### 3.714 $$\int \frac{\cosh ^3(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx$$

Optimal. Leaf size=212 $-\frac{b \sinh ^5(x)}{5 \left (a^2-b^2\right )}-\frac{b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac{a^2 b \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac{a^2 b^3 \sinh (x)}{\left (a^2-b^2\right )^3}+\frac{a \cosh ^5(x)}{5 \left (a^2-b^2\right )}-\frac{a \cosh ^3(x)}{3 \left (a^2-b^2\right )}-\frac{a b^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac{a^3 b^2 \cosh (x)}{\left (a^2-b^2\right )^3}+\frac{a^3 b^3 \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}$

[Out]

(a^3*b^3*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (a^3*b^2*Cosh[x])/(a^2 - b^2)^3
- (a*b^2*Cosh[x]^3)/(3*(a^2 - b^2)^2) - (a*Cosh[x]^3)/(3*(a^2 - b^2)) + (a*Cosh[x]^5)/(5*(a^2 - b^2)) - (a^2*b
^3*Sinh[x])/(a^2 - b^2)^3 + (a^2*b*Sinh[x]^3)/(3*(a^2 - b^2)^2) - (b*Sinh[x]^3)/(3*(a^2 - b^2)) - (b*Sinh[x]^5
)/(5*(a^2 - b^2))

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Rubi [A]  time = 0.432444, antiderivative size = 212, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 9, integrand size = 20, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.45, Rules used = {3109, 2564, 14, 2565, 30, 2637, 2638, 3074, 206} $-\frac{b \sinh ^5(x)}{5 \left (a^2-b^2\right )}-\frac{b \sinh ^3(x)}{3 \left (a^2-b^2\right )}+\frac{a^2 b \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac{a^2 b^3 \sinh (x)}{\left (a^2-b^2\right )^3}+\frac{a \cosh ^5(x)}{5 \left (a^2-b^2\right )}-\frac{a \cosh ^3(x)}{3 \left (a^2-b^2\right )}-\frac{a b^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}+\frac{a^3 b^2 \cosh (x)}{\left (a^2-b^2\right )^3}+\frac{a^3 b^3 \tan ^{-1}\left (\frac{a \sinh (x)+b \cosh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(a^3*b^3*ArcTan[(b*Cosh[x] + a*Sinh[x])/Sqrt[a^2 - b^2]])/(a^2 - b^2)^(7/2) + (a^3*b^2*Cosh[x])/(a^2 - b^2)^3
- (a*b^2*Cosh[x]^3)/(3*(a^2 - b^2)^2) - (a*Cosh[x]^3)/(3*(a^2 - b^2)) + (a*Cosh[x]^5)/(5*(a^2 - b^2)) - (a^2*b
^3*Sinh[x])/(a^2 - b^2)^3 + (a^2*b*Sinh[x]^3)/(3*(a^2 - b^2)^2) - (b*Sinh[x]^3)/(3*(a^2 - b^2)) - (b*Sinh[x]^5
)/(5*(a^2 - b^2))

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 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 2565

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

Rule 30

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

Rule 2637

Int[sin[Pi/2 + (c_.) + (d_.)*(x_)], x_Symbol] :> Simp[Sin[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 2638

Int[sin[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[Cos[c + d*x]/d, x] /; FreeQ[{c, d}, x]

