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

Optimal. Leaf size=215 \[ \frac {a b x \left (a^2+b^2\right )}{\left (a^2-b^2\right )^3}+\frac {a b x}{\left (a^2-b^2\right )^2}+\frac {a^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}+\frac {b^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}-\frac {a^2 b}{\left (a^2-b^2\right )^2 (a \coth (x)+b)}-\frac {a b \sinh (x) \cosh (x)}{\left (a^2-b^2\right )^2}-\frac {3 a^2 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}+\frac {a b^3 x}{\left (a^2-b^2\right )^3}-\frac {a^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}+\frac {a^3 b x}{\left (a^2-b^2\right )^3} \]

[Out]

a^3*b*x/(a^2-b^2)^3+a*b^3*x/(a^2-b^2)^3+a*b*x/(a^2-b^2)^2+a*b*(a^2+b^2)*x/(a^2-b^2)^3-a^2*b/(a^2-b^2)^2/(b+a*c
oth(x))-a^4*ln(a*cosh(x)+b*sinh(x))/(a^2-b^2)^3-3*a^2*b^2*ln(a*cosh(x)+b*sinh(x))/(a^2-b^2)^3-a*b*cosh(x)*sinh
(x)/(a^2-b^2)^2+1/2*a^2*sinh(x)^2/(a^2-b^2)^2+1/2*b^2*sinh(x)^2/(a^2-b^2)^2

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Rubi [A]  time = 0.54, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps used = 17, number of rules used = 13, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.722, Rules used = {3111, 3109, 2564, 30, 2635, 8, 3097, 3133, 3099, 3085, 3483, 3531, 3530} \[ \frac {a^3 b x}{\left (a^2-b^2\right )^3}+\frac {a b x \left (a^2+b^2\right )}{\left (a^2-b^2\right )^3}+\frac {a b x}{\left (a^2-b^2\right )^2}+\frac {a b^3 x}{\left (a^2-b^2\right )^3}+\frac {a^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}+\frac {b^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}-\frac {a^2 b}{\left (a^2-b^2\right )^2 (a \coth (x)+b)}-\frac {a b \sinh (x) \cosh (x)}{\left (a^2-b^2\right )^2}-\frac {a^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac {3 a^2 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3} \]

Antiderivative was successfully verified.

[In]

Int[(Cosh[x]*Sinh[x]^3)/(a*Cosh[x] + b*Sinh[x])^2,x]

[Out]

(a^3*b*x)/(a^2 - b^2)^3 + (a*b^3*x)/(a^2 - b^2)^3 + (a*b*x)/(a^2 - b^2)^2 + (a*b*(a^2 + b^2)*x)/(a^2 - b^2)^3
- (a^2*b)/((a^2 - b^2)^2*(b + a*Coth[x])) - (a^4*Log[a*Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 - (3*a^2*b^2*Log[a*
Cosh[x] + b*Sinh[x]])/(a^2 - b^2)^3 - (a*b*Cosh[x]*Sinh[x])/(a^2 - b^2)^2 + (a^2*Sinh[x]^2)/(2*(a^2 - b^2)^2)
+ (b^2*Sinh[x]^2)/(2*(a^2 - b^2)^2)

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rule 2564

Int[cos[(e_.) + (f_.)*(x_)]^(n_.)*((a_.)*sin[(e_.) + (f_.)*(x_)])^(m_.), x_Symbol] :> Dist[1/(a*f), Subst[Int[
x^m*(1 - x^2/a^2)^((n - 1)/2), x], x, a*Sin[e + f*x]], x] /; FreeQ[{a, e, f, m}, x] && IntegerQ[(n - 1)/2] &&
!(IntegerQ[(m - 1)/2] && LtQ[0, m, n])

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 3085

Int[sin[(c_.) + (d_.)*(x_)]^(m_)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(n_.), x_Symb
ol] :> Int[(b + a*Cot[c + d*x])^n, x] /; FreeQ[{a, b, c, d}, x] && EqQ[m + n, 0] && IntegerQ[n] && NeQ[a^2 + b
^2, 0]

