3.237 \(\int \frac {\tanh ^3(x)}{(a+b \sinh (x))^2} \, dx\)

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

[Out]

a*b*(3*a^2-b^2)*arctan(sinh(x))/(a^2+b^2)^3+a^2*(a^2-3*b^2)*ln(cosh(x))/(a^2+b^2)^3-a^2*(a^2-3*b^2)*ln(a+b*sin
h(x))/(a^2+b^2)^3+a^3/(a^2+b^2)^2/(a+b*sinh(x))+1/2*sech(x)^2*(a^2-b^2-2*a*b*sinh(x))/(a^2+b^2)^2

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Rubi [A]  time = 0.36, antiderivative size = 135, normalized size of antiderivative = 1.00, 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 = {2721, 1647, 1629, 635, 203, 260} \[ \frac {a^3}{\left (a^2+b^2\right )^2 (a+b \sinh (x))}-\frac {a^2 \left (a^2-3 b^2\right ) \log (a+b \sinh (x))}{\left (a^2+b^2\right )^3}+\frac {a b \left (3 a^2-b^2\right ) \tan ^{-1}(\sinh (x))}{\left (a^2+b^2\right )^3}+\frac {a^2 \left (a^2-3 b^2\right ) \log (\cosh (x))}{\left (a^2+b^2\right )^3}+\frac {\text {sech}^2(x) \left (a^2-2 a b \sinh (x)-b^2\right )}{2 \left (a^2+b^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[Tanh[x]^3/(a + b*Sinh[x])^2,x]

[Out]

(a*b*(3*a^2 - b^2)*ArcTan[Sinh[x]])/(a^2 + b^2)^3 + (a^2*(a^2 - 3*b^2)*Log[Cosh[x]])/(a^2 + b^2)^3 - (a^2*(a^2
- 3*b^2)*Log[a + b*Sinh[x]])/(a^2 + b^2)^3 + a^3/((a^2 + b^2)^2*(a + b*Sinh[x])) + (Sech[x]^2*(a^2 - b^2 - 2*
a*b*Sinh[x]))/(2*(a^2 + b^2)^2)

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]

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 1629

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1647

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[((a*g - c*f*x)*(a + c*x^2)^(p + 1))/(2*a*c*(p + 1))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rule 2721

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(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] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rubi steps

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

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Mathematica [C]  time = 0.70, size = 150, normalized size = 1.11 \[ \frac {\left (a^4-b^4\right ) \text {sech}^2(x)-2 a^2 \left (a^2-3 b^2\right ) \log (a+b \sinh (x))-2 a b \left (a^2+b^2\right ) \tan ^{-1}(\sinh (x))-2 a b \left (a^2+b^2\right ) \tanh (x) \text {sech}(x)+a^2 (a-i b) (a-3 i b) \log (-\sinh (x)+i)+a^2 (a+i b) (a+3 i b) \log (\sinh (x)+i)+\frac {2 a^3 \left (a^2+b^2\right )}{a+b \sinh (x)}}{2 \left (a^2+b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Integrate[Tanh[x]^3/(a + b*Sinh[x])^2,x]

[Out]

(-2*a*b*(a^2 + b^2)*ArcTan[Sinh[x]] + a^2*(a - I*b)*(a - (3*I)*b)*Log[I - Sinh[x]] + a^2*(a + I*b)*(a + (3*I)*
b)*Log[I + Sinh[x]] - 2*a^2*(a^2 - 3*b^2)*Log[a + b*Sinh[x]] + (a^4 - b^4)*Sech[x]^2 + (2*a^3*(a^2 + b^2))/(a
+ b*Sinh[x]) - 2*a*b*(a^2 + b^2)*Sech[x]*Tanh[x])/(2*(a^2 + b^2)^3)

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fricas [B]  time = 1.66, size = 2850, normalized size = 21.11 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^3/(a+b*sinh(x))^2,x, algorithm="fricas")

[Out]

