\(\int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx\) [188]

Optimal result

Integrand size = 13, antiderivative size = 145 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=-\frac {a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}-\frac {2 \left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^6}+\frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5} \]

[Out]

-1/8*a*(8*a^4+20*a^2*b^2+15*b^4)*x/b^6-2*(a^2+b^2)^(5/2)*arctanh((b-a*tanh(1/2*x))/(a^2+b^2)^(1/2))/b^6+1/5*co
sh(x)^5/b+1/12*cosh(x)^3*(4*a^2+4*b^2-3*a*b*sinh(x))/b^3+1/8*cosh(x)*(8*(a^2+b^2)^2-a*b*(4*a^2+7*b^2)*sinh(x))
/b^5

Rubi [A] (verified)

Time = 0.29 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.462, Rules used = {2774, 2944, 2814, 2739, 632, 212} \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=-\frac {2 \left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^6}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}-\frac {a x \left (8 a^4+20 a^2 b^2+15 b^4\right )}{8 b^6}+\frac {\cosh ^5(x)}{5 b} \]

[In]

Int[Cosh[x]^6/(a + b*Sinh[x]),x]

[Out]

-1/8*(a*(8*a^4 + 20*a^2*b^2 + 15*b^4)*x)/b^6 - (2*(a^2 + b^2)^(5/2)*ArcTanh[(b - a*Tanh[x/2])/Sqrt[a^2 + b^2]]
)/b^6 + Cosh[x]^5/(5*b) + (Cosh[x]^3*(4*(a^2 + b^2) - 3*a*b*Sinh[x]))/(12*b^3) + (Cosh[x]*(8*(a^2 + b^2)^2 - a
*b*(4*a^2 + 7*b^2)*Sinh[x]))/(8*b^5)

Rule 212

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))*ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 632

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 2739

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 2774

Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_), x_Symbol] :> Simp[g*(g*C
os[e + f*x])^(p - 1)*((a + b*Sin[e + f*x])^(m + 1)/(b*f*(m + p))), x] + Dist[g^2*((p - 1)/(b*(m + p))), Int[(g
*Cos[e + f*x])^(p - 2)*(a + b*Sin[e + f*x])^m*(b + a*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, m}, x] &&
NeQ[a^2 - b^2, 0] && GtQ[p, 1] && NeQ[m + p, 0] && IntegersQ[2*m, 2*p]

Rule 2814

Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[b*(x/d)
, x] - Dist[(b*c - a*d)/d, Int[1/(c + d*Sin[e + f*x]), x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d
, 0]

Rule 2944

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

Rubi steps \begin{align*} \text {integral}& = \frac {\cosh ^5(x)}{5 b}+\frac {i \int \frac {\cosh ^4(x) (-i b+i a \sinh (x))}{a+b \sinh (x)} \, dx}{b} \\ & = \frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}-\frac {i \int \frac {\cosh ^2(x) \left (i b \left (a^2+4 b^2\right )-i a \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)} \, dx}{4 b^3} \\ & = \frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}+\frac {i \int \frac {-i b \left (4 a^4+9 a^2 b^2+8 b^4\right )+i a \left (8 a^4+20 a^2 b^2+15 b^4\right ) \sinh (x)}{a+b \sinh (x)} \, dx}{8 b^5} \\ & = -\frac {a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}+\frac {\left (a^2+b^2\right )^3 \int \frac {1}{a+b \sinh (x)} \, dx}{b^6} \\ & = -\frac {a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}+\frac {\left (2 \left (a^2+b^2\right )^3\right ) \text {Subst}\left (\int \frac {1}{a+2 b x-a x^2} \, dx,x,\tanh \left (\frac {x}{2}\right )\right )}{b^6} \\ & = -\frac {a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}+\frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5}-\frac {\left (4 \left (a^2+b^2\right )^3\right ) \text {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 )}{b^6} \\ & = -\frac {a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}-\frac {2 \left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^6}+\frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5} \\ \end{align*}

Mathematica [C] (verified)

Result contains complex when optimal does not.

