3.124 \(\int \frac{\coth ^5(x)}{a+b \text{sech}(x)} \, dx\)

Optimal. Leaf size=178 \[ -\frac{b^6 \log (a+b \text{sech}(x))}{a \left (a^2-b^2\right )^3}+\frac{\left (8 a^2+21 a b+15 b^2\right ) \log (1-\text{sech}(x))}{16 (a+b)^3}+\frac{\left (8 a^2-21 a b+15 b^2\right ) \log (\text{sech}(x)+1)}{16 (a-b)^3}-\frac{5 a+7 b}{16 (a+b)^2 (1-\text{sech}(x))}-\frac{5 a-7 b}{16 (a-b)^2 (\text{sech}(x)+1)}-\frac{1}{16 (a+b) (1-\text{sech}(x))^2}-\frac{1}{16 (a-b) (\text{sech}(x)+1)^2}+\frac{\log (\cosh (x))}{a} \]

[Out]

Log[Cosh[x]]/a + ((8*a^2 + 21*a*b + 15*b^2)*Log[1 - Sech[x]])/(16*(a + b)^3) + ((8*a^2 - 21*a*b + 15*b^2)*Log[
1 + Sech[x]])/(16*(a - b)^3) - (b^6*Log[a + b*Sech[x]])/(a*(a^2 - b^2)^3) - 1/(16*(a + b)*(1 - Sech[x])^2) - (
5*a + 7*b)/(16*(a + b)^2*(1 - Sech[x])) - 1/(16*(a - b)*(1 + Sech[x])^2) - (5*a - 7*b)/(16*(a - b)^2*(1 + Sech
[x]))

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Rubi [A]  time = 0.320137, antiderivative size = 178, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {3885, 894} \[ -\frac{b^6 \log (a+b \text{sech}(x))}{a \left (a^2-b^2\right )^3}+\frac{\left (8 a^2+21 a b+15 b^2\right ) \log (1-\text{sech}(x))}{16 (a+b)^3}+\frac{\left (8 a^2-21 a b+15 b^2\right ) \log (\text{sech}(x)+1)}{16 (a-b)^3}-\frac{5 a+7 b}{16 (a+b)^2 (1-\text{sech}(x))}-\frac{5 a-7 b}{16 (a-b)^2 (\text{sech}(x)+1)}-\frac{1}{16 (a+b) (1-\text{sech}(x))^2}-\frac{1}{16 (a-b) (\text{sech}(x)+1)^2}+\frac{\log (\cosh (x))}{a} \]

Antiderivative was successfully verified.

[In]

Int[Coth[x]^5/(a + b*Sech[x]),x]

[Out]

Log[Cosh[x]]/a + ((8*a^2 + 21*a*b + 15*b^2)*Log[1 - Sech[x]])/(16*(a + b)^3) + ((8*a^2 - 21*a*b + 15*b^2)*Log[
1 + Sech[x]])/(16*(a - b)^3) - (b^6*Log[a + b*Sech[x]])/(a*(a^2 - b^2)^3) - 1/(16*(a + b)*(1 - Sech[x])^2) - (
5*a + 7*b)/(16*(a + b)^2*(1 - Sech[x])) - 1/(16*(a - b)*(1 + Sech[x])^2) - (5*a - 7*b)/(16*(a - b)^2*(1 + Sech
[x]))

Rule 3885

Int[cot[(c_.) + (d_.)*(x_)]^(m_.)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> -Dist[(-1)^((m - 1
)/2)/(d*b^(m - 1)), Subst[Int[((b^2 - x^2)^((m - 1)/2)*(a + x)^n)/x, x], x, b*Csc[c + d*x]], x] /; FreeQ[{a, b
, c, d, n}, x] && IntegerQ[(m - 1)/2] && NeQ[a^2 - b^2, 0]

