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

Optimal. Leaf size=135 \[ \frac{b^5 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac{a \left (10 a^2 b^2+3 a^4+15 b^4\right ) \tan ^{-1}(\sinh (x))}{8 \left (a^2+b^2\right )^3}-\frac{b^5 \log (\cosh (x))}{\left (a^2+b^2\right )^3}+\frac{\text{sech}^4(x) (a \sinh (x)+b)}{4 \left (a^2+b^2\right )}+\frac{\text{sech}^2(x) \left (a \left (3 a^2+7 b^2\right ) \sinh (x)+4 b^3\right )}{8 \left (a^2+b^2\right )^2} \]

[Out]

(a*(3*a^4 + 10*a^2*b^2 + 15*b^4)*ArcTan[Sinh[x]])/(8*(a^2 + b^2)^3) - (b^5*Log[Cosh[x]])/(a^2 + b^2)^3 + (b^5*
Log[a + b*Sinh[x]])/(a^2 + b^2)^3 + (Sech[x]^4*(b + a*Sinh[x]))/(4*(a^2 + b^2)) + (Sech[x]^2*(4*b^3 + a*(3*a^2
+ 7*b^2)*Sinh[x]))/(8*(a^2 + b^2)^2)

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Rubi [A]  time = 0.19725, antiderivative size = 135, 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 = {2668, 741, 823, 801, 635, 203, 260} \[ \frac{b^5 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac{a \left (10 a^2 b^2+3 a^4+15 b^4\right ) \tan ^{-1}(\sinh (x))}{8 \left (a^2+b^2\right )^3}-\frac{b^5 \log (\cosh (x))}{\left (a^2+b^2\right )^3}+\frac{\text{sech}^4(x) (a \sinh (x)+b)}{4 \left (a^2+b^2\right )}+\frac{\text{sech}^2(x) \left (a \left (3 a^2+7 b^2\right ) \sinh (x)+4 b^3\right )}{8 \left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^5/(a + b*Sinh[x]),x]

[Out]

(a*(3*a^4 + 10*a^2*b^2 + 15*b^4)*ArcTan[Sinh[x]])/(8*(a^2 + b^2)^3) - (b^5*Log[Cosh[x]])/(a^2 + b^2)^3 + (b^5*
Log[a + b*Sinh[x]])/(a^2 + b^2)^3 + (Sech[x]^4*(b + a*Sinh[x]))/(4*(a^2 + b^2)) + (Sech[x]^2*(4*b^3 + a*(3*a^2
+ 7*b^2)*Sinh[x]))/(8*(a^2 + b^2)^2)

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]

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

Rule 203

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTan[(Rt[b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[b, 2]), x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

