3.176 \(\int \frac{\text{csch}^5(x)}{a+b \cosh (x)} \, dx\)
Optimal. Leaf size=151 \[ \frac{b^5 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^3}+\frac{\left (3 a^2+9 a b+8 b^2\right ) \log (1-\cosh (x))}{16 (a+b)^3}-\frac{\left (3 a^2-9 a b+8 b^2\right ) \log (\cosh (x)+1)}{16 (a-b)^3}+\frac{\text{csch}^4(x) (b-a \cosh (x))}{4 \left (a^2-b^2\right )}+\frac{\text{csch}^2(x) \left (a \left (3 a^2-7 b^2\right ) \cosh (x)+4 b^3\right )}{8 \left (a^2-b^2\right )^2} \]
[Out]
((4*b^3 + a*(3*a^2 - 7*b^2)*Cosh[x])*Csch[x]^2)/(8*(a^2 - b^2)^2) + ((b - a*Cosh[x])*Csch[x]^4)/(4*(a^2 - b^2)
) + ((3*a^2 + 9*a*b + 8*b^2)*Log[1 - Cosh[x]])/(16*(a + b)^3) - ((3*a^2 - 9*a*b + 8*b^2)*Log[1 + Cosh[x]])/(16
*(a - b)^3) + (b^5*Log[a + b*Cosh[x]])/(a^2 - b^2)^3
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Rubi [A] time = 0.253579, antiderivative size = 151, normalized size of antiderivative = 1.,
number of steps used = 5, number of rules used = 4, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.308, Rules used =
{2668, 741, 823, 801} \[ \frac{b^5 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^3}+\frac{\left (3 a^2+9 a b+8 b^2\right ) \log (1-\cosh (x))}{16 (a+b)^3}-\frac{\left (3 a^2-9 a b+8 b^2\right ) \log (\cosh (x)+1)}{16 (a-b)^3}+\frac{\text{csch}^4(x) (b-a \cosh (x))}{4 \left (a^2-b^2\right )}+\frac{\text{csch}^2(x) \left (a \left (3 a^2-7 b^2\right ) \cosh (x)+4 b^3\right )}{8 \left (a^2-b^2\right )^2} \]
Antiderivative was successfully verified.
[In]
Int[Csch[x]^5/(a + b*Cosh[x]),x]
[Out]
((4*b^3 + a*(3*a^2 - 7*b^2)*Cosh[x])*Csch[x]^2)/(8*(a^2 - b^2)^2) + ((b - a*Cosh[x])*Csch[x]^4)/(4*(a^2 - b^2)
) + ((3*a^2 + 9*a*b + 8*b^2)*Log[1 - Cosh[x]])/(16*(a + b)^3) - ((3*a^2 - 9*a*b + 8*b^2)*Log[1 + Cosh[x]])/(16
*(a - b)^3) + (b^5*Log[a + b*Cosh[x]])/(a^2 - b^2)^3
Rule 2668
Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(b^p*f), S
ubst[Int[(a + x)^m*(b^2 - x^2)^((p - 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && Integer
Q[(p - 1)/2] && NeQ[a^2 - b^2, 0]
Rule 741
Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(m + 1)*(a*e + c*d*x)*(
a + c*x^2)^(p + 1))/(2*a*(p + 1)*(c*d^2 + a*e^2)), x] + Dist[1/(2*a*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*
Simp[c*d^2*(2*p + 3) + a*e^2*(m + 2*p + 3) + c*e*d*(m + 2*p + 4)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a
, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]
Rule 823
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^(
m + 1)*(f*a*c*e - a*g*c*d + c*(c*d*f + a*e*g)*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), x] + Di
st[1/(2*a*c*(p + 1)*(c*d^2 + a*e^2)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Simp[f*(c^2*d^2*(2*p + 3) + a*c*e^2*
(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f + a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x]
&& NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
Rule 801
Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]
Rubi steps
\begin{align*} \int \frac{\text{csch}^5(x)}{a+b \cosh (x)} \, dx &=-\left (b^5 \operatorname{Subst}\left (\int \frac{1}{(a+x) \left (b^2-x^2\right )^3} \, dx,x,b \cosh (x)\right )\right )\\ &=\frac{(b-a \cosh (x)) \text{csch}^4(x)}{4 \left (a^2-b^2\right )}-\frac{b^3 \operatorname{Subst}\left (\int \frac{3 a^2-4 b^2+3 a x}{(a+x) \left (b^2-x^2\right )^2} \, dx,x,b \cosh (x)\right )}{4 \left (a^2-b^2\right )}\\ &=\frac{\left (4 b^3+a \left (3 a^2-7 b^2\right ) \cosh (x)\right ) \text{csch}^2(x)}{8 \left (a^2-b^2\right )^2}+\frac{(b-a \cosh (x)) \text{csch}^4(x)}{4 \left (a^2-b^2\right )}+\frac{b \operatorname{Subst}\left (\int \frac{-3 a^4+7 a^2 b^2-8 b^4-a \left (3 a^2-7 b^2\right ) x}{(a+x) \left (b^2-x^2\right )} \, dx,x,b \cosh (x)\right )}{8 \left (a^2-b^2\right )^2}\\ &=\frac{\left (4 b^3+a \left (3 a^2-7 b^2\right ) \cosh (x)\right ) \text{csch}^2(x)}{8 \left (a^2-b^2\right )^2}+\frac{(b-a \cosh (x)) \text{csch}^4(x)}{4 \left (a^2-b^2\right )}+\frac{b \operatorname{Subst}\left (\int \left (-\frac{(a-b)^2 \left (3 a^2+9 a b+8 b^2\right )}{2 b (a+b) (b-x)}+\frac{8 b^4}{(a-b) (a+b) (a+x)}-\frac{(a+b)^2 \left (3 a^2-9 a b+8 b^2\right )}{2 (a-b) b (b+x)}\right ) \, dx,x,b \cosh (x)\right )}{8 \left (a^2-b^2\right )^2}\\ &=\frac{\left (4 b^3+a \left (3 a^2-7 b^2\right ) \cosh (x)\right ) \text{csch}^2(x)}{8 \left (a^2-b^2\right )^2}+\frac{(b-a \cosh (x)) \text{csch}^4(x)}{4 \left (a^2-b^2\right )}+\frac{\left (3 a^2+9 a b+8 b^2\right ) \log (1-\cosh (x))}{16 (a+b)^3}-\frac{\left (3 a^2-9 a b+8 b^2\right ) \log (1+\cosh (x))}{16 (a-b)^3}+\frac{b^5 \log (a+b \cosh (x))}{\left (a^2-b^2\right )^3}\\ \end{align*}
Mathematica [A] time = 0.861324, size = 148, normalized size = 0.98 \[ \frac{1}{64} \left (\frac{2 \left (3 a^2-8 a b+5 b^2\right ) \text{sech}^2\left (\frac{x}{2}\right )+\frac{8 \left (a \left (-10 a^2 b^2+3 a^4+15 b^4\right ) \log \left (\tanh \left (\frac{x}{2}\right )\right )+8 b^5 \log (a+b \cosh (x))-8 b^5 \log (\sinh (x))\right )}{(a+b)^3}+(a-b)^2 \text{sech}^4\left (\frac{x}{2}\right )}{(a-b)^3}-\frac{\text{csch}^4\left (\frac{x}{2}\right )}{a+b}+\frac{2 (3 a+5 b) \text{csch}^2\left (\frac{x}{2}\right )}{(a+b)^2}\right ) \]
Antiderivative was successfully verified.
[In]
Integrate[Csch[x]^5/(a + b*Cosh[x]),x]
[Out]
((2*(3*a + 5*b)*Csch[x/2]^2)/(a + b)^2 - Csch[x/2]^4/(a + b) + ((8*(8*b^5*Log[a + b*Cosh[x]] - 8*b^5*Log[Sinh[
x]] + a*(3*a^4 - 10*a^2*b^2 + 15*b^4)*Log[Tanh[x/2]]))/(a + b)^3 + 2*(3*a^2 - 8*a*b + 5*b^2)*Sech[x/2]^2 + (a
- b)^2*Sech[x/2]^4)/(a - b)^3)/64
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Maple [A] time = 0.026, size = 191, normalized size = 1.3 \begin{align*}{\frac{a}{64\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{4}}-{\frac{b}{64\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{4}}-{\frac{a}{8\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{3\,b}{16\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{{b}^{5}}{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3}}\ln \left ( a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}- \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}b-a-b \right ) }-{\frac{1}{64\,a+64\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-4}}+{\frac{3\,b}{16\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{a}{8\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{3\,{a}^{2}}{8\, \left ( a+b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }+{\frac{9\,ab}{8\, \left ( a+b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }+{\frac{{b}^{2}}{ \left ( a+b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(csch(x)^5/(a+b*cosh(x)),x)
[Out]
1/64/(a-b)^2*tanh(1/2*x)^4*a-1/64/(a-b)^2*tanh(1/2*x)^4*b-1/8/(a-b)^2*tanh(1/2*x)^2*a+3/16/(a-b)^2*tanh(1/2*x)
^2*b+1/(a-b)^3*b^5/(a+b)^3*ln(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b-a-b)-1/64/(a+b)/tanh(1/2*x)^4+3/16/(a+b)^2/tanh(
1/2*x)^2*b+1/8/(a+b)^2/tanh(1/2*x)^2*a+3/8/(a+b)^3*ln(tanh(1/2*x))*a^2+9/8/(a+b)^3*ln(tanh(1/2*x))*a*b+1/(a+b)
^3*ln(tanh(1/2*x))*b^2