Rule 894

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIn
tegrand[(d + e*x)^m*(f + g*x)^n*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] &&
NeQ[c*d^2 + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && IntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \frac{\coth ^5(x)}{a+b \text{sech}(x)} \, dx &=-\left (b^6 \operatorname{Subst}\left (\int \frac{1}{x (a+x) \left (b^2-x^2\right )^3} \, dx,x,b \text{sech}(x)\right )\right )\\ &=-\left (b^6 \operatorname{Subst}\left (\int \left (\frac{1}{8 b^4 (a+b) (b-x)^3}+\frac{5 a+7 b}{16 b^5 (a+b)^2 (b-x)^2}+\frac{8 a^2+21 a b+15 b^2}{16 b^6 (a+b)^3 (b-x)}+\frac{1}{a b^6 x}+\frac{1}{a (a-b)^3 (a+b)^3 (a+x)}+\frac{1}{8 b^4 (-a+b) (b+x)^3}+\frac{-5 a+7 b}{16 (a-b)^2 b^5 (b+x)^2}+\frac{8 a^2-21 a b+15 b^2}{16 b^6 (-a+b)^3 (b+x)}\right ) \, dx,x,b \text{sech}(x)\right )\right )\\ &=\frac{\log (\cosh (x))}{a}+\frac{\left (8 a^2+21 a b+15 b^2\right ) \log (1-\text{sech}(x))}{16 (a+b)^3}+\frac{\left (8 a^2-21 a b+15 b^2\right ) \log (1+\text{sech}(x))}{16 (a-b)^3}-\frac{b^6 \log (a+b \text{sech}(x))}{a \left (a^2-b^2\right )^3}-\frac{1}{16 (a+b) (1-\text{sech}(x))^2}-\frac{5 a+7 b}{16 (a+b)^2 (1-\text{sech}(x))}-\frac{1}{16 (a-b) (1+\text{sech}(x))^2}-\frac{5 a-7 b}{16 (a-b)^2 (1+\text{sech}(x))}\\ \end{align*}

Mathematica [A]  time = 0.997587, size = 167, normalized size = 0.94 \[ \frac{1}{64} \left (-\frac{8 \left (a \left (b \left (-10 a^2 b^2+3 a^4+15 b^4\right ) \log \left (\tanh \left (\frac{x}{2}\right )\right )-8 a \left (-3 a^2 b^2+a^4+3 b^4\right ) \log (\sinh (x))\right )+8 b^6 \log (a \cosh (x)+b)\right )}{a (a-b)^3 (a+b)^3}-\frac{\text{csch}^4\left (\frac{x}{2}\right )}{a+b}-\frac{2 (7 a+9 b) \text{csch}^2\left (\frac{x}{2}\right )}{(a+b)^2}-\frac{\text{sech}^4\left (\frac{x}{2}\right )}{a-b}+\frac{2 (7 a-9 b) \text{sech}^2\left (\frac{x}{2}\right )}{(a-b)^2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Coth[x]^5/(a + b*Sech[x]),x]

[Out]

((-2*(7*a + 9*b)*Csch[x/2]^2)/(a + b)^2 - Csch[x/2]^4/(a + b) - (8*(8*b^6*Log[b + a*Cosh[x]] + a*(-8*a*(a^4 -
3*a^2*b^2 + 3*b^4)*Log[Sinh[x]] + b*(3*a^4 - 10*a^2*b^2 + 15*b^4)*Log[Tanh[x/2]])))/(a*(a - b)^3*(a + b)^3) +
(2*(7*a - 9*b)*Sech[x/2]^2)/(a - b)^2 - Sech[x/2]^4/(a - b))/64

