3.207 \(\int \frac{\text{sech}^4(x)}{(a+b \sinh (x))^2} \, dx\)

Optimal. Leaf size=144 \[ -\frac{10 a b^4 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac{b \text{sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{\text{sech}^3(x) \left (\left (a^2-4 b^2\right ) \sinh (x)+5 a b\right )}{3 \left (a^2+b^2\right )^2}+\frac{\text{sech}(x) \left (\left (9 a^2 b^2+2 a^4-8 b^4\right ) \sinh (x)+15 a b^3\right )}{3 \left (a^2+b^2\right )^3} \]

[Out]

(-10*a*b^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - (b*Sech[x]^3)/((a^2 + b^2)*(a + b*S
inh[x])) + (Sech[x]^3*(5*a*b + (a^2 - 4*b^2)*Sinh[x]))/(3*(a^2 + b^2)^2) + (Sech[x]*(15*a*b^3 + (2*a^4 + 9*a^2
*b^2 - 8*b^4)*Sinh[x]))/(3*(a^2 + b^2)^3)

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Rubi [A]  time = 0.310433, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.462, Rules used = {2694, 2866, 12, 2660, 618, 206} \[ -\frac{10 a b^4 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac{b \text{sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{\text{sech}^3(x) \left (\left (a^2-4 b^2\right ) \sinh (x)+5 a b\right )}{3 \left (a^2+b^2\right )^2}+\frac{\text{sech}(x) \left (\left (9 a^2 b^2+2 a^4-8 b^4\right ) \sinh (x)+15 a b^3\right )}{3 \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[Sech[x]^4/(a + b*Sinh[x])^2,x]

[Out]

(-10*a*b^4*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]])/(a^2 + b^2)^(7/2) - (b*Sech[x]^3)/((a^2 + b^2)*(a + b*S
inh[x])) + (Sech[x]^3*(5*a*b + (a^2 - 4*b^2)*Sinh[x]))/(3*(a^2 + b^2)^2) + (Sech[x]*(15*a*b^3 + (2*a^4 + 9*a^2
*b^2 - 8*b^4)*Sinh[x]))/(3*(a^2 + b^2)^3)

Rule 2694

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> -Simp[(b*(g
*Cos[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1))/(f*g*(a^2 - b^2)*(m + 1)), x] + Dist[1/((a^2 - b^2)*(m +
1)), Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x], x] /; F
reeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -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 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{\text{sech}^4(x)}{(a+b \sinh (x))^2} \, dx &=-\frac{b \text{sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}-\frac{\int \frac{\text{sech}^4(x) (-a+4 b \sinh (x))}{a+b \sinh (x)} \, dx}{a^2+b^2}\\ &=-\frac{b \text{sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{\text{sech}^3(x) \left (5 a b+\left (a^2-4 b^2\right ) \sinh (x)\right )}{3 \left (a^2+b^2\right )^2}+\frac{\int \frac{\text{sech}^2(x) \left (a \left (2 a^2+7 b^2\right )+2 b \left (a^2-4 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)} \, dx}{3 \left (a^2+b^2\right )^2}\\ &=-\frac{b \text{sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{\text{sech}^3(x) \left (5 a b+\left (a^2-4 b^2\right ) \sinh (x)\right )}{3 \left (a^2+b^2\right )^2}+\frac{\text{sech}(x) \left (15 a b^3+\left (2 a^4+9 a^2 b^2-8 b^4\right ) \sinh (x)\right )}{3 \left (a^2+b^2\right )^3}-\frac{\int -\frac{15 a b^4}{a+b \sinh (x)} \, dx}{3 \left (a^2+b^2\right )^3}\\ &=-\frac{b \text{sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{\text{sech}^3(x) \left (5 a b+\left (a^2-4 b^2\right ) \sinh (x)\right )}{3 \left (a^2+b^2\right )^2}+\frac{\text{sech}(x) \left (15 a b^3+\left (2 a^4+9 a^2 b^2-8 b^4\right ) \sinh (x)\right )}{3 \left (a^2+b^2\right )^3}+\frac{\left (5 a b^4\right ) \int \frac{1}{a+b \sinh (x)} \, dx}{\left (a^2+b^2\right )^3}\\ &=-\frac{b \text{sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{\text{sech}^3(x) \left (5 a b+\left (a^2-4 b^2\right ) \sinh (x)\right )}{3 \left (a^2+b^2\right )^2}+\frac{\text{sech}(x) \left (15 a b^3+\left (2 a^4+9 a^2 b^2-8 b^4\right ) \sinh (x)\right )}{3 \left (a^2+b^2\right )^3}+\frac{\left (10 a b^4\right ) \operatorname{Subst}\left (\int \frac{1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac{x}{2}\right )\right )}{\left (a^2+b^2\right )^3}\\ &=-\frac{b \text{sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{\text{sech}^3(x) \left (5 a b+\left (a^2-4 b^2\right ) \sinh (x)\right )}{3 \left (a^2+b^2\right )^2}+\frac{\text{sech}(x) \left (15 a b^3+\left (2 a^4+9 a^2 b^2-8 b^4\right ) \sinh (x)\right )}{3 \left (a^2+b^2\right )^3}-\frac{\left (20 a b^4\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 )}{\left (a^2+b^2\right )^3}\\ &=-\frac{10 a b^4 \tanh ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{a^2+b^2}}\right )}{\left (a^2+b^2\right )^{7/2}}-\frac{b \text{sech}^3(x)}{\left (a^2+b^2\right ) (a+b \sinh (x))}+\frac{\text{sech}^3(x) \left (5 a b+\left (a^2-4 b^2\right ) \sinh (x)\right )}{3 \left (a^2+b^2\right )^2}+\frac{\text{sech}(x) \left (15 a b^3+\left (2 a^4+9 a^2 b^2-8 b^4\right ) \sinh (x)\right )}{3 \left (a^2+b^2\right )^3}\\ \end{align*}