Time = 4.68 (sec) , antiderivative size = 463, normalized size of antiderivative = 3.19 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\frac {\cosh (x) \left (8 \left (15 a^4+35 a^2 b^2+23 b^4\right )-15 a b \left (4 a^2+9 b^2\right ) \sinh (x)+8 b^2 \left (5 a^2+11 b^2\right ) \sinh ^2(x)-30 a b^3 \sinh ^3(x)+24 b^4 \sinh ^4(x)-\frac {30 (-1)^{3/4} \sqrt {b} \left (8 a^4-4 i a^3 b+16 a^2 b^2-7 i a b^3+8 b^4\right ) \arcsin \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a-i b} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {b}}\right )}{\sqrt {a-i b} \sqrt {1+i \sinh (x)} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}-\frac {240 \left (a^2+b^2\right )^2 \text {arctanh}\left (\frac {\sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {-\frac {b (-i+\sinh (x))}{a+i b}}}\right )}{\sqrt {-\frac {b (-i+\sinh (x))}{a+i b}} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}+\frac {240 (a-i b)^{5/2} (a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a-i b} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {a+i b} \sqrt {-\frac {b (-i+\sinh (x))}{a+i b}}}\right )}{\sqrt {-\frac {b (-i+\sinh (x))}{a+i b}} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}\right )}{120 b^5} \]

[In]

Integrate[Cosh[x]^6/(a + b*Sinh[x]),x]

[Out]

(Cosh[x]*(8*(15*a^4 + 35*a^2*b^2 + 23*b^4) - 15*a*b*(4*a^2 + 9*b^2)*Sinh[x] + 8*b^2*(5*a^2 + 11*b^2)*Sinh[x]^2
- 30*a*b^3*Sinh[x]^3 + 24*b^4*Sinh[x]^4 - (30*(-1)^(3/4)*Sqrt[b]*(8*a^4 - (4*I)*a^3*b + 16*a^2*b^2 - (7*I)*a*
b^3 + 8*b^4)*ArcSin[((1/2 + I/2)*Sqrt[a - I*b]*Sqrt[-((b*(I + Sinh[x]))/(a - I*b))])/Sqrt[b]])/(Sqrt[a - I*b]*
Sqrt[1 + I*Sinh[x]]*Sqrt[-((b*(I + Sinh[x]))/(a - I*b))]) - (240*(a^2 + b^2)^2*ArcTanh[Sqrt[-((b*(I + Sinh[x])
)/(a - I*b))]/Sqrt[-((b*(-I + Sinh[x]))/(a + I*b))]])/(Sqrt[-((b*(-I + Sinh[x]))/(a + I*b))]*Sqrt[-((b*(I + Si
nh[x]))/(a - I*b))]) + (240*(a - I*b)^(5/2)*(a + I*b)^(3/2)*ArcTanh[(Sqrt[a - I*b]*Sqrt[-((b*(I + Sinh[x]))/(a
- I*b))])/(Sqrt[a + I*b]*Sqrt[-((b*(-I + Sinh[x]))/(a + I*b))])])/(Sqrt[-((b*(-I + Sinh[x]))/(a + I*b))]*Sqrt
[-((b*(I + Sinh[x]))/(a - I*b))])))/(120*b^5)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(343\) vs. \(2(132)=264\).

Time = 57.61 (sec) , antiderivative size = 344, normalized size of antiderivative = 2.37