-(2*(a^5 - a*b^4)*cosh(x)^5 + 2*(a^5 - a*b^4)*sinh(x)^5 - 2*(a^4*b + 2*a^2*b^3 + b^5)*cosh(x)^4 - 2*(a^4*b + 2
*a^2*b^3 + b^5 - 5*(a^5 - a*b^4)*cosh(x))*sinh(x)^4 + 8*(a^5 + a^3*b^2)*cosh(x)^3 + 4*(2*a^5 + 2*a^3*b^2 + 5*(
a^5 - a*b^4)*cosh(x)^2 - 2*(a^4*b + 2*a^2*b^3 + b^5)*cosh(x))*sinh(x)^3 + 2*(a^4*b + 2*a^2*b^3 + b^5)*cosh(x)^
2 + 2*(a^4*b + 2*a^2*b^3 + b^5 + 10*(a^5 - a*b^4)*cosh(x)^3 - 6*(a^4*b + 2*a^2*b^3 + b^5)*cosh(x)^2 + 12*(a^5
+ a^3*b^2)*cosh(x))*sinh(x)^2 + 2*((3*a^3*b^2 - a*b^4)*cosh(x)^6 + (3*a^3*b^2 - a*b^4)*sinh(x)^6 + 2*(3*a^4*b
- a^2*b^3)*cosh(x)^5 + 2*(3*a^4*b - a^2*b^3 + 3*(3*a^3*b^2 - a*b^4)*cosh(x))*sinh(x)^5 - 3*a^3*b^2 + a*b^4 + (
3*a^3*b^2 - a*b^4)*cosh(x)^4 + (3*a^3*b^2 - a*b^4 + 15*(3*a^3*b^2 - a*b^4)*cosh(x)^2 + 10*(3*a^4*b - a^2*b^3)*
cosh(x))*sinh(x)^4 + 4*(3*a^4*b - a^2*b^3)*cosh(x)^3 + 4*(3*a^4*b - a^2*b^3 + 5*(3*a^3*b^2 - a*b^4)*cosh(x)^3
+ 5*(3*a^4*b - a^2*b^3)*cosh(x)^2 + (3*a^3*b^2 - a*b^4)*cosh(x))*sinh(x)^3 - (3*a^3*b^2 - a*b^4)*cosh(x)^2 - (
3*a^3*b^2 - a*b^4 - 15*(3*a^3*b^2 - a*b^4)*cosh(x)^4 - 20*(3*a^4*b - a^2*b^3)*cosh(x)^3 - 6*(3*a^3*b^2 - a*b^4
)*cosh(x)^2 - 12*(3*a^4*b - a^2*b^3)*cosh(x))*sinh(x)^2 + 2*(3*a^4*b - a^2*b^3)*cosh(x) + 2*(3*(3*a^3*b^2 - a*
b^4)*cosh(x)^5 + 3*a^4*b - a^2*b^3 + 5*(3*a^4*b - a^2*b^3)*cosh(x)^4 + 2*(3*a^3*b^2 - a*b^4)*cosh(x)^3 + 6*(3*
a^4*b - a^2*b^3)*cosh(x)^2 - (3*a^3*b^2 - a*b^4)*cosh(x))*sinh(x))*arctan(cosh(x) + sinh(x)) + 2*(a^5 - a*b^4)
*cosh(x) - ((a^4*b - 3*a^2*b^3)*cosh(x)^6 + (a^4*b - 3*a^2*b^3)*sinh(x)^6 + 2*(a^5 - 3*a^3*b^2)*cosh(x)^5 + 2*
(a^5 - 3*a^3*b^2 + 3*(a^4*b - 3*a^2*b^3)*cosh(x))*sinh(x)^5 - a^4*b + 3*a^2*b^3 + (a^4*b - 3*a^2*b^3)*cosh(x)^
4 + (a^4*b - 3*a^2*b^3 + 15*(a^4*b - 3*a^2*b^3)*cosh(x)^2 + 10*(a^5 - 3*a^3*b^2)*cosh(x))*sinh(x)^4 + 4*(a^5 -
3*a^3*b^2)*cosh(x)^3 + 4*(a^5 - 3*a^3*b^2 + 5*(a^4*b - 3*a^2*b^3)*cosh(x)^3 + 5*(a^5 - 3*a^3*b^2)*cosh(x)^2 +
(a^4*b - 3*a^2*b^3)*cosh(x))*sinh(x)^3 - (a^4*b - 3*a^2*b^3)*cosh(x)^2 - (a^4*b - 3*a^2*b^3 - 15*(a^4*b - 3*a
^2*b^3)*cosh(x)^4 - 20*(a^5 - 3*a^3*b^2)*cosh(x)^3 - 6*(a^4*b - 3*a^2*b^3)*cosh(x)^2 - 12*(a^5 - 3*a^3*b^2)*co
sh(x))*sinh(x)^2 + 2*(a^5 - 3*a^3*b^2)*cosh(x) + 2*(3*(a^4*b - 3*a^2*b^3)*cosh(x)^5 + a^5 - 3*a^3*b^2 + 5*(a^5
- 3*a^3*b^2)*cosh(x)^4 + 2*(a^4*b - 3*a^2*b^3)*cosh(x)^3 + 6*(a^5 - 3*a^3*b^2)*cosh(x)^2 - (a^4*b - 3*a^2*b^3
)*cosh(x))*sinh(x))*log(2*(b*sinh(x) + a)/(cosh(x) - sinh(x))) + ((a^4*b - 3*a^2*b^3)*cosh(x)^6 + (a^4*b - 3*a
^2*b^3)*sinh(x)^6 + 2*(a^5 - 3*a^3*b^2)*cosh(x)^5 + 2*(a^5 - 3*a^3*b^2 + 3*(a^4*b - 3*a^2*b^3)*cosh(x))*sinh(x
)^5 - a^4*b + 3*a^2*b^3 + (a^4*b - 3*a^2*b^3)*cosh(x)^4 + (a^4*b - 3*a^2*b^3 + 15*(a^4*b - 3*a^2*b^3)*cosh(x)^
2 + 10*(a^5 - 3*a^3*b^2)*cosh(x))*sinh(x)^4 + 4*(a^5 - 3*a^3*b^2)*cosh(x)^3 + 4*(a^5 - 3*a^3*b^2 + 5*(a^4*b -
3*a^2*b^3)*cosh(x)^3 + 5*(a^5 - 3*a^3*b^2)*cosh(x)^2 + (a^4*b - 3*a^2*b^3)*cosh(x))*sinh(x)^3 - (a^4*b - 3*a^2
*b^3)*cosh(x)^2 - (a^4*b - 3*a^2*b^3 - 15*(a^4*b - 3*a^2*b^3)*cosh(x)^4 - 20*(a^5 - 3*a^3*b^2)*cosh(x)^3 - 6*(
a^4*b - 3*a^2*b^3)*cosh(x)^2 - 12*(a^5 - 3*a^3*b^2)*cosh(x))*sinh(x)^2 + 2*(a^5 - 3*a^3*b^2)*cosh(x) + 2*(3*(a
^4*b - 3*a^2*b^3)*cosh(x)^5 + a^5 - 3*a^3*b^2 + 5*(a^5 - 3*a^3*b^2)*cosh(x)^4 + 2*(a^4*b - 3*a^2*b^3)*cosh(x)^
3 + 6*(a^5 - 3*a^3*b^2)*cosh(x)^2 - (a^4*b - 3*a^2*b^3)*cosh(x))*sinh(x))*log(2*cosh(x)/(cosh(x) - sinh(x))) +
2*(a^5 - a*b^4 + 5*(a^5 - a*b^4)*cosh(x)^4 - 4*(a^4*b + 2*a^2*b^3 + b^5)*cosh(x)^3 + 12*(a^5 + a^3*b^2)*cosh(
x)^2 + 2*(a^4*b + 2*a^2*b^3 + b^5)*cosh(x))*sinh(x))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7 - (a^6*b + 3*a^4*b^3
+ 3*a^2*b^5 + b^7)*cosh(x)^6 - (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*sinh(x)^6 - 2*(a^7 + 3*a^5*b^2 + 3*a^3*b
^4 + a*b^6)*cosh(x)^5 - 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + 3*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(
x))*sinh(x)^5 - (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^4 - (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7 + 15*(a
^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^2 + 10*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x))*sinh(x)^4 -
4*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^3 - 4*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + 5*(a^6*b + 3*a^4*
b^3 + 3*a^2*b^5 + b^7)*cosh(x)^3 + 5*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^2 + (a^6*b + 3*a^4*b^3 + 3*
a^2*b^5 + b^7)*cosh(x))*sinh(x)^3 + (a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^2 + (a^6*b + 3*a^4*b^3 + 3*a
^2*b^5 + b^7 - 15*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^4 - 20*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*c
osh(x)^3 - 6*(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^2 - 12*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x
))*sinh(x)^2 - 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x) - 2*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6 + 3*(a
^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^5 + 5*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^4 + 2*(a^6*b +
3*a^4*b^3 + 3*a^2*b^5 + b^7)*cosh(x)^3 + 6*(a^7 + 3*a^5*b^2 + 3*a^3*b^4 + a*b^6)*cosh(x)^2 - (a^6*b + 3*a^4*b
^3 + 3*a^2*b^5 + b^7)*cosh(x))*sinh(x))