Rule 3097

Int[sin[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp
[(b*x)/(a^2 + b^2), x] - Dist[a/(a^2 + b^2), Int[(b*Cos[c + d*x] - a*Sin[c + d*x])/(a*Cos[c + d*x] + b*Sin[c +
d*x]), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0]

Rule 3099

Int[sin[(c_.) + (d_.)*(x_)]^(m_)/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :>
-Simp[(a*Sin[c + d*x]^(m - 1))/(d*(a^2 + b^2)*(m - 1)), x] + (Dist[a^2/(a^2 + b^2), Int[Sin[c + d*x]^(m - 2)/
(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x] + Dist[b/(a^2 + b^2), Int[Sin[c + d*x]^(m - 1), x], x]) /; FreeQ[{a,
b, c, d}, x] && NeQ[a^2 + b^2, 0] && GtQ[m, 1]

Rule 3109

Int[(cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.))/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(
c_.) + (d_.)*(x_)]), x_Symbol] :> Dist[b/(a^2 + b^2), Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1), x], x] + (Dist[
a/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n, x], x] - Dist[(a*b)/(a^2 + b^2), Int[(Cos[c + d*x]^(m
- 1)*Sin[c + d*x]^(n - 1))/(a*Cos[c + d*x] + b*Sin[c + d*x]), x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b
^2, 0] && IGtQ[m, 0] && IGtQ[n, 0]

Rule 3111

Int[cos[(c_.) + (d_.)*(x_)]^(m_.)*sin[(c_.) + (d_.)*(x_)]^(n_.)*(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_
.) + (d_.)*(x_)])^(p_), x_Symbol] :> Dist[b/(a^2 + b^2), Int[Cos[c + d*x]^m*Sin[c + d*x]^(n - 1)*(a*Cos[c + d*
x] + b*Sin[c + d*x])^(p + 1), x], x] + (Dist[a/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^n*(a*Cos[c +
d*x] + b*Sin[c + d*x])^(p + 1), x], x] - Dist[(a*b)/(a^2 + b^2), Int[Cos[c + d*x]^(m - 1)*Sin[c + d*x]^(n - 1
)*(a*Cos[c + d*x] + b*Sin[c + d*x])^p, x], x]) /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && IGtQ[m, 0] &&
IGtQ[n, 0] && ILtQ[p, 0]

Rule 3133

Int[((A_.) + cos[(d_.) + (e_.)*(x_)]*(B_.) + (C_.)*sin[(d_.) + (e_.)*(x_)])/((a_.) + cos[(d_.) + (e_.)*(x_)]*(
b_.) + (c_.)*sin[(d_.) + (e_.)*(x_)]), x_Symbol] :> Simp[((b*B + c*C)*x)/(b^2 + c^2), x] + Simp[((c*B - b*C)*L
og[a + b*Cos[d + e*x] + c*Sin[d + e*x]])/(e*(b^2 + c^2)), x] /; FreeQ[{a, b, c, d, e, A, B, C}, x] && NeQ[b^2
+ c^2, 0] && EqQ[A*(b^2 + c^2) - a*(b*B + c*C), 0]

Rule 3483

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n + 1))/(d*(n + 1)
*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a - b*Tan[c + d*x])*(a + b*Tan[c + d*x])^(n + 1), x], x] /; FreeQ
[{a, b, c, d}, x] && NeQ[a^2 + b^2, 0] && LtQ[n, -1]

Rule 3530

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(c*Log[Re
moveContent[a*Cos[e + f*x] + b*Sin[e + f*x], x]])/(b*f), x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d,
0] && NeQ[a^2 + b^2, 0] && EqQ[a*c + b*d, 0]

Rule 3531

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((a*c +
b*d)*x)/(a^2 + b^2), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rubi steps