method	result	size
risch	\(-\frac {a^{5} x}{b^{6}}-\frac {5 a^{3} x}{2 b^{4}}-\frac {15 a x}{8 b^{2}}+\frac {{\mathrm e}^{5 x}}{160 b}-\frac {a \,{\mathrm e}^{4 x}}{64 b^{2}}+\frac {{\mathrm e}^{3 x} a^{2}}{24 b^{3}}+\frac {7 \,{\mathrm e}^{3 x}}{96 b}-\frac {a^{3} {\mathrm e}^{2 x}}{8 b^{4}}-\frac {a \,{\mathrm e}^{2 x}}{4 b^{2}}+\frac {{\mathrm e}^{x} a^{4}}{2 b^{5}}+\frac {9 \,{\mathrm e}^{x} a^{2}}{8 b^{3}}+\frac {11 \,{\mathrm e}^{x}}{16 b}+\frac {{\mathrm e}^{-x} a^{4}}{2 b^{5}}+\frac {9 \,{\mathrm e}^{-x} a^{2}}{8 b^{3}}+\frac {11 \,{\mathrm e}^{-x}}{16 b}+\frac {a^{3} {\mathrm e}^{-2 x}}{8 b^{4}}+\frac {a \,{\mathrm e}^{-2 x}}{4 b^{2}}+\frac {{\mathrm e}^{-3 x} a^{2}}{24 b^{3}}+\frac {7 \,{\mathrm e}^{-3 x}}{96 b}+\frac {a \,{\mathrm e}^{-4 x}}{64 b^{2}}+\frac {{\mathrm e}^{-5 x}}{160 b}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \ln \left ({\mathrm e}^{x}-\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}}-a^{5}-2 a^{3} b^{2}-a \,b^{4}}{b \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\right )}{b^{6}}-\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}} \ln \left ({\mathrm e}^{x}+\frac {\left (a^{2}+b^{2}\right )^{\frac {5}{2}}+a^{5}+2 a^{3} b^{2}+a \,b^{4}}{b \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}\right )}{b^{6}}\)	\(344\)
default	\(-\frac {2 \left (-a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{6} \sqrt {a^{2}+b^{2}}}-\frac {1}{5 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {2 b +a}{4 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {4 a^{2}+6 a b +13 b^{2}}{12 b^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {4 a^{3}+4 a^{2} b +11 a \,b^{2}+9 b^{3}}{8 b^{4} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {8 a^{4}+4 a^{3} b +20 a^{2} b^{2}+9 b^{3} a +15 b^{4}}{8 b^{5} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \left (8 a^{4}+20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 b^{6}}+\frac {1}{5 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {2 b -a}{4 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {-4 a^{2}+6 a b -13 b^{2}}{12 b^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-4 a^{3}+4 a^{2} b -11 a \,b^{2}+9 b^{3}}{8 b^{4} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {-8 a^{4}+4 a^{3} b -20 a^{2} b^{2}+9 b^{3} a -15 b^{4}}{8 b^{5} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \left (8 a^{4}+20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 b^{6}}\)	\(410\)

[In]

int(cosh(x)^6/(a+b*sinh(x)),x,method=_RETURNVERBOSE)

[Out]

-a^5*x/b^6-5/2*a^3*x/b^4-15/8*a*x/b^2+1/160/b*exp(x)^5-1/64*a/b^2*exp(x)^4+1/24/b^3*exp(x)^3*a^2+7/96/b*exp(x)
^3-1/8*a^3/b^4*exp(x)^2-1/4*a/b^2*exp(x)^2+1/2/b^5*exp(x)*a^4+9/8/b^3*exp(x)*a^2+11/16/b*exp(x)+1/2/b^5/exp(x)
*a^4+9/8/b^3/exp(x)*a^2+11/16/b/exp(x)+1/8*a^3/b^4/exp(x)^2+1/4*a/b^2/exp(x)^2+1/24/b^3/exp(x)^3*a^2+7/96/b/ex
p(x)^3+1/64*a/b^2/exp(x)^4+1/160/b/exp(x)^5+(a^2+b^2)^(5/2)/b^6*ln(exp(x)-((a^2+b^2)^(5/2)-a^5-2*a^3*b^2-a*b^4
)/b/(a^4+2*a^2*b^2+b^4))-(a^2+b^2)^(5/2)/b^6*ln(exp(x)+((a^2+b^2)^(5/2)+a^5+2*a^3*b^2+a*b^4)/b/(a^4+2*a^2*b^2+
b^4))

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1486 vs. \(2 (133) = 266\).