Rule 3074

Int[(cos[(c_.) + (d_.)*(x_)]*(a_.) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> -Dist[d^(-1), Subst[Int
[1/(a^2 + b^2 - x^2), x], x, b*Cos[c + d*x] - a*Sin[c + d*x]], x] /; FreeQ[{a, b, c, d}, x] && NeQ[a^2 + b^2,
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{\cosh ^3(x) \sinh ^3(x)}{a \cosh (x)+b \sinh (x)} \, dx &=\frac{a \int \cosh ^2(x) \sinh ^3(x) \, dx}{a^2-b^2}-\frac{b \int \cosh ^3(x) \sinh ^2(x) \, dx}{a^2-b^2}+\frac{(a b) \int \frac{\cosh ^2(x) \sinh ^2(x)}{a \cosh (x)+b \sinh (x)} \, dx}{a^2-b^2}\\ &=\frac{\left (a^2 b\right ) \int \cosh (x) \sinh ^2(x) \, dx}{\left (a^2-b^2\right )^2}-\frac{\left (a b^2\right ) \int \cosh ^2(x) \sinh (x) \, dx}{\left (a^2-b^2\right )^2}+\frac{\left (a^2 b^2\right ) \int \frac{\cosh (x) \sinh (x)}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^2}-\frac{a \operatorname{Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,\cosh (x)\right )}{a^2-b^2}-\frac{(i b) \operatorname{Subst}\left (\int x^2 \left (1-x^2\right ) \, dx,x,i \sinh (x)\right )}{a^2-b^2}\\ &=\frac{\left (a^3 b^2\right ) \int \sinh (x) \, dx}{\left (a^2-b^2\right )^3}-\frac{\left (a^2 b^3\right ) \int \cosh (x) \, dx}{\left (a^2-b^2\right )^3}+\frac{\left (a^3 b^3\right ) \int \frac{1}{a \cosh (x)+b \sinh (x)} \, dx}{\left (a^2-b^2\right )^3}+\frac{\left (i a^2 b\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,i \sinh (x)\right )}{\left (a^2-b^2\right )^2}-\frac{\left (a b^2\right ) \operatorname{Subst}\left (\int x^2 \, dx,x,\cosh (x)\right )}{\left (a^2-b^2\right )^2}-\frac{a \operatorname{Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,\cosh (x)\right )}{a^2-b^2}-\frac{(i b) \operatorname{Subst}\left (\int \left (x^2-x^4\right ) \, dx,x,i \sinh (x)\right )}{a^2-b^2}\\ &=\frac{a^3 b^2 \cosh (x)}{\left (a^2-b^2\right )^3}-\frac{a b^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac{a \cosh ^3(x)}{3 \left (a^2-b^2\right )}+\frac{a \cosh ^5(x)}{5 \left (a^2-b^2\right )}-\frac{a^2 b^3 \sinh (x)}{\left (a^2-b^2\right )^3}+\frac{a^2 b \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac{b \sinh ^3(x)}{3 \left (a^2-b^2\right )}-\frac{b \sinh ^5(x)}{5 \left (a^2-b^2\right )}+\frac{\left (i a^3 b^3\right ) \operatorname{Subst}\left (\int \frac{1}{a^2-b^2-x^2} \, dx,x,-i b \cosh (x)-i a \sinh (x)\right )}{\left (a^2-b^2\right )^3}\\ &=\frac{a^3 b^3 \tan ^{-1}\left (\frac{b \cosh (x)+a \sinh (x)}{\sqrt{a^2-b^2}}\right )}{\left (a^2-b^2\right )^{7/2}}+\frac{a^3 b^2 \cosh (x)}{\left (a^2-b^2\right )^3}-\frac{a b^2 \cosh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac{a \cosh ^3(x)}{3 \left (a^2-b^2\right )}+\frac{a \cosh ^5(x)}{5 \left (a^2-b^2\right )}-\frac{a^2 b^3 \sinh (x)}{\left (a^2-b^2\right )^3}+\frac{a^2 b \sinh ^3(x)}{3 \left (a^2-b^2\right )^2}-\frac{b \sinh ^3(x)}{3 \left (a^2-b^2\right )}-\frac{b \sinh ^5(x)}{5 \left (a^2-b^2\right )}\\ \end{align*}

Mathematica [A]  time = 2.2181, size = 325, normalized size = 1.53 $\frac{1}{32} \left (\frac{2 b \left (10 a^2 b^2+5 a^4+b^4\right ) \sinh (x)}{(b-a)^3 (a+b)^3}+\frac{2 b \left (3 a^2+b^2\right ) \sinh (3 x)}{3 (a-b)^2 (a+b)^2}+\frac{2 a \left (10 a^2 b^2+a^4+5 b^4\right ) \cosh (x)}{(a-b)^3 (a+b)^3}-\frac{2 a \left (a^2+3 b^2\right ) \cosh (3 x)}{3 (a-b)^2 (a+b)^2}+\frac{4 a b \left (10 a^2 b^2+3 a^4+3 b^4\right ) \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )}{(a-b)^{7/2} (a+b)^{7/2}}-3 \left (\frac{2 b \sinh (x)}{b^2-a^2}+\frac{2 a \cosh (x)}{a^2-b^2}+\frac{4 a b \tan ^{-1}\left (\frac{a \tanh \left (\frac{x}{2}\right )+b}{\sqrt{a-b} \sqrt{a+b}}\right )}{(a-b)^{3/2} (a+b)^{3/2}}\right )-\frac{2 b \sinh (5 x)}{5 (a-b) (a+b)}+\frac{2 a \cosh (5 x)}{5 (a-b) (a+b)}\right )$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

