### 3.721 $$\int \frac{\cosh ^3(x) \sinh (x)}{(a \cosh (x)+b \sinh (x))^2} \, dx$$

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

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

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Rubi [A]  time = 0.558087, antiderivative size = 215, normalized size of antiderivative = 1., 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, 3100, 2635, 8, 3098, 3133, 3109, 2564, 30, 3086, 3483, 3531, 3530} $\frac{a b^3 x}{\left (a^2-b^2\right )^3}+\frac{a b x \left (a^2+b^2\right )}{\left (a^2-b^2\right )^3}-\frac{a b x}{\left (a^2-b^2\right )^2}+\frac{a^3 b x}{\left (a^2-b^2\right )^3}+\frac{a^2 \sinh ^2(x)}{2 \left (a^2-b^2\right )^2}+\frac{b^2 \cosh ^2(x)}{2 \left (a^2-b^2\right )^2}+\frac{a b^2}{\left (a^2-b^2\right )^2 (a+b \tanh (x))}-\frac{a b \sinh (x) \cosh (x)}{\left (a^2-b^2\right )^2}-\frac{b^4 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}-\frac{3 a^2 b^2 \log (a \cosh (x)+b \sinh (x))}{\left (a^2-b^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

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

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 3100

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

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 8

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

Rule 3098

Int[cos[(c_.) + (d_.)*(x_)]/(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)]), x_Symbol] :> Simp
[(a*x)/(a^2 + b^2), x] + Dist[b/(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 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 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 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 30

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

Rule 3086

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

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]

Rubi steps

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

Mathematica [A]  time = 1.19962, size = 183, normalized size = 0.85 $\frac{a \cosh (x) \left (\left (a^4-b^4\right ) \cosh (2 x)-4 b \left (-a x \left (a^2+3 b^2\right )+a \left (a^2-b^2\right ) \sinh (x) \cosh (x)+b \left (3 a^2+b^2\right ) \log (a \cosh (x)+b \sinh (x))\right )\right )+b \sinh (x) \left (-2 a b \left (a^2-b^2\right ) \sinh (2 x)+\left (a^4-b^4\right ) \cosh (2 x)+4 b \left (-b \left (3 a^2+b^2\right ) \log (a \cosh (x)+b \sinh (x))-a^2 b+a^3 x+3 a b^2 x+b^3\right )\right )}{4 (a-b)^3 (a+b)^3 (a \cosh (x)+b \sinh (x))}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.079, size = 253, normalized size = 1.2 \begin{align*}{\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 ) ^{-1}}+{\frac{b}{ \left ( a-b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) }-2\,{\frac{{a}^{2}{b}^{3}\tanh \left ( x/2 \right ) }{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3} \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) }}+2\,{\frac{{b}^{5}\tanh \left ( x/2 \right ) }{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3} \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ) }}-3\,{\frac{{a}^{2}\ln \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ( x/2 \right ) \right ) ^{2} \right ){b}^{2}}{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3}}}-{\frac{{b}^{4}}{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3}}\ln \left ( a+2\,\tanh \left ( x/2 \right ) b+a \left ( \tanh \left ({\frac{x}{2}} \right ) \right ) ^{2} \right ) }+{\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 ) ^{-1}}-{\frac{b}{ \left ( a+b \right ) ^{3}}\ln \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) } \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cosh(x)^3*sinh(x)/(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)+1/(a-b)^3*ln(tanh(1/2*x)+1)*b-2*a^2/(a-b)^3/(a+b)^3*
b^3*tanh(1/2*x)/(a+2*tanh(1/2*x)*b+a*tanh(1/2*x)^2)+2*b^5/(a-b)^3/(a+b)^3*tanh(1/2*x)/(a+2*tanh(1/2*x)*b+a*tan
h(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-b^4/(a-b)^3/(a+b)^3*ln(a+2*tanh(1/
2*x)*b+a*tanh(1/2*x)^2)+1/2/(a+b)^2/(tanh(1/2*x)-1)^2+1/2/(a+b)^2/(tanh(1/2*x)-1)-1/(a+b)^3*ln(tanh(1/2*x)-1)*
b

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Maxima [A]  time = 1.222, size = 324, normalized size = 1.51 \begin{align*} \frac{b x}{a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}} - \frac{{\left (3 \, a^{2} b^{2} + b^{4}\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} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 20 \, 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 )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x/(a^3 + 3*a^2*b + 3*a*b^2 + b^3) - (3*a^2*b^2 + b^4)*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 - 4*a^3*b + 6*a^2*b^2 - 20*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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Fricas [B]  time = 2.63856, size = 3650, normalized size = 16.98 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cosh(x)^3*sinh(x)/(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^4*b + 4*a
^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*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^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 +
b^5)*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^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*x)*cosh(x))*sinh(x)^3 +
(a^5 + 3*a^4*b + 2*a^3*b^2 + 14*a^2*b^3 - 19*a*b^4 - b^5 + 8*(a^4*b + 2*a^3*b^2 - 2*a*b^4 - b^5)*x)*cosh(x)^2
+ (a^5 + 3*a^4*b + 2*a^3*b^2 + 14*a^2*b^3 - 19*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^4*b + 4*a^3*b^2 + 6*a^2*b^3
+ 4*a*b^4 + b^5)*x)*cosh(x)^2 + 8*(a^4*b + 2*a^3*b^2 - 2*a*b^4 - b^5)*x)*sinh(x)^2 - 8*((3*a^3*b^2 + 3*a^2*b^3
+ a*b^4 + b^5)*cosh(x)^4 + 4*(3*a^3*b^2 + 3*a^2*b^3 + a*b^4 + b^5)*cosh(x)*sinh(x)^3 + (3*a^3*b^2 + 3*a^2*b^3
+ a*b^4 + b^5)*sinh(x)^4 + (3*a^3*b^2 - 3*a^2*b^3 + a*b^4 - b^5)*cosh(x)^2 + (3*a^3*b^2 - 3*a^2*b^3 + a*b^4 -
b^5 + 6*(3*a^3*b^2 + 3*a^2*b^3 + a*b^4 + b^5)*cosh(x)^2)*sinh(x)^2 + 2*(2*(3*a^3*b^2 + 3*a^2*b^3 + a*b^4 + b^
5)*cosh(x)^3 + (3*a^3*b^2 - 3*a^2*b^3 + a*b^4 - b^5)*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^4*b + 4*a^3*b^2 + 6*a^2*b^3 + 4*a*b^4 + b^5)*x)*cosh(x)^3 + (a^5 + 3*a^4*
b + 2*a^3*b^2 + 14*a^2*b^3 - 19*a*b^4 - b^5 + 8*(a^4*b + 2*a^3*b^2 - 2*a*b^4 - b^5)*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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]  time = 1.15, size = 324, normalized size = 1.51 \begin{align*} \frac{b x}{a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}} - \frac{{\left (3 \, a^{2} b^{2} + b^{4}\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^{2} b e^{\left (4 \, x\right )} - 4 \, a b^{2} e^{\left (4 \, x\right )} + 2 \, b^{3} e^{\left (4 \, x\right )} - a^{3} e^{\left (2 \, x\right )} - a^{2} b e^{\left (2 \, x\right )} - 11 \, a b^{2} e^{\left (2 \, x\right )} - 3 \, 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 )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

b*x/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) - (3*a^2*b^2 + b^4)*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^2*b*e^(4*x) - 4*a*b^2*e^(4*x) + 2*b^3*e^(4*
x) - a^3*e^(2*x) - a^2*b*e^(2*x) - 11*a*b^2*e^(2*x) - 3*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)))