Mathematica [A]  time = 0.430103, size = 137, normalized size = 0.95 \[ \frac{\left (9 a^2 b^2+2 a^4-5 b^4\right ) \tanh (x)+\frac{30 a b^4 \tan ^{-1}\left (\frac{b-a \tanh \left (\frac{x}{2}\right )}{\sqrt{-a^2-b^2}}\right )}{\sqrt{-a^2-b^2}}+\left (a^2+b^2\right ) \text{sech}^3(x) \left (\left (a^2-b^2\right ) \sinh (x)+2 a b\right )+12 a b^3 \text{sech}(x)-\frac{3 b^5 \cosh (x)}{a+b \sinh (x)}}{3 \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Sech[x]^4/(a + b*Sinh[x])^2,x]

[Out]

((30*a*b^4*ArcTan[(b - a*Tanh[x/2])/Sqrt[-a^2 - b^2]])/Sqrt[-a^2 - b^2] + 12*a*b^3*Sech[x] - (3*b^5*Cosh[x])/(
a + b*Sinh[x]) + (a^2 + b^2)*Sech[x]^3*(2*a*b + (a^2 - b^2)*Sinh[x]) + (2*a^4 + 9*a^2*b^2 - 5*b^4)*Tanh[x])/(3
*(a^2 + b^2)^3)

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Maple [A]  time = 0.072, size = 266, normalized size = 1.9 \begin{align*} -2\,{\frac{ \left ( -{a}^{4}-3\,{a}^{2}{b}^{2}+2\,{b}^{4} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{5}+ \left ( -2\,{a}^{3}b-6\,a{b}^{3} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{4}+ \left ( -2/3\,{a}^{4}-6\,{a}^{2}{b}^{2}+8/3\,{b}^{4} \right ) \left ( \tanh \left ( x/2 \right ) \right ) ^{3}-8\,a{b}^{3} \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+ \left ( -{a}^{4}-3\,{a}^{2}{b}^{2}+2\,{b}^{4} \right ) \tanh \left ( x/2 \right ) -2/3\,{a}^{3}b-14/3\,a{b}^{3}}{ \left ({a}^{2}+{b}^{2} \right ) \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) \left ( \left ( \tanh \left ( x/2 \right ) \right ) ^{2}+1 \right ) ^{3}}}-2\,{\frac{{b}^{4}}{ \left ({a}^{2}+{b}^{2} \right ) \left ({a}^{4}+2\,{a}^{2}{b}^{2}+{b}^{4} \right ) } \left ({\frac{1}{a \left ( \tanh \left ( x/2 \right ) \right ) ^{2}-2\,\tanh \left ( x/2 \right ) b-a} \left ( -{\frac{{b}^{2}\tanh \left ( x/2 \right ) }{a}}-b \right ) }-5\,{\frac{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 ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sech(x)^4/(a+b*sinh(x))^2,x)