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Maxima [B] time = 1.14881, size = 470, normalized size = 3.11 \begin{align*} \frac{b^{5} \log \left (2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} + b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} - \frac{{\left (3 \, a^{2} - 9 \, a b + 8 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + 1\right )}{8 \,{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac{{\left (3 \, a^{2} + 9 \, a b + 8 \, b^{2}\right )} \log \left (e^{\left (-x\right )} - 1\right )}{8 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac{8 \, b^{3} e^{\left (-2 \, x\right )} + 8 \, b^{3} e^{\left (-6 \, x\right )} +{\left (3 \, a^{3} - 7 \, a b^{2}\right )} e^{\left (-x\right )} -{\left (11 \, a^{3} - 15 \, a b^{2}\right )} e^{\left (-3 \, x\right )} + 16 \,{\left (a^{2} b - 2 \, b^{3}\right )} e^{\left (-4 \, x\right )} -{\left (11 \, a^{3} - 15 \, a b^{2}\right )} e^{\left (-5 \, x\right )} +{\left (3 \, a^{3} - 7 \, a b^{2}\right )} e^{\left (-7 \, x\right )}}{4 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4} - 4 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-2 \, x\right )} + 6 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-4 \, x\right )} - 4 \,{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-6 \, x\right )} +{\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} e^{\left (-8 \, x\right )}\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)^5/(a+b*cosh(x)),x, algorithm="maxima")
[Out]
b^5*log(2*a*e^(-x) + b*e^(-2*x) + b)/(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6) - 1/8*(3*a^2 - 9*a*b + 8*b^2)*log(e^(
-x) + 1)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/8*(3*a^2 + 9*a*b + 8*b^2)*log(e^(-x) - 1)/(a^3 + 3*a^2*b + 3*a*b^
2 + b^3) + 1/4*(8*b^3*e^(-2*x) + 8*b^3*e^(-6*x) + (3*a^3 - 7*a*b^2)*e^(-x) - (11*a^3 - 15*a*b^2)*e^(-3*x) + 16
*(a^2*b - 2*b^3)*e^(-4*x) - (11*a^3 - 15*a*b^2)*e^(-5*x) + (3*a^3 - 7*a*b^2)*e^(-7*x))/(a^4 - 2*a^2*b^2 + b^4
- 4*(a^4 - 2*a^2*b^2 + b^4)*e^(-2*x) + 6*(a^4 - 2*a^2*b^2 + b^4)*e^(-4*x) - 4*(a^4 - 2*a^2*b^2 + b^4)*e^(-6*x)
+ (a^4 - 2*a^2*b^2 + b^4)*e^(-8*x))
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Fricas [B] time = 2.76161, size = 8195, normalized size = 54.27 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)^5/(a+b*cosh(x)),x, algorithm="fricas")
[Out]
1/8*(2*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x)^7 + 2*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*sinh(x)^7 + 16*(a^2*b^3 - b
^5)*cosh(x)^6 + 2*(8*a^2*b^3 - 8*b^5 + 7*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x))*sinh(x)^6 - 2*(11*a^5 - 26*a^
3*b^2 + 15*a*b^4)*cosh(x)^5 - 2*(11*a^5 - 26*a^3*b^2 + 15*a*b^4 - 21*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x)^2
- 48*(a^2*b^3 - b^5)*cosh(x))*sinh(x)^5 + 32*(a^4*b - 3*a^2*b^3 + 2*b^5)*cosh(x)^4 + 2*(16*a^4*b - 48*a^2*b^3
+ 32*b^5 + 35*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x)^3 + 120*(a^2*b^3 - b^5)*cosh(x)^2 - 5*(11*a^5 - 26*a^3*b^
2 + 15*a*b^4)*cosh(x))*sinh(x)^4 - 2*(11*a^5 - 26*a^3*b^2 + 15*a*b^4)*cosh(x)^3 - 2*(11*a^5 - 26*a^3*b^2 + 15*
a*b^4 - 35*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x)^4 - 160*(a^2*b^3 - b^5)*cosh(x)^3 + 10*(11*a^5 - 26*a^3*b^2
+ 15*a*b^4)*cosh(x)^2 - 64*(a^4*b - 3*a^2*b^3 + 2*b^5)*cosh(x))*sinh(x)^3 + 16*(a^2*b^3 - b^5)*cosh(x)^2 + 2*(
21*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x)^5 + 8*a^2*b^3 - 8*b^5 + 120*(a^2*b^3 - b^5)*cosh(x)^4 - 10*(11*a^5 -
26*a^3*b^2 + 15*a*b^4)*cosh(x)^3 + 96*(a^4*b - 3*a^2*b^3 + 2*b^5)*cosh(x)^2 - 3*(11*a^5 - 26*a^3*b^2 + 15*a*b
^4)*cosh(x))*sinh(x)^2 + 2*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x) + 8*(b^5*cosh(x)^8 + 8*b^5*cosh(x)*sinh(x)^7
+ b^5*sinh(x)^8 - 4*b^5*cosh(x)^6 + 6*b^5*cosh(x)^4 - 4*b^5*cosh(x)^2 + 4*(7*b^5*cosh(x)^2 - b^5)*sinh(x)^6 +
8*(7*b^5*cosh(x)^3 - 3*b^5*cosh(x))*sinh(x)^5 + b^5 + 2*(35*b^5*cosh(x)^4 - 30*b^5*cosh(x)^2 + 3*b^5)*sinh(x)
^4 + 8*(7*b^5*cosh(x)^5 - 10*b^5*cosh(x)^3 + 3*b^5*cosh(x))*sinh(x)^3 + 4*(7*b^5*cosh(x)^6 - 15*b^5*cosh(x)^4