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

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Mathematica [A]  time = 1.03, size = 176, normalized size = 0.82 \[ \frac {a \left (a^2-b^2\right )^2 \cosh (3 x)-2 b \sinh (x) \left (\left (a^2-b^2\right )^2 \cosh (2 x)+2 a \left (3 a^3+2 a \left (a^2+3 b^2\right ) \log (a \cosh (x)+b \sinh (x))-6 a^2 b x-3 a b^2-2 b^3 x\right )\right )+a \cosh (x) \left (a^4+24 a^3 b x-8 a^2 \left (a^2+3 b^2\right ) \log (a \cosh (x)+b \sinh (x))+2 a^2 b^2+8 a b^3 x-3 b^4\right )}{8 (a-b)^3 (a+b)^3 (a \cosh (x)+b \sinh (x))} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(a^2 - b^2)^2*Cosh[3*x] + a*Cosh[x]*(a^4 + 2*a^2*b^2 - 3*b^4 + 24*a^3*b*x + 8*a*b^3*x - 8*a^2*(a^2 + 3*b^2)
*Log[a*Cosh[x] + b*Sinh[x]]) - 2*b*((a^2 - b^2)^2*Cosh[2*x] + 2*a*(3*a^3 - 3*a*b^2 - 6*a^2*b*x - 2*b^3*x + 2*a
*(a^2 + 3*b^2)*Log[a*Cosh[x] + b*Sinh[x]]))*Sinh[x])/(8*(a - b)^3*(a + b)^3*(a*Cosh[x] + b*Sinh[x]))

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fricas [B]  time = 0.58, size = 1655, normalized size = 7.70 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*((a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^6 + 6*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 +
a*b^4 - b^5)*cosh(x)*sinh(x)^5 + (a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*sinh(x)^6 + a^5 + a^4*b -
2*a^3*b^2 - 2*a^2*b^3 + a*b^4 + b^5 + (a^5 - 3*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5 + 8*(a^5 + 4*a^4
*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*x)*cosh(x)^4 + (a^5 - 3*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5 + 15
*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^2 + 8*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*
b^4)*x)*sinh(x)^4 + 4*(5*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^3 + (a^5 - 3*a^4*b + 2*a^
3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5 + 8*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*x)*cosh(x))*sinh(x)^3 +
(a^5 + 19*a^4*b - 14*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5 + 8*(a^5 + 2*a^4*b - 2*a^2*b^3 - a*b^4)*x)*cosh(x)^2
+ (a^5 + 19*a^4*b - 14*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5 + 15*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 -
b^5)*cosh(x)^4 + 6*(a^5 - 3*a^4*b + 2*a^3*b^2 + 2*a^2*b^3 - 3*a*b^4 + b^5 + 8*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*
a^2*b^3 + a*b^4)*x)*cosh(x)^2 + 8*(a^5 + 2*a^4*b - 2*a^2*b^3 - a*b^4)*x)*sinh(x)^2 - 8*((a^5 + a^4*b + 3*a^3*b
^2 + 3*a^2*b^3)*cosh(x)^4 + 4*(a^5 + a^4*b + 3*a^3*b^2 + 3*a^2*b^3)*cosh(x)*sinh(x)^3 + (a^5 + a^4*b + 3*a^3*b
^2 + 3*a^2*b^3)*sinh(x)^4 + (a^5 - a^4*b + 3*a^3*b^2 - 3*a^2*b^3)*cosh(x)^2 + (a^5 - a^4*b + 3*a^3*b^2 - 3*a^2
*b^3 + 6*(a^5 + a^4*b + 3*a^3*b^2 + 3*a^2*b^3)*cosh(x)^2)*sinh(x)^2 + 2*(2*(a^5 + a^4*b + 3*a^3*b^2 + 3*a^2*b^
3)*cosh(x)^3 + (a^5 - a^4*b + 3*a^3*b^2 - 3*a^2*b^3)*cosh(x))*sinh(x))*log(2*(a*cosh(x) + b*sinh(x))/(cosh(x)
- sinh(x))) + 2*(3*(a^5 - a^4*b - 2*a^3*b^2 + 2*a^2*b^3 + a*b^4 - b^5)*cosh(x)^5 + 2*(a^5 - 3*a^4*b + 2*a^3*b^
2 + 2*a^2*b^3 - 3*a*b^4 + b^5 + 8*(a^5 + 4*a^4*b + 6*a^3*b^2 + 4*a^2*b^3 + a*b^4)*x)*cosh(x)^3 + (a^5 + 19*a^4
*b - 14*a^3*b^2 - 2*a^2*b^3 - 3*a*b^4 - b^5 + 8*(a^5 + 2*a^4*b - 2*a^2*b^3 - a*b^4)*x)*cosh(x))*sinh(x))/((a^7
+ a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 - a*b^6 - b^7)*cosh(x)^4 + 4*(a^7 + a^6*b - 3*a^5*b^2
- 3*a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 - a*b^6 - b^7)*cosh(x)*sinh(x)^3 + (a^7 + a^6*b - 3*a^5*b^2 - 3*a^4*b^3 +
3*a^3*b^4 + 3*a^2*b^5 - a*b^6 - b^7)*sinh(x)^4 + (a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5
- a*b^6 + b^7)*cosh(x)^2 + (a^7 - a^6*b - 3*a^5*b^2 + 3*a^4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7 + 6*(a^
7 + a^6*b - 3*a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 - a*b^6 - b^7)*cosh(x)^2)*sinh(x)^2 + 2*(2*(a^7 + a^
6*b - 3*a^5*b^2 - 3*a^4*b^3 + 3*a^3*b^4 + 3*a^2*b^5 - a*b^6 - b^7)*cosh(x)^3 + (a^7 - a^6*b - 3*a^5*b^2 + 3*a^
4*b^3 + 3*a^3*b^4 - 3*a^2*b^5 - a*b^6 + b^7)*cosh(x))*sinh(x))