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giac [B]  time = 0.45, size = 307, normalized size = 2.27 \[ \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, x\right )} - 1\right )} e^{\left (-x\right )}\right )\right )} {\left (3 \, a^{3} b - a b^{3}\right )}}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} + \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \log \left ({\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4\right )}{2 \, {\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )}} - \frac {{\left (a^{4} b - 3 \, a^{2} b^{3}\right )} \log \left ({\left | -b {\left (e^{\left (-x\right )} - e^{x}\right )} + 2 \, a \right |}\right )}{a^{6} b + 3 \, a^{4} b^{3} + 3 \, a^{2} b^{5} + b^{7}} - \frac {2 \, {\left (a^{3} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} - a b^{2} {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + a^{2} b {\left (e^{\left (-x\right )} - e^{x}\right )} + b^{3} {\left (e^{\left (-x\right )} - e^{x}\right )} + 6 \, a^{3} - 2 \, a b^{2}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left (b {\left (e^{\left (-x\right )} - e^{x}\right )}^{3} - 2 \, a {\left (e^{\left (-x\right )} - e^{x}\right )}^{2} + 4 \, b {\left (e^{\left (-x\right )} - e^{x}\right )} - 8 \, a\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^3/(a+b*sinh(x))^2,x, algorithm="giac")