+ 9*b^5*cosh(x)^2 - b^5)*sinh(x)^2 + 8*(b^5*cosh(x)^7 - 3*b^5*cosh(x)^5 + 3*b^5*cosh(x)^3 - b^5*cosh(x))*sinh(
x))*log(2*(b*cosh(x) + a)/(cosh(x) - sinh(x))) - ((3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^8 + 8*(3*a^5
- 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)*sinh(x)^7 + (3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*sinh(x)^8 - 4*(3
*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^6 - 4*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5 - 7*(3*a^5 - 10*a^3
*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^2)*sinh(x)^6 + 8*(7*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^3 - 3*(3*
a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x))*sinh(x)^5 + 3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5 + 6*(3*a^5 - 1
0*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^4 + 2*(9*a^5 - 30*a^3*b^2 + 45*a*b^4 + 24*b^5 + 35*(3*a^5 - 10*a^3*b^2 +
15*a*b^4 + 8*b^5)*cosh(x)^4 - 30*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^2)*sinh(x)^4 + 8*(7*(3*a^5 -
10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^5 - 10*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^3 + 3*(3*a^5 -
10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x))*sinh(x)^3 - 4*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^2 + 4*(7
*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^6 - 3*a^5 + 10*a^3*b^2 - 15*a*b^4 - 8*b^5 - 15*(3*a^5 - 10*a^
3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^4 + 9*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^2)*sinh(x)^2 + 8*((3*a
^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^7 - 3*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^5 + 3*(3*a^5
- 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x)^3 - (3*a^5 - 10*a^3*b^2 + 15*a*b^4 + 8*b^5)*cosh(x))*sinh(x))*log(co
sh(x) + sinh(x) + 1) + ((3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)^8 + 8*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 -
8*b^5)*cosh(x)*sinh(x)^7 + (3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*sinh(x)^8 - 4*(3*a^5 - 10*a^3*b^2 + 15*a*b
^4 - 8*b^5)*cosh(x)^6 - 4*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5 - 7*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*c
osh(x)^2)*sinh(x)^6 + 8*(7*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)^3 - 3*(3*a^5 - 10*a^3*b^2 + 15*a*b^
4 - 8*b^5)*cosh(x))*sinh(x)^5 + 3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5 + 6*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b
^5)*cosh(x)^4 + 2*(9*a^5 - 30*a^3*b^2 + 45*a*b^4 - 24*b^5 + 35*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)
^4 - 30*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)^2)*sinh(x)^4 + 8*(7*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8
*b^5)*cosh(x)^5 - 10*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)^3 + 3*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*
b^5)*cosh(x))*sinh(x)^3 - 4*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)^2 + 4*(7*(3*a^5 - 10*a^3*b^2 + 15*
a*b^4 - 8*b^5)*cosh(x)^6 - 3*a^5 + 10*a^3*b^2 - 15*a*b^4 + 8*b^5 - 15*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*
cosh(x)^4 + 9*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)^2)*sinh(x)^2 + 8*((3*a^5 - 10*a^3*b^2 + 15*a*b^4
- 8*b^5)*cosh(x)^7 - 3*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x)^5 + 3*(3*a^5 - 10*a^3*b^2 + 15*a*b^4 -
8*b^5)*cosh(x)^3 - (3*a^5 - 10*a^3*b^2 + 15*a*b^4 - 8*b^5)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) - 1) + 2*(
7*(3*a^5 - 10*a^3*b^2 + 7*a*b^4)*cosh(x)^6 + 48*(a^2*b^3 - b^5)*cosh(x)^5 + 3*a^5 - 10*a^3*b^2 + 7*a*b^4 - 5*(