\begin{align*} \int \frac{\text{sech}^5(x)}{a+b \sinh (x)} \, dx &=-\left (b^5 \operatorname{Subst}\left (\int \frac{1}{(a+x) \left (-b^2-x^2\right )^3} \, dx,x,b \sinh (x)\right )\right )\\ &=\frac{\text{sech}^4(x) (b+a \sinh (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 \sinh (x)\right )}{4 \left (a^2+b^2\right )}\\ &=\frac{\text{sech}^4(x) (b+a \sinh (x))}{4 \left (a^2+b^2\right )}+\frac{\text{sech}^2(x) \left (4 b^3+a \left (3 a^2+7 b^2\right ) \sinh (x)\right )}{8 \left (a^2+b^2\right )^2}+\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 \sinh (x)\right )}{8 \left (a^2+b^2\right )^2}\\ &=\frac{\text{sech}^4(x) (b+a \sinh (x))}{4 \left (a^2+b^2\right )}+\frac{\text{sech}^2(x) \left (4 b^3+a \left (3 a^2+7 b^2\right ) \sinh (x)\right )}{8 \left (a^2+b^2\right )^2}+\frac{b \operatorname{Subst}\left (\int \left (\frac{8 b^4}{\left (a^2+b^2\right ) (a+x)}+\frac{3 a^5+10 a^3 b^2+15 a b^4-8 b^4 x}{\left (a^2+b^2\right ) \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (x)\right )}{8 \left (a^2+b^2\right )^2}\\ &=\frac{b^5 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac{\text{sech}^4(x) (b+a \sinh (x))}{4 \left (a^2+b^2\right )}+\frac{\text{sech}^2(x) \left (4 b^3+a \left (3 a^2+7 b^2\right ) \sinh (x)\right )}{8 \left (a^2+b^2\right )^2}+\frac{b \operatorname{Subst}\left (\int \frac{3 a^5+10 a^3 b^2+15 a b^4-8 b^4 x}{b^2+x^2} \, dx,x,b \sinh (x)\right )}{8 \left (a^2+b^2\right )^3}\\ &=\frac{b^5 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac{\text{sech}^4(x) (b+a \sinh (x))}{4 \left (a^2+b^2\right )}+\frac{\text{sech}^2(x) \left (4 b^3+a \left (3 a^2+7 b^2\right ) \sinh (x)\right )}{8 \left (a^2+b^2\right )^2}-\frac{b^5 \operatorname{Subst}\left (\int \frac{x}{b^2+x^2} \, dx,x,b \sinh (x)\right )}{\left (a^2+b^2\right )^3}+\frac{\left (a b \left (3 a^4+10 a^2 b^2+15 b^4\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2+x^2} \, dx,x,b \sinh (x)\right )}{8 \left (a^2+b^2\right )^3}\\ &=\frac{a \left (3 a^4+10 a^2 b^2+15 b^4\right ) \tan ^{-1}(\sinh (x))}{8 \left (a^2+b^2\right )^3}-\frac{b^5 \log (\cosh (x))}{\left (a^2+b^2\right )^3}+\frac{b^5 \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac{\text{sech}^4(x) (b+a \sinh (x))}{4 \left (a^2+b^2\right )}+\frac{\text{sech}^2(x) \left (4 b^3+a \left (3 a^2+7 b^2\right ) \sinh (x)\right )}{8 \left (a^2+b^2\right )^2}\\ \end{align*}

Mathematica [A]  time = 0.225598, size = 135, normalized size = 1. \[ \frac{4 b^3 \left (a^2+b^2\right ) \text{sech}^2(x)+2 b \left (a^2+b^2\right )^2 \text{sech}^4(x)+\left (20 a^3 b^2+6 a^5+30 a b^4\right ) \tan ^{-1}\left (\tanh \left (\frac{x}{2}\right )\right )+2 a \left (a^2+b^2\right )^2 \tanh (x) \text{sech}^3(x)+a \left (10 a^2 b^2+3 a^4+7 b^4\right ) \tanh (x) \text{sech}(x)+8 b^5 (\log (a+b \sinh (x))-\log (\cosh (x)))}{8 \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((6*a^5 + 20*a^3*b^2 + 30*a*b^4)*ArcTan[Tanh[x/2]] + 8*b^5*(-Log[Cosh[x]] + Log[a + b*Sinh[x]]) + 4*b^3*(a^2 +
b^2)*Sech[x]^2 + 2*b*(a^2 + b^2)^2*Sech[x]^4 + a*(3*a^4 + 10*a^2*b^2 + 7*b^4)*Sech[x]*Tanh[x] + 2*a*(a^2 + b^
2)^2*Sech[x]^3*Tanh[x])/(8*(a^2 + b^2)^3)

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Maple [B]  time = 0.043, size = 1140, normalized size = 8.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^5/(a+b*sinh(x)),x)

[Out]

-4/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^2*b^5+5/4/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh
(1/2*x)^2+1)^4*tanh(1/2*x)*a^5+5/2/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*arctan(tanh(1/2*x))*a^3*b^2+15/4/(a^6+3*a^4*b
^2+3*a^2*b^4+b^6)*arctan(tanh(1/2*x))*a*b^4-5/4/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^
7*a^5-4/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^6*b^5+3/4/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/
(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^5*a^5-4/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^4*b^5-3/
4/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^3*a^5+b^5/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*ln(a*t
anh(1/2*x)^2-2*tanh(1/2*x)*b-a)-1/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)*b^5*ln(tanh(1/2*x)^2+1)+3/4/(a^6+3*a^4*b^2+3*a
^2*b^4+b^6)*arctan(tanh(1/2*x))*a^5-2/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^2*a^4*b-6/
(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^2*a^2*b^3+7/2/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tan
h(1/2*x)^2+1)^4*tanh(1/2*x)*a^3*b^2+9/4/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)*a*b^4-7/
2/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^7*a^3*b^2-9/4/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(t
anh(1/2*x)^2+1)^4*tanh(1/2*x)^7*a*b^4+1/2/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^5*a^3*
b^2-1/4/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^5*a*b^4-4/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/
(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^4*a^2*b^3-1/2/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^3*
a^3*b^2+1/4/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^3*a*b^4-2/(a^6+3*a^4*b^2+3*a^2*b^4+b
^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^6*a^4*b-6/(a^6+3*a^4*b^2+3*a^2*b^4+b^6)/(tanh(1/2*x)^2+1)^4*tanh(1/2*x)^6*
a^2*b^3