[Out]

1/2*(pi + 2*arctan(1/2*(e^(2*x) - 1)*e^(-x)))*(3*a^3*b - a*b^3)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + 1/2*(a^4
- 3*a^2*b^2)*log((e^(-x) - e^x)^2 + 4)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (a^4*b - 3*a^2*b^3)*log(abs(-b*(
e^(-x) - e^x) + 2*a))/(a^6*b + 3*a^4*b^3 + 3*a^2*b^5 + b^7) - 2*(a^3*(e^(-x) - e^x)^2 - a*b^2*(e^(-x) - e^x)^2
+ a^2*b*(e^(-x) - e^x) + b^3*(e^(-x) - e^x) + 6*a^3 - 2*a*b^2)/((a^4 + 2*a^2*b^2 + b^4)*(b*(e^(-x) - e^x)^3 -
2*a*(e^(-x) - e^x)^2 + 4*b*(e^(-x) - e^x) - 8*a))

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maple [B]  time = 0.10, size = 491, normalized size = 3.64 \[ \frac {2 a^{4} \tanh \left (\frac {x}{2}\right ) b}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right ) \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )}+\frac {2 a^{2} \tanh \left (\frac {x}{2}\right ) b^{3}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right ) \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )}-\frac {a^{4} \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right )}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {3 a^{2} \ln \left (a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )-2 \tanh \left (\frac {x}{2}\right ) b -a \right ) b^{2}}{\left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (a^{2}+b^{2}\right )}+\frac {2 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right ) a^{3} b}{\left (a^{2}+b^{2}\right )^{3} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {2 \left (\tanh ^{3}\left (\frac {x}{2}\right )\right ) a \,b^{3}}{\left (a^{2}+b^{2}\right )^{3} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}-\frac {2 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) a^{4}}{\left (a^{2}+b^{2}\right )^{3} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {2 \left (\tanh ^{2}\left (\frac {x}{2}\right )\right ) b^{4}}{\left (a^{2}+b^{2}\right )^{3} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}-\frac {2 \tanh \left (\frac {x}{2}\right ) a^{3} b}{\left (a^{2}+b^{2}\right )^{3} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}-\frac {2 \tanh \left (\frac {x}{2}\right ) a \,b^{3}}{\left (a^{2}+b^{2}\right )^{3} \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right )^{2}}+\frac {\ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) a^{4}}{\left (a^{2}+b^{2}\right )^{3}}-\frac {3 \ln \left (\tanh ^{2}\left (\frac {x}{2}\right )+1\right ) a^{2} b^{2}}{\left (a^{2}+b^{2}\right )^{3}}+\frac {6 \arctan \left (\tanh \left (\frac {x}{2}\right )\right ) a^{3} b}{\left (a^{2}+b^{2}\right )^{3}}-\frac {2 a \arctan \left (\tanh \left (\frac {x}{2}\right )\right ) b^{3}}{\left (a^{2}+b^{2}\right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^3/(a+b*sinh(x))^2,x)

[Out]