11*a^5 - 26*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 64*(a^4*b - 3*a^2*b^3 + 2*b^5)*cosh(x)^3 - 3*(11*a^5 - 26*a^3*b^2
+ 15*a*b^4)*cosh(x)^2 + 16*(a^2*b^3 - b^5)*cosh(x))*sinh(x))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^8 +
8*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)*sinh(x)^7 + (a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sinh(x)^8 - 4*(a
^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^6 - 4*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 - 7*(a^6 - 3*a^4*b^2 + 3*a^
2*b^4 - b^6)*cosh(x)^2)*sinh(x)^6 + a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6 + 8*(7*(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)^5 + 6*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6
)*cosh(x)^4 + 2*(3*a^6 - 9*a^4*b^2 + 9*a^2*b^4 - 3*b^6 + 35*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 - 30
*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^5
- 10*(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
- 4*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^2 + 4*(7*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^6 - a^6
+ 3*a^4*b^2 - 3*a^2*b^4 + b^6 - 15*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^4 + 9*(a^6 - 3*a^4*b^2 + 3*a^2*
b^4 - b^6)*cosh(x)^2)*sinh(x)^2 + 8*((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*cosh(x)^7 - 3*(a^6 - 3*a^4*b^2 + 3*a^
2*b^4 - b^6)*cosh(x)^5 + 3*(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)**5/(a+b*cosh(x)),x)
[Out]
Timed out
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Giac [B] time = 1.16045, size = 456, normalized size = 3.02 \begin{align*} \frac{b^{6} \log \left ({\left | b{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, a \right |}\right )}{a^{6} b - 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} - b^{7}} - \frac{{\left (3 \, a^{2} - 9 \, a b + 8 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x} + 2\right )}{16 \,{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac{{\left (3 \, a^{2} + 9 \, a b + 8 \, b^{2}\right )} \log \left (e^{\left (-x\right )} + e^{x} - 2\right )}{16 \,{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )}} + \frac{3 \, b^{5}{\left (e^{\left (-x\right )} + e^{x}\right )}^{4} + 3 \, a^{5}{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 10 \, a^{3} b^{2}{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 7 \, a b^{4}{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 8 \, a^{2} b^{3}{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 32 \, b^{5}{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 20 \, a^{5}{\left (e^{\left (-x\right )} + e^{x}\right )} + 56 \, a^{3} b^{2}{\left (e^{\left (-x\right )} + e^{x}\right )} - 36 \, a b^{4}{\left (e^{\left (-x\right )} + e^{x}\right )} + 16 \, a^{4} b - 64 \, a^{2} b^{3} + 96 \, b^{5}}{4 \,{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )}{\left ({\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 4\right )}^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(csch(x)^5/(a+b*cosh(x)),x, algorithm="giac")
[Out]
b^6*log(abs(b*(e^(-x) + e^x) + 2*a))/(a^6*b - 3*a^4*b^3 + 3*a^2*b^5 - b^7) - 1/16*(3*a^2 - 9*a*b + 8*b^2)*log(
e^(-x) + e^x + 2)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/16*(3*a^2 + 9*a*b + 8*b^2)*log(e^(-x) + e^x - 2)/(a^3 +
3*a^2*b + 3*a*b^2 + b^3) + 1/4*(3*b^5*(e^(-x) + e^x)^4 + 3*a^5*(e^(-x) + e^x)^3 - 10*a^3*b^2*(e^(-x) + e^x)^3
+ 7*a*b^4*(e^(-x) + e^x)^3 + 8*a^2*b^3*(e^(-x) + e^x)^2 - 32*b^5*(e^(-x) + e^x)^2 - 20*a^5*(e^(-x) + e^x) + 56
*a^3*b^2*(e^(-x) + e^x) - 36*a*b^4*(e^(-x) + e^x) + 16*a^4*b - 64*a^2*b^3 + 96*b^5)/((a^6 - 3*a^4*b^2 + 3*a^2*
b^4 - b^6)*((e^(-x) + e^x)^2 - 4)^2)