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giac [A]  time = 0.14, size = 238, normalized size = 1.11 \[ \frac {a x}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac {{\left (a^{4} + 3 \, a^{2} b^{2}\right )} \log \left ({\left | a e^{\left (2 \, x\right )} + b e^{\left (2 \, x\right )} + a - b \right |}\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {e^{\left (2 \, x\right )}}{8 \, {\left (a^{2} + 2 \, a b + b^{2}\right )}} + \frac {2 \, a^{3} e^{\left (4 \, x\right )} - 4 \, a^{2} b e^{\left (4 \, x\right )} + 2 \, a b^{2} e^{\left (4 \, x\right )} + 3 \, a^{3} e^{\left (2 \, x\right )} + 11 \, a^{2} b e^{\left (2 \, x\right )} + a b^{2} e^{\left (2 \, x\right )} + b^{3} e^{\left (2 \, x\right )} + a^{3} + a^{2} b - a b^{2} - b^{3}}{8 \, {\left (a^{4} - 2 \, a^{2} b^{2} + b^{4}\right )} {\left (a e^{\left (4 \, x\right )} + b e^{\left (4 \, x\right )} + a e^{\left (2 \, x\right )} - b e^{\left (2 \, x\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*x/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) - (a^4 + 3*a^2*b^2)*log(abs(a*e^(2*x) + b*e^(2*x) + a - b))/(a^6 - 3*a^4*b
^2 + 3*a^2*b^4 - b^6) + 1/8*e^(2*x)/(a^2 + 2*a*b + b^2) + 1/8*(2*a^3*e^(4*x) - 4*a^2*b*e^(4*x) + 2*a*b^2*e^(4*
x) + 3*a^3*e^(2*x) + 11*a^2*b*e^(2*x) + a*b^2*e^(2*x) + b^3*e^(2*x) + a^3 + a^2*b - a*b^2 - b^3)/((a^4 - 2*a^2
*b^2 + b^4)*(a*e^(4*x) + b*e^(4*x) + a*e^(2*x) - b*e^(2*x)))