2*a^4/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*tanh(1/2*x)/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)*b+2*a^2/(a^4+2*a^2*b^2+b^4
)/(a^2+b^2)*tanh(1/2*x)/(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)*b^3-a^4/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*ln(a*tanh(1/
2*x)^2-2*tanh(1/2*x)*b-a)+3*a^2/(a^4+2*a^2*b^2+b^4)/(a^2+b^2)*ln(a*tanh(1/2*x)^2-2*tanh(1/2*x)*b-a)*b^2+2/(a^2
+b^2)^3/(tanh(1/2*x)^2+1)^2*tanh(1/2*x)^3*a^3*b+2/(a^2+b^2)^3/(tanh(1/2*x)^2+1)^2*tanh(1/2*x)^3*a*b^3-2/(a^2+b
^2)^3/(tanh(1/2*x)^2+1)^2*tanh(1/2*x)^2*a^4+2/(a^2+b^2)^3/(tanh(1/2*x)^2+1)^2*tanh(1/2*x)^2*b^4-2/(a^2+b^2)^3/
(tanh(1/2*x)^2+1)^2*tanh(1/2*x)*a^3*b-2/(a^2+b^2)^3/(tanh(1/2*x)^2+1)^2*tanh(1/2*x)*a*b^3+1/(a^2+b^2)^3*ln(tan
h(1/2*x)^2+1)*a^4-3/(a^2+b^2)^3*ln(tanh(1/2*x)^2+1)*a^2*b^2+6/(a^2+b^2)^3*arctan(tanh(1/2*x))*a^3*b-2/(a^2+b^2
)^3*a*arctan(tanh(1/2*x))*b^3

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maxima [B]  time = 0.47, size = 375, normalized size = 2.78 \[ -\frac {2 \, {\left (3 \, a^{3} b - a b^{3}\right )} \arctan \left (e^{\left (-x\right )}\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} - \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \log \left (-2 \, a e^{\left (-x\right )} + b e^{\left (-2 \, x\right )} - b\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {{\left (a^{4} - 3 \, a^{2} b^{2}\right )} \log \left (e^{\left (-2 \, x\right )} + 1\right )}{a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}} + \frac {2 \, {\left (4 \, a^{3} e^{\left (-3 \, x\right )} + {\left (a^{3} - a b^{2}\right )} e^{\left (-x\right )} - {\left (a^{2} b + b^{3}\right )} e^{\left (-2 \, x\right )} + {\left (a^{2} b + b^{3}\right )} e^{\left (-4 \, x\right )} + {\left (a^{3} - a b^{2}\right )} e^{\left (-5 \, x\right )}\right )}}{a^{4} b + 2 \, a^{2} b^{3} + b^{5} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-x\right )} + {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-2 \, x\right )} + 4 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-3 \, x\right )} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-4 \, x\right )} + 2 \, {\left (a^{5} + 2 \, a^{3} b^{2} + a b^{4}\right )} e^{\left (-5 \, x\right )} - {\left (a^{4} b + 2 \, a^{2} b^{3} + b^{5}\right )} e^{\left (-6 \, x\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)^3/(a+b*sinh(x))^2,x, algorithm="maxima")

[Out]

-2*(3*a^3*b - a*b^3)*arctan(e^(-x))/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) - (a^4 - 3*a^2*b^2)*log(-2*a*e^(-x) +
b*e^(-2*x) - b)/(a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6) + (a^4 - 3*a^2*b^2)*log(e^(-2*x) + 1)/(a^6 + 3*a^4*b^2 + 3
*a^2*b^4 + b^6) + 2*(4*a^3*e^(-3*x) + (a^3 - a*b^2)*e^(-x) - (a^2*b + b^3)*e^(-2*x) + (a^2*b + b^3)*e^(-4*x) +
(a^3 - a*b^2)*e^(-5*x))/(a^4*b + 2*a^2*b^3 + b^5 + 2*(a^5 + 2*a^3*b^2 + a*b^4)*e^(-x) + (a^4*b + 2*a^2*b^3 +
b^5)*e^(-2*x) + 4*(a^5 + 2*a^3*b^2 + a*b^4)*e^(-3*x) - (a^4*b + 2*a^2*b^3 + b^5)*e^(-4*x) + 2*(a^5 + 2*a^3*b^2
+ a*b^4)*e^(-5*x) - (a^4*b + 2*a^2*b^3 + b^5)*e^(-6*x))