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Maple [A]  time = 0.041, size = 215, normalized size = 1.2 \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{3\,a}{16\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}+{\frac{b}{4\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2}}-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-{\frac{1}{64\,a+64\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-4}}-{\frac{3\,a}{16\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}-{\frac{b}{4\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{-2}}+{\frac{{a}^{2}}{ \left ( a+b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }+{\frac{21\,ab}{8\, \left ( a+b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }+{\frac{15\,{b}^{2}}{8\, \left ( a+b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) }-{\frac{1}{a}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) }-{\frac{{b}^{6}}{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3}a}\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 ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(coth(x)^5/(a+b*sech(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-3/16/(a-b)^2*tanh(1/2*x)^2*a+1/4/(a-b)^2*tanh(1/2*x
)^2*b-1/a*ln(tanh(1/2*x)+1)-1/64/(a+b)/tanh(1/2*x)^4-3/16/(a+b)^2/tanh(1/2*x)^2*a-1/4/(a+b)^2/tanh(1/2*x)^2*b+
1/(a+b)^3*ln(tanh(1/2*x))*a^2+21/8/(a+b)^3*ln(tanh(1/2*x))*a*b+15/8/(a+b)^3*ln(tanh(1/2*x))*b^2-1/a*ln(tanh(1/
2*x)-1)-1/(a-b)^3*b^6/(a+b)^3/a*ln(a*tanh(1/2*x)^2-tanh(1/2*x)^2*b+a+b)

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Maxima [B]  time = 1.14141, size = 494, normalized size = 2.78 \begin{align*} -\frac{b^{6} \log \left (2 \, b e^{\left (-x\right )} + a e^{\left (-2 \, x\right )} + a\right )}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}} + \frac{{\left (8 \, a^{2} - 21 \, a b + 15 \, 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 (8 \, a^{2} + 21 \, a b + 15 \, 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 (5 \, a^{2} b - 9 \, b^{3}\right )} e^{\left (-x\right )} - 8 \,{\left (2 \, a^{3} - 3 \, a b^{2}\right )} e^{\left (-2 \, x\right )} +{\left (3 \, a^{2} b + b^{3}\right )} e^{\left (-3 \, x\right )} + 16 \,{\left (a^{3} - 2 \, a b^{2}\right )} e^{\left (-4 \, x\right )} +{\left (3 \, a^{2} b + b^{3}\right )} e^{\left (-5 \, x\right )} - 8 \,{\left (2 \, a^{3} - 3 \, a b^{2}\right )} e^{\left (-6 \, x\right )} +{\left (5 \, a^{2} b - 9 \, b^{3}\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 )}} + \frac{x}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^6*log(2*b*e^(-x) + a*e^(-2*x) + a)/(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6) + 1/8*(8*a^2 - 21*a*b + 15*b^2)*lo
g(e^(-x) + 1)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/8*(8*a^2 + 21*a*b + 15*b^2)*log(e^(-x) - 1)/(a^3 + 3*a^2*b +
3*a*b^2 + b^3) + 1/4*((5*a^2*b - 9*b^3)*e^(-x) - 8*(2*a^3 - 3*a*b^2)*e^(-2*x) + (3*a^2*b + b^3)*e^(-3*x) + 16
*(a^3 - 2*a*b^2)*e^(-4*x) + (3*a^2*b + b^3)*e^(-5*x) - 8*(2*a^3 - 3*a*b^2)*e^(-6*x) + (5*a^2*b - 9*b^3)*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)) + x/a