((4*a*b*(3*a^4 + 10*a^2*b^2 + 3*b^4)*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])])/((a - b)^(7/2)*(a +
b)^(7/2)) + (2*a*(a^4 + 10*a^2*b^2 + 5*b^4)*Cosh[x])/((a - b)^3*(a + b)^3) - (2*a*(a^2 + 3*b^2)*Cosh[3*x])/(3*
(a - b)^2*(a + b)^2) + (2*a*Cosh[5*x])/(5*(a - b)*(a + b)) + (2*b*(5*a^4 + 10*a^2*b^2 + b^4)*Sinh[x])/((-a + b
)^3*(a + b)^3) - 3*((4*a*b*ArcTan[(b + a*Tanh[x/2])/(Sqrt[a - b]*Sqrt[a + b])])/((a - b)^(3/2)*(a + b)^(3/2))
+ (2*a*Cosh[x])/(a^2 - b^2) + (2*b*Sinh[x])/(-a^2 + b^2)) + (2*b*(3*a^2 + b^2)*Sinh[3*x])/(3*(a - b)^2*(a + b)
^2) - (2*b*Sinh[5*x])/(5*(a - b)*(a + b)))/32

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Maple [A]  time = 0.056, size = 344, normalized size = 1.6 \begin{align*} -4\,{\frac{1}{ \left ( 8\,a-8\,b \right ) \left ( \tanh \left ( x/2 \right ) +1 \right ) ^{4}}}+{\frac{16}{80\,a-80\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-5}}+{\frac{5\,a}{12\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}-{\frac{7\,b}{12\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-3}}-{\frac{a}{8\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}+{\frac{3\,b}{8\, \left ( a-b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-2}}-{\frac{{a}^{2}}{8\, \left ( a-b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+{\frac{3\,ab}{8\, \left ( a-b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) +1 \right ) ^{-1}}+2\,{\frac{{a}^{3}{b}^{3}}{ \left ( a-b \right ) ^{3} \left ( a+b \right ) ^{3}\sqrt{{a}^{2}-{b}^{2}}}\arctan \left ( 1/2\,{\frac{2\,a\tanh \left ( x/2 \right ) +2\,b}{\sqrt{{a}^{2}-{b}^{2}}}} \right ) }-{\frac{16}{80\,a+80\,b} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-5}}-4\,{\frac{1}{ \left ( 8\,a+8\,b \right ) \left ( \tanh \left ( x/2 \right ) -1 \right ) ^{4}}}-{\frac{a}{8\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}-{\frac{3\,b}{8\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-2}}-{\frac{5\,a}{12\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}-{\frac{7\,b}{12\, \left ( a+b \right ) ^{2}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-3}}+{\frac{{a}^{2}}{8\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}}+{\frac{3\,ab}{8\, \left ( a+b \right ) ^{3}} \left ( \tanh \left ({\frac{x}{2}} \right ) -1 \right ) ^{-1}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4/(8*a-8*b)/(tanh(1/2*x)+1)^4+16/5/(16*a-16*b)/(tanh(1/2*x)+1)^5+5/12/(a-b)^2/(tanh(1/2*x)+1)^3*a-7/12/(a-b)^
2/(tanh(1/2*x)+1)^3*b-1/8/(a-b)^2/(tanh(1/2*x)+1)^2*a+3/8/(a-b)^2/(tanh(1/2*x)+1)^2*b-1/8*a^2/(a-b)^3/(tanh(1/
2*x)+1)+3/8*a/(a-b)^3/(tanh(1/2*x)+1)*b+2*a^3*b^3/(a-b)^3/(a+b)^3/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tanh(1/2*x)+
2*b)/(a^2-b^2)^(1/2))-16/5/(16*a+16*b)/(tanh(1/2*x)-1)^5-4/(8*a+8*b)/(tanh(1/2*x)-1)^4-1/8/(a+b)^2/(tanh(1/2*x
)-1)^2*a-3/8/(a+b)^2/(tanh(1/2*x)-1)^2*b-5/12/(a+b)^2/(tanh(1/2*x)-1)^3*a-7/12/(a+b)^2/(tanh(1/2*x)-1)^3*b+1/8
*a^2/(a+b)^3/(tanh(1/2*x)-1)+3/8*a/(a+b)^3/(tanh(1/2*x)-1)*b