[Out]

-2/(a^2+b^2)/(a^4+2*a^2*b^2+b^4)*((-a^4-3*a^2*b^2+2*b^4)*tanh(1/2*x)^5+(-2*a^3*b-6*a*b^3)*tanh(1/2*x)^4+(-2/3*
a^4-6*a^2*b^2+8/3*b^4)*tanh(1/2*x)^3-8*a*b^3*tanh(1/2*x)^2+(-a^4-3*a^2*b^2+2*b^4)*tanh(1/2*x)-2/3*a^3*b-14/3*a
*b^3)/(tanh(1/2*x)^2+1)^3-2*b^4/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*((-b^2/a*tanh(1/2*x)-b)/(a*tanh(1/2*x)^2-2*tanh(
1/2*x)*b-a)-5*a/(a^2+b^2)^(1/2)*arctanh(1/2*(2*a*tanh(1/2*x)-2*b)/(a^2+b^2)^(1/2)))

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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(sech(x)^4/(a+b*sinh(x))^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.45743, size = 7135, normalized size = 49.55 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^4/(a+b*sinh(x))^2,x, algorithm="fricas")

[Out]

-1/3*(30*(a^3*b^4 + a*b^6)*cosh(x)^7 + 30*(a^3*b^4 + a*b^6)*sinh(x)^7 + 4*a^6*b + 22*a^4*b^3 + 2*a^2*b^5 - 16*
b^7 + 30*(a^4*b^3 + a^2*b^5)*cosh(x)^6 + 30*(a^4*b^3 + a^2*b^5 + 7*(a^3*b^4 + a*b^6)*cosh(x))*sinh(x)^6 - 10*(
2*a^5*b^2 - 5*a^3*b^4 - 7*a*b^6)*cosh(x)^5 - 10*(2*a^5*b^2 - 5*a^3*b^4 - 7*a*b^6 - 63*(a^3*b^4 + a*b^6)*cosh(x
)^2 - 18*(a^4*b^3 + a^2*b^5)*cosh(x))*sinh(x)^5 + 10*(2*a^6*b + 13*a^4*b^3 + 11*a^2*b^5)*cosh(x)^4 + 10*(2*a^6
*b + 13*a^4*b^3 + 11*a^2*b^5 + 105*(a^3*b^4 + a*b^6)*cosh(x)^3 + 45*(a^4*b^3 + a^2*b^5)*cosh(x)^2 - 5*(2*a^5*b
^2 - 5*a^3*b^4 - 7*a*b^6)*cosh(x))*sinh(x)^4 - 2*(12*a^7 + 56*a^5*b^2 + 31*a^3*b^4 - 13*a*b^6)*cosh(x)^3 - 2*(
12*a^7 + 56*a^5*b^2 + 31*a^3*b^4 - 13*a*b^6 - 525*(a^3*b^4 + a*b^6)*cosh(x)^4 - 300*(a^4*b^3 + a^2*b^5)*cosh(x
)^3 + 50*(2*a^5*b^2 - 5*a^3*b^4 - 7*a*b^6)*cosh(x)^2 - 20*(2*a^6*b + 13*a^4*b^3 + 11*a^2*b^5)*cosh(x))*sinh(x)
^3 + 2*(4*a^6*b + 37*a^4*b^3 + 17*a^2*b^5 - 16*b^7)*cosh(x)^2 + 2*(4*a^6*b + 37*a^4*b^3 + 17*a^2*b^5 - 16*b^7
+ 315*(a^3*b^4 + a*b^6)*cosh(x)^5 + 225*(a^4*b^3 + a^2*b^5)*cosh(x)^4 - 50*(2*a^5*b^2 - 5*a^3*b^4 - 7*a*b^6)*c
osh(x)^3 + 30*(2*a^6*b + 13*a^4*b^3 + 11*a^2*b^5)*cosh(x)^2 - 3*(12*a^7 + 56*a^5*b^2 + 31*a^3*b^4 - 13*a*b^6)*
cosh(x))*sinh(x)^2 + 15*(a*b^5*cosh(x)^8 + a*b^5*sinh(x)^8 + 2*a^2*b^4*cosh(x)^7 + 2*a*b^5*cosh(x)^6 + 6*a^2*b
^4*cosh(x)^5 + 6*a^2*b^4*cosh(x)^3 - 2*a*b^5*cosh(x)^2 + 2*(4*a*b^5*cosh(x) + a^2*b^4)*sinh(x)^7 + 2*a^2*b^4*c
osh(x) + 2*(14*a*b^5*cosh(x)^2 + 7*a^2*b^4*cosh(x) + a*b^5)*sinh(x)^6 - a*b^5 + 2*(28*a*b^5*cosh(x)^3 + 21*a^2
*b^4*cosh(x)^2 + 6*a*b^5*cosh(x) + 3*a^2*b^4)*sinh(x)^5 + 10*(7*a*b^5*cosh(x)^4 + 7*a^2*b^4*cosh(x)^3 + 3*a*b^
5*cosh(x)^2 + 3*a^2*b^4*cosh(x))*sinh(x)^4 + 2*(28*a*b^5*cosh(x)^5 + 35*a^2*b^4*cosh(x)^4 + 20*a*b^5*cosh(x)^3
+ 30*a^2*b^4*cosh(x)^2 + 3*a^2*b^4)*sinh(x)^3 + 2*(14*a*b^5*cosh(x)^6 + 21*a^2*b^4*cosh(x)^5 + 15*a*b^5*cosh(
x)^4 + 30*a^2*b^4*cosh(x)^3 + 9*a^2*b^4*cosh(x) - a*b^5)*sinh(x)^2 + 2*(4*a*b^5*cosh(x)^7 + 7*a^2*b^4*cosh(x)^
6 + 6*a*b^5*cosh(x)^5 + 15*a^2*b^4*cosh(x)^4 + 9*a^2*b^4*cosh(x)^2 - 2*a*b^5*cosh(x) + a^2*b^4)*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)*si
nh(x) - b)) - 2*(4*a^7 + 22*a^5*b^2 + 17*a^3*b^4 - a*b^6)*cosh(x) - 2*(4*a^7 + 22*a^5*b^2 + 17*a^3*b^4 - a*b^6
- 105*(a^3*b^4 + a*b^6)*cosh(x)^6 - 90*(a^4*b^3 + a^2*b^5)*cosh(x)^5 + 25*(2*a^5*b^2 - 5*a^3*b^4 - 7*a*b^6)*c