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Fricas [B]  time = 3.65215, size = 11934, normalized size = 67.04 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/8*(8*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cosh(x)^8 + 8*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*sinh(x)^8 -
2*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x)^7 - 2*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5 - 32*(a^6 - 3*a^4*b^2 + 3*a^2
*b^4 - b^6)*x*cosh(x))*sinh(x)^7 + 16*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x
)*cosh(x)^6 + 2*(16*a^6 - 40*a^4*b^2 + 24*a^2*b^4 + 112*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cosh(x)^2 - 16*(
a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x - 7*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x))*sinh(x)^6 - 2*(3*a^5*b - 2*
a^3*b^3 - a*b^5)*cosh(x)^5 - 2*(3*a^5*b - 2*a^3*b^3 - a*b^5 - 224*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cosh(x
)^3 + 21*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x)^2 - 48*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 +
3*a^2*b^4 - b^6)*x)*cosh(x))*sinh(x)^5 - 16*(2*a^6 - 6*a^4*b^2 + 4*a^2*b^4 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 -
b^6)*x)*cosh(x)^4 - 2*(16*a^6 - 48*a^4*b^2 + 32*a^2*b^4 - 280*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cosh(x)^4
+ 35*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x)^3 - 120*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 + 3
*a^2*b^4 - b^6)*x)*cosh(x)^2 - 24*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x + 5*(3*a^5*b - 2*a^3*b^3 - a*b^5)*cosh
(x))*sinh(x)^4 - 2*(3*a^5*b - 2*a^3*b^3 - a*b^5)*cosh(x)^3 + 2*(224*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x*cosh
(x)^5 - 3*a^5*b + 2*a^3*b^3 + a*b^5 - 35*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x)^4 + 160*(2*a^6 - 5*a^4*b^2 +
3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x)*cosh(x)^3 - 10*(3*a^5*b - 2*a^3*b^3 - a*b^5)*cosh(x)^2 -
32*(2*a^6 - 6*a^4*b^2 + 4*a^2*b^4 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x)*cosh(x))*sinh(x)^3 + 16*(2*a^6 -
5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x)*cosh(x)^2 + 2*(112*(a^6 - 3*a^4*b^2 + 3*a^2*
b^4 - b^6)*x*cosh(x)^6 + 16*a^6 - 40*a^4*b^2 + 24*a^2*b^4 - 21*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x)^5 + 12
0*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x)*cosh(x)^4 - 10*(3*a^5*b - 2*a^3*b^
3 - a*b^5)*cosh(x)^3 - 48*(2*a^6 - 6*a^4*b^2 + 4*a^2*b^4 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x)*cosh(x)^2
- 16*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x - 3*(3*a^5*b - 2*a^3*b^3 - a*b^5)*cosh(x))*sinh(x)^2 + 8*(a^6 - 3*a
^4*b^2 + 3*a^2*b^4 - b^6)*x - 2*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x) + 8*(b^6*cosh(x)^8 + 8*b^6*cosh(x)*si
nh(x)^7 + b^6*sinh(x)^8 - 4*b^6*cosh(x)^6 + 6*b^6*cosh(x)^4 - 4*b^6*cosh(x)^2 + 4*(7*b^6*cosh(x)^2 - b^6)*sinh
(x)^6 + b^6 + 8*(7*b^6*cosh(x)^3 - 3*b^6*cosh(x))*sinh(x)^5 + 2*(35*b^6*cosh(x)^4 - 30*b^6*cosh(x)^2 + 3*b^6)*
sinh(x)^4 + 8*(7*b^6*cosh(x)^5 - 10*b^6*cosh(x)^3 + 3*b^6*cosh(x))*sinh(x)^3 + 4*(7*b^6*cosh(x)^6 - 15*b^6*cos
h(x)^4 + 9*b^6*cosh(x)^2 - b^6)*sinh(x)^2 + 8*(b^6*cosh(x)^7 - 3*b^6*cosh(x)^5 + 3*b^6*cosh(x)^3 - b^6*cosh(x)
)*sinh(x))*log(2*(a*cosh(x) + b)/(cosh(x) - sinh(x))) - ((8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b
^4 + 15*a*b^5)*cosh(x)^8 + 8*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)*sinh(
x)^7 + (8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*sinh(x)^8 - 4*(8*a^6 + 3*a^5*b - 24
*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^6 - 4*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a
^2*b^4 + 15*a*b^5 - 7*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^2)*sinh(x)^6
+ 8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5 + 8*(7*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10
*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^3 - 3*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a
*b^5)*cosh(x))*sinh(x)^5 + 6*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^4 + 2
*(24*a^6 + 9*a^5*b - 72*a^4*b^2 - 30*a^3*b^3 + 72*a^2*b^4 + 45*a*b^5 + 35*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a
^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^4 - 30*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*
b^5)*cosh(x)^2)*sinh(x)^4 + 8*(7*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^5
- 10*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^3 + 3*(8*a^6 + 3*a^5*b - 24*
a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x))*sinh(x)^3 - 4*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^
3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^2 + 4*(7*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5
)*cosh(x)^6 - 8*a^6 - 3*a^5*b + 24*a^4*b^2 + 10*a^3*b^3 - 24*a^2*b^4 - 15*a*b^5 - 15*(8*a^6 + 3*a^5*b - 24*a^4