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 3.24966, size = 10982, normalized size = 51.8 \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)^3/(a*cosh(x)+b*sinh(x)),x, algorithm="fricas")

[Out]

[1/480*(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)^10 + 30*(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)^9 + 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)^10 - 5*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 -
5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^8 - 5*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5
+ 3*a*b^6 - b^7 - 27*(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)*si
nh(x)^8 + 40*(9*(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 -
3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x))*sinh(x)^7 + 3*a^7 + 3*a^6*b - 9
*a^5*b^2 - 9*a^4*b^3 + 9*a^3*b^4 + 9*a^2*b^5 - 3*a*b^6 - 3*b^7 - 30*(a^7 + a^6*b - 9*a^5*b^2 + 7*a^4*b^3 + 7*a
^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cosh(x)^6 - 10*(3*a^7 + 3*a^6*b - 27*a^5*b^2 + 21*a^4*b^3 + 21*a^3*b^4 - 27*
a^2*b^5 + 3*a*b^6 + 3*b^7 - 63*(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)*cos
h(x)^4 + 14*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^2)*sinh(x)^6 +
4*(189*(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)^5 - 70*(a^7 - 3*a^
6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^3 - 45*(a^7 + a^6*b - 9*a^5*b^2 + 7*a
^4*b^3 + 7*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cosh(x))*sinh(x)^5 - 30*(a^7 - a^6*b - 9*a^5*b^2 - 7*a^4*b^3 + 7
*a^3*b^4 + 9*a^2*b^5 + a*b^6 - b^7)*cosh(x)^4 - 10*(3*a^7 - 3*a^6*b - 27*a^5*b^2 - 21*a^4*b^3 + 21*a^3*b^4 + 2
7*a^2*b^5 + 3*a*b^6 - 3*b^7 - 63*(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)*c
osh(x)^6 + 35*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^4 + 45*(a^7
+ a^6*b - 9*a^5*b^2 + 7*a^4*b^3 + 7*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cosh(x)^2)*sinh(x)^4 + 40*(9*(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)^7 - 7*(a^7 - 3*a^6*b + a^5*b^2 + 5*a
^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^5 - 15*(a^7 + a^6*b - 9*a^5*b^2 + 7*a^4*b^3 + 7*a^3*b^4
- 9*a^2*b^5 + a*b^6 + b^7)*cosh(x)^3 - 3*(a^7 - a^6*b - 9*a^5*b^2 - 7*a^4*b^3 + 7*a^3*b^4 + 9*a^2*b^5 + a*b^6
- b^7)*cosh(x))*sinh(x)^3 - 5*(a^7 + 3*a^6*b + a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + a^2*b^5 + 3*a*b^6 + b^7)*cosh
(x)^2 + 5*(27*(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)^8 - a^7 - 3*
a^6*b - a^5*b^2 + 5*a^4*b^3 + 5*a^3*b^4 - a^2*b^5 - 3*a*b^6 - b^7 - 28*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 -
5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^6 - 90*(a^7 + a^6*b - 9*a^5*b^2 + 7*a^4*b^3 + 7*a^3*b^4 - 9*a^2*b
^5 + a*b^6 + b^7)*cosh(x)^4 - 36*(a^7 - a^6*b - 9*a^5*b^2 - 7*a^4*b^3 + 7*a^3*b^4 + 9*a^2*b^5 + a*b^6 - b^7)*c
osh(x)^2)*sinh(x)^2 + 480*(a^3*b^3*cosh(x)^5 + 5*a^3*b^3*cosh(x)^4*sinh(x) + 10*a^3*b^3*cosh(x)^3*sinh(x)^2 +
10*a^3*b^3*cosh(x)^2*sinh(x)^3 + 5*a^3*b^3*cosh(x)*sinh(x)^4 + a^3*b^3*sinh(x)^5)*sqrt(-a^2 + b^2)*log(((a + b
)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + 2*sqrt(-a^2 + b^2)*(cosh(x) + sinh(x)) - a + b)/
((a + b)*cosh(x)^2 + 2*(a + b)*cosh(x)*sinh(x) + (a + b)*sinh(x)^2 + a - b)) + 10*(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)^9 - 4*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*
b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^7 - 18*(a^7 + a^6*b - 9*a^5*b^2 + 7*a^4*b^3 + 7*a^3*b^4 - 9*a^2*b^5 + a
*b^6 + b^7)*cosh(x)^5 - 12*(a^7 - a^6*b - 9*a^5*b^2 - 7*a^4*b^3 + 7*a^3*b^4 + 9*a^2*b^5 + a*b^6 - b^7)*cosh(x)
^3 - (a^7 + 3*a^6*b + a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + a^2*b^5 + 3*a*b^6 + b^7)*cosh(x))*sinh(x))/((a^8 - 4*a
^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*cosh(x)^5 + 5*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*cosh(x)^4*
sinh(x) + 10*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*cosh(x)^3*sinh(x)^2 + 10*(a^8 - 4*a^6*b^2 + 6*a^4