osh(x)^4 - 20*(2*a^6*b + 13*a^4*b^3 + 11*a^2*b^5)*cosh(x)^3 + 3*(12*a^7 + 56*a^5*b^2 + 31*a^3*b^4 - 13*a*b^6)*
cosh(x)^2 - 2*(4*a^6*b + 37*a^4*b^3 + 17*a^2*b^5 - 16*b^7)*cosh(x))*sinh(x))/(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 +
4*a^2*b^7 + b^9 - (a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^8 - (a^8*b + 4*a^6*b^3 + 6*a^4*b^5
+ 4*a^2*b^7 + b^9)*sinh(x)^8 - 2*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^7 - 2*(a^9 + 4*a^7
*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8 + 4*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x))*sinh(x)^7
- 2*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^6 - 2*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7
+ b^9 + 14*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^2 + 7*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a
^3*b^6 + a*b^8)*cosh(x))*sinh(x)^6 - 6*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^5 - 2*(3*a^9
+ 12*a^7*b^2 + 18*a^5*b^4 + 12*a^3*b^6 + 3*a*b^8 + 28*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x
)^3 + 21*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^2 + 6*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^
2*b^7 + b^9)*cosh(x))*sinh(x)^5 - 10*(7*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^4 + 7*(a^9 +
4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^3 + 3*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*co
sh(x)^2 + 3*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x))*sinh(x)^4 - 6*(a^9 + 4*a^7*b^2 + 6*a^5*
b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^3 - 2*(3*a^9 + 12*a^7*b^2 + 18*a^5*b^4 + 12*a^3*b^6 + 3*a*b^8 + 28*(a^8*b + 4
*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^5 + 35*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(
x)^4 + 20*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^3 + 30*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^
3*b^6 + a*b^8)*cosh(x)^2)*sinh(x)^3 + 2*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^2 + 2*(a^8*b
+ 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9 - 14*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x)^6 -
21*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^5 - 15*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7
+ b^9)*cosh(x)^4 - 30*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^3 - 9*(a^9 + 4*a^7*b^2 + 6*a^
5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x))*sinh(x)^2 - 2*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x) -
2*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8 + 4*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(
x)^7 + 7*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^6 + 6*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^
2*b^7 + b^9)*cosh(x)^5 + 15*(a^9 + 4*a^7*b^2 + 6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^4 + 9*(a^9 + 4*a^7*b^2 +
6*a^5*b^4 + 4*a^3*b^6 + a*b^8)*cosh(x)^2 - 2*(a^8*b + 4*a^6*b^3 + 6*a^4*b^5 + 4*a^2*b^7 + b^9)*cosh(x))*sinh(
x))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\operatorname{sech}^{4}{\left (x \right )}}{\left (a + b \sinh{\left (x \right )}\right )^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)**4/(a+b*sinh(x))**2,x)