*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^4 + 9*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b
^4 + 15*a*b^5)*cosh(x)^2)*sinh(x)^2 + 8*((8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*c
osh(x)^7 - 3*(8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^5 + 3*(8*a^6 + 3*a^5*
b - 24*a^4*b^2 - 10*a^3*b^3 + 24*a^2*b^4 + 15*a*b^5)*cosh(x)^3 - (8*a^6 + 3*a^5*b - 24*a^4*b^2 - 10*a^3*b^3 +
24*a^2*b^4 + 15*a*b^5)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) + 1) - ((8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*
b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^8 + 8*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)
*cosh(x)*sinh(x)^7 + (8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*sinh(x)^8 - 4*(8*a^6
- 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^6 - 4*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*
a^3*b^3 + 24*a^2*b^4 - 15*a*b^5 - 7*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x
)^2)*sinh(x)^6 + 8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5 + 8*(7*(8*a^6 - 3*a^5*b - 2
4*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^3 - 3*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*
a^2*b^4 - 15*a*b^5)*cosh(x))*sinh(x)^5 + 6*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)
*cosh(x)^4 + 2*(24*a^6 - 9*a^5*b - 72*a^4*b^2 + 30*a^3*b^3 + 72*a^2*b^4 - 45*a*b^5 + 35*(8*a^6 - 3*a^5*b - 24*
a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^4 - 30*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a
^2*b^4 - 15*a*b^5)*cosh(x)^2)*sinh(x)^4 + 8*(7*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*
b^5)*cosh(x)^5 - 10*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^3 + 3*(8*a^6 -
3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x))*sinh(x)^3 - 4*(8*a^6 - 3*a^5*b - 24*a^4*b
^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^2 + 4*(7*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*
b^4 - 15*a*b^5)*cosh(x)^6 - 8*a^6 + 3*a^5*b + 24*a^4*b^2 - 10*a^3*b^3 - 24*a^2*b^4 + 15*a*b^5 - 15*(8*a^6 - 3*
a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^4 + 9*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*
b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^2)*sinh(x)^2 + 8*((8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4
- 15*a*b^5)*cosh(x)^7 - 3*(8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^5 + 3*(
8*a^6 - 3*a^5*b - 24*a^4*b^2 + 10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x)^3 - (8*a^6 - 3*a^5*b - 24*a^4*b^2 +
10*a^3*b^3 + 24*a^2*b^4 - 15*a*b^5)*cosh(x))*sinh(x))*log(cosh(x) + sinh(x) - 1) + 2*(32*(a^6 - 3*a^4*b^2 + 3
*a^2*b^4 - b^6)*x*cosh(x)^7 - 7*(5*a^5*b - 14*a^3*b^3 + 9*a*b^5)*cosh(x)^6 - 5*a^5*b + 14*a^3*b^3 - 9*a*b^5 +
48*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x)*cosh(x)^5 - 5*(3*a^5*b - 2*a^3*b^
3 - a*b^5)*cosh(x)^4 - 32*(2*a^6 - 6*a^4*b^2 + 4*a^2*b^4 - 3*(a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*x)*cosh(x)^3
- 3*(3*a^5*b - 2*a^3*b^3 - a*b^5)*cosh(x)^2 + 16*(2*a^6 - 5*a^4*b^2 + 3*a^2*b^4 - 2*(a^6 - 3*a^4*b^2 + 3*a^2*b
^4 - b^6)*x)*cosh(x))*sinh(x))/((a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^8 + 8*(a^7 - 3*a^5*b^2 + 3*a^3*b
^4 - a*b^6)*cosh(x)*sinh(x)^7 + (a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*sinh(x)^8 + a^7 - 3*a^5*b^2 + 3*a^3*b^4
- a*b^6 - 4*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^6 - 4*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6 - 7*(a^7
- 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^2)*sinh(x)^6 + 8*(7*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^3 -
3*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x))*sinh(x)^5 + 6*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^
4 + 2*(3*a^7 - 9*a^5*b^2 + 9*a^3*b^4 - 3*a*b^6 + 35*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^4 - 30*(a^7
- 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^2)*sinh(x)^4 + 8*(7*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^5 -
10*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^3 + 3*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x))*sinh(x)
^3 - 4*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^2 - 4*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6 - 7*(a^7 - 3*a
^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^6 + 15*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^4 - 9*(a^7 - 3*a^5*b^
2 + 3*a^3*b^4 - a*b^6)*cosh(x)^2)*sinh(x)^2 + 8*((a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^7 - 3*(a^7 - 3*
a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^5 + 3*(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6)*cosh(x)^3 - (a^7 - 3*a^5*b^2
+ 3*a^3*b^4 - a*b^6)*cosh(x))*sinh(x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\coth ^{5}{\left (x \right )}}{a + b \operatorname{sech}{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(coth(x)**5/(a+b*sech(x)),x)