*b^4 - 4*a^2*b^6 + b^8)*cosh(x)^2*sinh(x)^3 + 5*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*cosh(x)*sinh(x
)^4 + (a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*sinh(x)^5), 1/480*(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)^10 + 30*(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)^9 + 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)^10 - 5*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^
8 - 5*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7 - 27*(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)^8 + 40*(9*(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 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 -
a^2*b^5 + 3*a*b^6 - b^7)*cosh(x))*sinh(x)^7 + 3*a^7 + 3*a^6*b - 9*a^5*b^2 - 9*a^4*b^3 + 9*a^3*b^4 + 9*a^2*b^5
- 3*a*b^6 - 3*b^7 - 30*(a^7 + a^6*b - 9*a^5*b^2 + 7*a^4*b^3 + 7*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cosh(x)^6 -
10*(3*a^7 + 3*a^6*b - 27*a^5*b^2 + 21*a^4*b^3 + 21*a^3*b^4 - 27*a^2*b^5 + 3*a*b^6 + 3*b^7 - 63*(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 + 14*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*
b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^2)*sinh(x)^6 + 4*(189*(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)^5 - 70*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b
^5 + 3*a*b^6 - b^7)*cosh(x)^3 - 45*(a^7 + a^6*b - 9*a^5*b^2 + 7*a^4*b^3 + 7*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)
*cosh(x))*sinh(x)^5 - 30*(a^7 - a^6*b - 9*a^5*b^2 - 7*a^4*b^3 + 7*a^3*b^4 + 9*a^2*b^5 + a*b^6 - b^7)*cosh(x)^4
- 10*(3*a^7 - 3*a^6*b - 27*a^5*b^2 - 21*a^4*b^3 + 21*a^3*b^4 + 27*a^2*b^5 + 3*a*b^6 - 3*b^7 - 63*(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)^6 + 35*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^
4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^4 + 45*(a^7 + a^6*b - 9*a^5*b^2 + 7*a^4*b^3 + 7*a^3*b^4 -
9*a^2*b^5 + a*b^6 + b^7)*cosh(x)^2)*sinh(x)^4 + 40*(9*(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)^7 - 7*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)
*cosh(x)^5 - 15*(a^7 + a^6*b - 9*a^5*b^2 + 7*a^4*b^3 + 7*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cosh(x)^3 - 3*(a^7
- a^6*b - 9*a^5*b^2 - 7*a^4*b^3 + 7*a^3*b^4 + 9*a^2*b^5 + a*b^6 - b^7)*cosh(x))*sinh(x)^3 - 5*(a^7 + 3*a^6*b
+ a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + a^2*b^5 + 3*a*b^6 + b^7)*cosh(x)^2 + 5*(27*(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)^8 - a^7 - 3*a^6*b - a^5*b^2 + 5*a^4*b^3 + 5*a^3*b^4 - a^2
*b^5 - 3*a*b^6 - b^7 - 28*(a^7 - 3*a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^
6 - 90*(a^7 + a^6*b - 9*a^5*b^2 + 7*a^4*b^3 + 7*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cosh(x)^4 - 36*(a^7 - a^6*b
- 9*a^5*b^2 - 7*a^4*b^3 + 7*a^3*b^4 + 9*a^2*b^5 + a*b^6 - b^7)*cosh(x)^2)*sinh(x)^2 - 960*(a^3*b^3*cosh(x)^5
+ 5*a^3*b^3*cosh(x)^4*sinh(x) + 10*a^3*b^3*cosh(x)^3*sinh(x)^2 + 10*a^3*b^3*cosh(x)^2*sinh(x)^3 + 5*a^3*b^3*co
sh(x)*sinh(x)^4 + a^3*b^3*sinh(x)^5)*sqrt(a^2 - b^2)*arctan(sqrt(a^2 - b^2)/((a + b)*cosh(x) + (a + b)*sinh(x)
)) + 10*(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)^9 - 4*(a^7 - 3*
a^6*b + a^5*b^2 + 5*a^4*b^3 - 5*a^3*b^4 - a^2*b^5 + 3*a*b^6 - b^7)*cosh(x)^7 - 18*(a^7 + a^6*b - 9*a^5*b^2 + 7
*a^4*b^3 + 7*a^3*b^4 - 9*a^2*b^5 + a*b^6 + b^7)*cosh(x)^5 - 12*(a^7 - a^6*b - 9*a^5*b^2 - 7*a^4*b^3 + 7*a^3*b^
4 + 9*a^2*b^5 + a*b^6 - b^7)*cosh(x)^3 - (a^7 + 3*a^6*b + a^5*b^2 - 5*a^4*b^3 - 5*a^3*b^4 + a^2*b^5 + 3*a*b^6
+ b^7)*cosh(x))*sinh(x))/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*cosh(x)^5 + 5*(a^8 - 4*a^6*b^2 + 6*a
^4*b^4 - 4*a^2*b^6 + b^8)*cosh(x)^4*sinh(x) + 10*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*cosh(x)^3*sin
h(x)^2 + 10*(a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*cosh(x)^2*sinh(x)^3 + 5*(a^8 - 4*a^6*b^2 + 6*a^4*b
^4 - 4*a^2*b^6 + b^8)*cosh(x)*sinh(x)^4 + (a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*sinh(x)^5)]

