Out]
Timed out
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Giac [A] time = 1.19373, size = 211, normalized size = 1.92 \begin{align*} \frac{2 \, b^{4} \arctan \left (\frac{b e^{x} + a}{\sqrt{-a^{2} + b^{2}}}\right )}{{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} \sqrt{-a^{2} + b^{2}}} + \frac{2 \,{\left (3 \, b^{3} e^{\left (5 \, x\right )} - 3 \, a b^{2} e^{\left (4 \, x\right )} + 4 \, a^{2} b e^{\left (3 \, x\right )} - 10 \, b^{3} e^{\left (3 \, x\right )} - 6 \, a^{3} e^{\left (2 \, x\right )} + 12 \, a b^{2} e^{\left (2 \, x\right )} + 3 \, b^{3} e^{x} + 2 \, a^{3} - 5 \, a b^{2}\right )}}{3 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )}{\left (e^{\left (2 \, x\right )} - 1\right )}^{3}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)^4/(a+b*cosh(x)),x, algorithm="giac")
[Out]
2*b^4*arctan((b*e^x + a)/sqrt(-a^2 + b^2))/((a^4 - 2*a^2*b^2 + b^4)*sqrt(-a^2 + b^2)) + 2/3*(3*b^3*e^(5*x) - 3
*a*b^2*e^(4*x) + 4*a^2*b*e^(3*x) - 10*b^3*e^(3*x) - 6*a^3*e^(2*x) + 12*a*b^2*e^(2*x) + 3*b^3*e^x + 2*a^3 - 5*a
*b^2)/((a^4 - 2*a^2*b^2 + b^4)*(e^(2*x) - 1)^3)