Time = 0.32 (sec) , antiderivative size = 1486, normalized size of antiderivative = 10.25 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\text {Too large to display} \]

[In]

integrate(cosh(x)^6/(a+b*sinh(x)),x, algorithm="fricas")

[Out]

1/960*(6*b^5*cosh(x)^10 + 6*b^5*sinh(x)^10 - 15*a*b^4*cosh(x)^9 + 15*(4*b^5*cosh(x) - a*b^4)*sinh(x)^9 + 10*(4
*a^2*b^3 + 7*b^5)*cosh(x)^8 + 5*(54*b^5*cosh(x)^2 - 27*a*b^4*cosh(x) + 8*a^2*b^3 + 14*b^5)*sinh(x)^8 - 120*(a^
3*b^2 + 2*a*b^4)*cosh(x)^7 + 20*(36*b^5*cosh(x)^3 - 27*a*b^4*cosh(x)^2 - 6*a^3*b^2 - 12*a*b^4 + 4*(4*a^2*b^3 +
7*b^5)*cosh(x))*sinh(x)^7 - 120*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*x*cosh(x)^5 + 60*(8*a^4*b + 18*a^2*b^3 + 11*b
^5)*cosh(x)^6 + 20*(63*b^5*cosh(x)^4 - 63*a*b^4*cosh(x)^3 + 24*a^4*b + 54*a^2*b^3 + 33*b^5 + 14*(4*a^2*b^3 + 7
*b^5)*cosh(x)^2 - 42*(a^3*b^2 + 2*a*b^4)*cosh(x))*sinh(x)^6 + 15*a*b^4*cosh(x) + 2*(756*b^5*cosh(x)^5 - 945*a*
b^4*cosh(x)^4 + 280*(4*a^2*b^3 + 7*b^5)*cosh(x)^3 - 1260*(a^3*b^2 + 2*a*b^4)*cosh(x)^2 - 60*(8*a^5 + 20*a^3*b^
2 + 15*a*b^4)*x + 180*(8*a^4*b + 18*a^2*b^3 + 11*b^5)*cosh(x))*sinh(x)^5 + 6*b^5 + 60*(8*a^4*b + 18*a^2*b^3 +
11*b^5)*cosh(x)^4 + 10*(126*b^5*cosh(x)^6 - 189*a*b^4*cosh(x)^5 + 48*a^4*b + 108*a^2*b^3 + 66*b^5 + 70*(4*a^2*
b^3 + 7*b^5)*cosh(x)^4 - 420*(a^3*b^2 + 2*a*b^4)*cosh(x)^3 - 60*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*x*cosh(x) + 90
*(8*a^4*b + 18*a^2*b^3 + 11*b^5)*cosh(x)^2)*sinh(x)^4 + 120*(a^3*b^2 + 2*a*b^4)*cosh(x)^3 + 20*(36*b^5*cosh(x)
^7 - 63*a*b^4*cosh(x)^6 + 28*(4*a^2*b^3 + 7*b^5)*cosh(x)^5 + 6*a^3*b^2 + 12*a*b^4 - 210*(a^3*b^2 + 2*a*b^4)*co
sh(x)^4 - 60*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*x*cosh(x)^2 + 60*(8*a^4*b + 18*a^2*b^3 + 11*b^5)*cosh(x)^3 + 12*(
8*a^4*b + 18*a^2*b^3 + 11*b^5)*cosh(x))*sinh(x)^3 + 10*(4*a^2*b^3 + 7*b^5)*cosh(x)^2 + 10*(27*b^5*cosh(x)^8 -
54*a*b^4*cosh(x)^7 + 28*(4*a^2*b^3 + 7*b^5)*cosh(x)^6 - 252*(a^3*b^2 + 2*a*b^4)*cosh(x)^5 + 4*a^2*b^3 + 7*b^5
- 120*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*x*cosh(x)^3 + 90*(8*a^4*b + 18*a^2*b^3 + 11*b^5)*cosh(x)^4 + 36*(8*a^4*b
+ 18*a^2*b^3 + 11*b^5)*cosh(x)^2 + 36*(a^3*b^2 + 2*a*b^4)*cosh(x))*sinh(x)^2 + 960*((a^4 + 2*a^2*b^2 + b^4)*c
osh(x)^5 + 5*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^4*sinh(x) + 10*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)^3*sinh(x)^2 + 10*(
a^4 + 2*a^2*b^2 + b^4)*cosh(x)^2*sinh(x)^3 + 5*(a^4 + 2*a^2*b^2 + b^4)*cosh(x)*sinh(x)^4 + (a^4 + 2*a^2*b^2 +
b^4)*sinh(x)^5)*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)*sinh(x) - b)) + 5*(12*b^5*cosh(x)^9 - 27*a*b^4*cosh(x)^8 + 16*(4*a^2*b^3 + 7*b^5)*cosh(x)^7
- 168*(a^3*b^2 + 2*a*b^4)*cosh(x)^6 - 120*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*x*cosh(x)^4 + 72*(8*a^4*b + 18*a^2*
b^3 + 11*b^5)*cosh(x)^5 + 3*a*b^4 + 48*(8*a^4*b + 18*a^2*b^3 + 11*b^5)*cosh(x)^3 + 72*(a^3*b^2 + 2*a*b^4)*cosh
(x)^2 + 4*(4*a^2*b^3 + 7*b^5)*cosh(x))*sinh(x))/(b^6*cosh(x)^5 + 5*b^6*cosh(x)^4*sinh(x) + 10*b^6*cosh(x)^3*si
nh(x)^2 + 10*b^6*cosh(x)^2*sinh(x)^3 + 5*b^6*cosh(x)*sinh(x)^4 + b^6*sinh(x)^5)