[Out]

Integral(coth(x)**5/(a + b*sech(x)), x)

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Giac [B]  time = 1.18878, size = 513, normalized size = 2.88 \begin{align*} -\frac{b^{6} \log \left ({\left | a{\left (e^{\left (-x\right )} + e^{x}\right )} + 2 \, b \right |}\right )}{a^{7} - 3 \, a^{5} b^{2} + 3 \, a^{3} b^{4} - a b^{6}} + \frac{{\left (8 \, a^{2} - 21 \, a b + 15 \, 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 (8 \, a^{2} + 21 \, a b + 15 \, 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 \, a^{5}{\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 9 \, a^{3} b^{2}{\left (e^{\left (-x\right )} + e^{x}\right )}^{4} + 9 \, a b^{4}{\left (e^{\left (-x\right )} + e^{x}\right )}^{4} - 5 \, a^{4} b{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} + 14 \, a^{2} b^{3}{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 9 \, b^{5}{\left (e^{\left (-x\right )} + e^{x}\right )}^{3} - 8 \, a^{5}{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 32 \, a^{3} b^{2}{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} - 48 \, a b^{4}{\left (e^{\left (-x\right )} + e^{x}\right )}^{2} + 12 \, a^{4} b{\left (e^{\left (-x\right )} + e^{x}\right )} - 40 \, a^{2} b^{3}{\left (e^{\left (-x\right )} + e^{x}\right )} + 28 \, b^{5}{\left (e^{\left (-x\right )} + e^{x}\right )} - 16 \, a^{3} b^{2} + 64 \, a b^{4}}{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(coth(x)^5/(a+b*sech(x)),x, algorithm="giac")

[Out]

-b^6*log(abs(a*(e^(-x) + e^x) + 2*b))/(a^7 - 3*a^5*b^2 + 3*a^3*b^4 - a*b^6) + 1/16*(8*a^2 - 21*a*b + 15*b^2)*l
og(e^(-x) + e^x + 2)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/16*(8*a^2 + 21*a*b + 15*b^2)*log(e^(-x) + e^x - 2)/(a
^3 + 3*a^2*b + 3*a*b^2 + b^3) - 1/4*(3*a^5*(e^(-x) + e^x)^4 - 9*a^3*b^2*(e^(-x) + e^x)^4 + 9*a*b^4*(e^(-x) + e
^x)^4 - 5*a^4*b*(e^(-x) + e^x)^3 + 14*a^2*b^3*(e^(-x) + e^x)^3 - 9*b^5*(e^(-x) + e^x)^3 - 8*a^5*(e^(-x) + e^x)
^2 + 32*a^3*b^2*(e^(-x) + e^x)^2 - 48*a*b^4*(e^(-x) + e^x)^2 + 12*a^4*b*(e^(-x) + e^x) - 40*a^2*b^3*(e^(-x) +
e^x) + 28*b^5*(e^(-x) + e^x) - 16*a^3*b^2 + 64*a*b^4)/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*((e^(-x) + e^x)^2 -
4)^2)