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maple [A]  time = 0.27, size = 253, normalized size = 1.18 \[ \frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}+\frac {1}{2 \left (a +b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{\left (a +b \right )^{3}}-\frac {2 a^{4} b \tanh \left (\frac {x}{2}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}+\frac {2 a^{2} b^{3} \tanh \left (\frac {x}{2}\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3} \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}-\frac {a^{4} \ln \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right )}{\left (a -b \right )^{3} \left (a +b \right )^{3}}-\frac {3 a^{2} \ln \left (a +2 \tanh \left (\frac {x}{2}\right ) b +a \left (\tanh ^{2}\left (\frac {x}{2}\right )\right )\right ) b^{2}}{\left (a -b \right )^{3} \left (a +b \right )^{3}}+\frac {1}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {1}{2 \left (a -b \right )^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )}+\frac {a \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{\left (a -b \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.52, size = 241, normalized size = 1.12 \[ -\frac {a x}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac {{\left (a^{4} + 3 \, a^{2} b^{2}\right )} \log \left (-{\left (a - b\right )} e^{\left (-2 \, x\right )} - a - b\right )}{a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}} + \frac {a^{4} - 2 \, a^{3} b + 2 \, a b^{3} - b^{4} + {\left (a^{4} - 20 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}\right )} e^{\left (-2 \, x\right )}}{8 \, {\left ({\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} e^{\left (-2 \, x\right )} + {\left (a^{6} - 2 \, a^{5} b - a^{4} b^{2} + 4 \, a^{3} b^{3} - a^{2} b^{4} - 2 \, a b^{5} + b^{6}\right )} e^{\left (-4 \, x\right )}\right )}} + \frac {e^{\left (-2 \, x\right )}}{8 \, {\left (a^{2} - 2 \, a b + b^{2}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a*x/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - (a^4 + 3*a^2*b^2)*log(-(a - b)*e^(-2*x) - a - b)/(a^6 - 3*a^4*b^2 + 3*a
^2*b^4 - b^6) + 1/8*(a^4 - 2*a^3*b + 2*a*b^3 - b^4 + (a^4 - 20*a^3*b + 6*a^2*b^2 - 4*a*b^3 + b^4)*e^(-2*x))/((
a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*e^(-2*x) + (a^6 - 2*a^5*b - a^4*b^2 + 4*a^3*b^3 - a^2*b^4 - 2*a*b^5 + b^6)*
e^(-4*x)) + 1/8*e^(-2*x)/(a^2 - 2*a*b + b^2)

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mupad [B]  time = 1.82, size = 127, normalized size = 0.59 \[ \frac {{\mathrm {e}}^{2\,x}}{8\,{\left (a+b\right )}^2}+\frac {{\mathrm {e}}^{-2\,x}}{8\,{\left (a-b\right )}^2}+\frac {a\,x}{{\left (a-b\right )}^3}-\frac {\ln \left (a-b+a\,{\mathrm {e}}^{2\,x}+b\,{\mathrm {e}}^{2\,x}\right )\,\left (a^4+3\,a^2\,b^2\right )}{a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6}+\frac {2\,a^3\,b}{{\left (a+b\right )}^3\,{\left (a-b\right )}^2\,\left (a-b+{\mathrm {e}}^{2\,x}\,\left (a+b\right )\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

exp(2*x)/(8*(a + b)^2) + exp(-2*x)/(8*(a - b)^2) + (a*x)/(a - b)^3 - (log(a - b + a*exp(2*x) + b*exp(2*x))*(a^
4 + 3*a^2*b^2))/(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2) + (2*a^3*b)/((a + b)^3*(a - b)^2*(a - b + exp(2*x)*(a + b)
))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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