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

[Out]

Timed out

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Giac [A]  time = 1.18073, size = 439, normalized size = 2.07 \begin{align*} \frac{2 \, a^{3} b^{3} \arctan \left (\frac{a e^{x} + b e^{x}}{\sqrt{a^{2} - b^{2}}}\right )}{{\left (a^{6} - 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} - b^{6}\right )} \sqrt{a^{2} - b^{2}}} - \frac{{\left (30 \, a^{2} e^{\left (4 \, x\right )} - 120 \, a b e^{\left (4 \, x\right )} + 30 \, b^{2} e^{\left (4 \, x\right )} + 5 \, a^{2} e^{\left (2 \, x\right )} - 5 \, b^{2} e^{\left (2 \, x\right )} - 3 \, a^{2} + 6 \, a b - 3 \, b^{2}\right )} e^{\left (-5 \, x\right )}}{480 \,{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )}} + \frac{3 \, a^{4} e^{\left (5 \, x\right )} + 12 \, a^{3} b e^{\left (5 \, x\right )} + 18 \, a^{2} b^{2} e^{\left (5 \, x\right )} + 12 \, a b^{3} e^{\left (5 \, x\right )} + 3 \, b^{4} e^{\left (5 \, x\right )} - 5 \, a^{4} e^{\left (3 \, x\right )} - 10 \, a^{3} b e^{\left (3 \, x\right )} + 10 \, a b^{3} e^{\left (3 \, x\right )} + 5 \, b^{4} e^{\left (3 \, x\right )} - 30 \, a^{4} e^{x} - 180 \, a^{3} b e^{x} - 300 \, a^{2} b^{2} e^{x} - 180 \, a b^{3} e^{x} - 30 \, b^{4} e^{x}}{480 \,{\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a^3*b^3*arctan((a*e^x + b*e^x)/sqrt(a^2 - b^2))/((a^6 - 3*a^4*b^2 + 3*a^2*b^4 - b^6)*sqrt(a^2 - b^2)) - 1/48
0*(30*a^2*e^(4*x) - 120*a*b*e^(4*x) + 30*b^2*e^(4*x) + 5*a^2*e^(2*x) - 5*b^2*e^(2*x) - 3*a^2 + 6*a*b - 3*b^2)*
e^(-5*x)/(a^3 - 3*a^2*b + 3*a*b^2 - b^3) + 1/480*(3*a^4*e^(5*x) + 12*a^3*b*e^(5*x) + 18*a^2*b^2*e^(5*x) + 12*a
*b^3*e^(5*x) + 3*b^4*e^(5*x) - 5*a^4*e^(3*x) - 10*a^3*b*e^(3*x) + 10*a*b^3*e^(3*x) + 5*b^4*e^(3*x) - 30*a^4*e^
x - 180*a^3*b*e^x - 300*a^2*b^2*e^x - 180*a*b^3*e^x - 30*b^4*e^x)/(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5
*a*b^4 + b^5)