[Out]

Integral(sech(x)**4/(a + b*sinh(x))**2, x)

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Giac [B]  time = 1.17021, size = 387, normalized size = 2.69 \begin{align*} \frac{5 \, a b^{4} \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 )}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \sqrt{a^{2} + b^{2}}} + \frac{2 \,{\left (a b^{4} e^{x} - b^{5}\right )}}{{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}{\left (b e^{\left (2 \, x\right )} + 2 \, a e^{x} - b\right )}} + \frac{2 \,{\left (12 \, a b^{3} e^{\left (5 \, x\right )} - 9 \, a^{2} b^{2} e^{\left (4 \, x\right )} + 3 \, b^{4} e^{\left (4 \, x\right )} + 8 \, a^{3} b e^{\left (3 \, x\right )} + 32 \, a b^{3} e^{\left (3 \, x\right )} - 6 \, a^{4} e^{\left (2 \, x\right )} - 18 \, a^{2} b^{2} e^{\left (2 \, x\right )} + 12 \, b^{4} e^{\left (2 \, x\right )} + 12 \, a b^{3} e^{x} - 2 \, a^{4} - 9 \, a^{2} b^{2} + 5 \, b^{4}\right )}}{3 \,{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}{\left (e^{\left (2 \, x\right )} + 1\right )}^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sech(x)^4/(a+b*sinh(x))^2,x, algorithm="giac")

[Out]

5*a*b^4*log(abs(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)))/((a^6 + 3*a^4*b^2 +
3*a^2*b^4 + b^6)*sqrt(a^2 + b^2)) + 2*(a*b^4*e^x - b^5)/((a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*(b*e^(2*x) + 2*a
*e^x - b)) + 2/3*(12*a*b^3*e^(5*x) - 9*a^2*b^2*e^(4*x) + 3*b^4*e^(4*x) + 8*a^3*b*e^(3*x) + 32*a*b^3*e^(3*x) -
6*a^4*e^(2*x) - 18*a^2*b^2*e^(2*x) + 12*b^4*e^(2*x) + 12*a*b^3*e^x - 2*a^4 - 9*a^2*b^2 + 5*b^4)/((a^6 + 3*a^4*
b^2 + 3*a^2*b^4 + b^6)*(e^(2*x) + 1)^3)