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mupad [B]  time = 3.89, size = 501, normalized size = 3.71 \[ \frac {\frac {2\,\left (a^8+2\,a^6\,b^2-2\,a^2\,b^6-b^8\right )}{\left (a^2+b^2\right )\,{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}-\frac {2\,{\mathrm {e}}^x\,\left (a^7\,b+3\,a^5\,b^3+3\,a^3\,b^5+a\,b^7\right )}{\left (a^2+b^2\right )\,{\left (a^4+2\,a^2\,b^2+b^4\right )}^2}}{{\mathrm {e}}^{2\,x}+1}-\frac {\frac {2\,\left (a^2-b^2\right )}{a^4+2\,a^2\,b^2+b^4}-\frac {4\,a\,b\,{\mathrm {e}}^x}{a^4+2\,a^2\,b^2+b^4}}{2\,{\mathrm {e}}^{2\,x}+{\mathrm {e}}^{4\,x}+1}-\frac {a\,\ln \left ({\mathrm {e}}^x+1{}\mathrm {i}\right )}{-a^3+a^2\,b\,3{}\mathrm {i}+3\,a\,b^2-b^3\,1{}\mathrm {i}}-\frac {\ln \left (15\,a^6\,b^3-a^2\,b^7-30\,a^4\,b^5-4\,a^8\,b+8\,a^9\,{\mathrm {e}}^x+a^2\,b^7\,{\mathrm {e}}^{2\,x}+30\,a^4\,b^5\,{\mathrm {e}}^{2\,x}-15\,a^6\,b^3\,{\mathrm {e}}^{2\,x}+4\,a^8\,b\,{\mathrm {e}}^{2\,x}+2\,a^3\,b^6\,{\mathrm {e}}^x+60\,a^5\,b^4\,{\mathrm {e}}^x-30\,a^7\,b^2\,{\mathrm {e}}^x\right )\,\left (a^4-3\,a^2\,b^2\right )}{a^6+3\,a^4\,b^2+3\,a^2\,b^4+b^6}+\frac {2\,{\mathrm {e}}^x\,\left (a^7\,b^2+2\,a^5\,b^4+a^3\,b^6\right )}{b\,\left (a^2\,b+b^3\right )\,\left (a^2+b^2\right )\,\left (2\,a\,{\mathrm {e}}^x-b+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a^4+2\,a^2\,b^2+b^4\right )}-\frac {a\,\ln \left (1+{\mathrm {e}}^x\,1{}\mathrm {i}\right )\,1{}\mathrm {i}}{-a^3\,1{}\mathrm {i}+3\,a^2\,b+a\,b^2\,3{}\mathrm {i}-b^3} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tanh(x)^3/(a + b*sinh(x))^2,x)

[Out]

((2*(a^8 - b^8 - 2*a^2*b^6 + 2*a^6*b^2))/((a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)^2) - (2*exp(x)*(a*b^7 + a^7*b +
3*a^3*b^5 + 3*a^5*b^3))/((a^2 + b^2)*(a^4 + b^4 + 2*a^2*b^2)^2))/(exp(2*x) + 1) - ((2*(a^2 - b^2))/(a^4 + b^4
+ 2*a^2*b^2) - (4*a*b*exp(x))/(a^4 + b^4 + 2*a^2*b^2))/(2*exp(2*x) + exp(4*x) + 1) - (a*log(exp(x) + 1i))/(3*a
*b^2 + a^2*b*3i - a^3 - b^3*1i) - (log(15*a^6*b^3 - a^2*b^7 - 30*a^4*b^5 - 4*a^8*b + 8*a^9*exp(x) + a^2*b^7*ex
p(2*x) + 30*a^4*b^5*exp(2*x) - 15*a^6*b^3*exp(2*x) + 4*a^8*b*exp(2*x) + 2*a^3*b^6*exp(x) + 60*a^5*b^4*exp(x) -
30*a^7*b^2*exp(x))*(a^4 - 3*a^2*b^2))/(a^6 + b^6 + 3*a^2*b^4 + 3*a^4*b^2) - (a*log(exp(x)*1i + 1)*1i)/(a*b^2*
3i + 3*a^2*b - a^3*1i - b^3) + (2*exp(x)*(a^3*b^6 + 2*a^5*b^4 + a^7*b^2))/(b*(a^2*b + b^3)*(a^2 + b^2)*(2*a*ex
p(x) - b + b*exp(2*x))*(a^4 + b^4 + 2*a^2*b^2))

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\tanh ^{3}{\relax (x )}}{\left (a + b \sinh {\relax (x )}\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tanh(x)**3/(a+b*sinh(x))**2,x)

[Out]

Integral(tanh(x)**3/(a + b*sinh(x))**2, x)

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