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Maxima [B]  time = 1.86502, size = 466, normalized size = 3.45 \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{b^{5} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac{{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )} \arctan \left (e^{\left (-x\right )}\right )}{4 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\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(sech(x)^5/(a+b*sinh(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) - b^5*log(e^(-2*x) + 1)/(a^6 + 3*a^4
*b^2 + 3*a^2*b^4 + b^6) - 1/4*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*arctan(e^(-x))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^
6) + 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 = 2.85386, size = 6700, normalized size = 49.63 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4*((3*a^5 + 10*a^3*b^2 + 7*a*b^4)*cosh(x)^7 + (3*a^5 + 10*a^3*b^2 + 7*a*b^4)*sinh(x)^7 + 8*(a^2*b^3 + b^5)*c
osh(x)^6 + (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 + (11*a^5 + 26*a^3*b^2 + 1
5*a*b^4)*cosh(x)^5 + (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 + 16*(a^4*b + 3*a^2*b^3 + 2*b^5)*cosh(x)^4 + (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 - (11*a^5 + 26*a^3*b^2 + 15*a*b^4)*cosh(x)^3 - (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 + 8*(a^2*b^3 + b^5)*cosh(x)^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 + ((3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^8 + 8*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)*sinh(x)^7 + (3*a^5
+ 10*a^3*b^2 + 15*a*b^4)*sinh(x)^8 + 4*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^6 + 4*(3*a^5 + 10*a^3*b^2 + 15
*a*b^4 + 7*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^6 + 8*(7*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)
^3 + 3*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x)^5 + 3*a^5 + 10*a^3*b^2 + 15*a*b^4 + 6*(3*a^5 + 10*a^3*
b^2 + 15*a*b^4)*cosh(x)^4 + 2*(9*a^5 + 30*a^3*b^2 + 45*a*b^4 + 35*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^4 +
30*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^4 + 8*(7*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^5 + 10*
(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^3 + 3*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x)^3 + 4*(3*a^5 +
10*a^3*b^2 + 15*a*b^4)*cosh(x)^2 + 4*(7*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^6 + 3*a^5 + 10*a^3*b^2 + 15*a*
b^4 + 15*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^4 + 9*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^2)*sinh(x)^2 +
8*((3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^7 + 3*(3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x)^5 + 3*(3*a^5 + 10*a^3
*b^2 + 15*a*b^4)*cosh(x)^3 + (3*a^5 + 10*a^3*b^2 + 15*a*b^4)*cosh(x))*sinh(x))*arctan(cosh(x) + sinh(x)) - (3*
a^5 + 10*a^3*b^2 + 7*a*b^4)*cosh(x) + 4*(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*co
sh(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*sinh(x) + a)/(cos
h(x) - sinh(x))) - 4*(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*cosh(x)/(cosh(x) - sinh(x))) + (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]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{5}{\left (x \right )}}{a + b \sinh{\left (x \right )}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.23793, size = 498, normalized size = 3.69 \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{b^{5} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac{{\left (\pi + 2 \, \arctan \left (\frac{1}{2} \,{\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )}{\left (3 \, a^{5} + 10 \, a^{3} b^{2} + 15 \, a b^{4}\right )}}{16 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\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(sech(x)^5/(a+b*sinh(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) - 1/2*b^5*log((e^(-x) - e^x)^2 + 4
)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/16*(pi + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))*(3*a^5 + 10*a^3*b^2 + 1
5*a*b^4)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 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)