Sympy [F(-1)]

Timed out. \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\text {Timed out} \]

[In]

integrate(cosh(x)**6/(a+b*sinh(x)),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (133) = 266\).

Time = 0.30 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.95 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=-\frac {{\left (15 \, a b^{3} e^{\left (-x\right )} - 6 \, b^{4} - 10 \, {\left (4 \, a^{2} b^{2} + 7 \, b^{4}\right )} e^{\left (-2 \, x\right )} + 120 \, {\left (a^{3} b + 2 \, a b^{3}\right )} e^{\left (-3 \, x\right )} - 60 \, {\left (8 \, a^{4} + 18 \, a^{2} b^{2} + 11 \, b^{4}\right )} e^{\left (-4 \, x\right )}\right )} e^{\left (5 \, x\right )}}{960 \, b^{5}} + \frac {15 \, a b^{3} e^{\left (-4 \, x\right )} + 6 \, b^{4} e^{\left (-5 \, x\right )} + 60 \, {\left (8 \, a^{4} + 18 \, a^{2} b^{2} + 11 \, b^{4}\right )} e^{\left (-x\right )} + 120 \, {\left (a^{3} b + 2 \, a b^{3}\right )} e^{\left (-2 \, x\right )} + 10 \, {\left (4 \, a^{2} b^{2} + 7 \, b^{4}\right )} e^{\left (-3 \, x\right )}}{960 \, b^{5}} - \frac {{\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x}{8 \, b^{6}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b^{6}} \]

[In]

integrate(cosh(x)^6/(a+b*sinh(x)),x, algorithm="maxima")

[Out]

-1/960*(15*a*b^3*e^(-x) - 6*b^4 - 10*(4*a^2*b^2 + 7*b^4)*e^(-2*x) + 120*(a^3*b + 2*a*b^3)*e^(-3*x) - 60*(8*a^4
+ 18*a^2*b^2 + 11*b^4)*e^(-4*x))*e^(5*x)/b^5 + 1/960*(15*a*b^3*e^(-4*x) + 6*b^4*e^(-5*x) + 60*(8*a^4 + 18*a^2
*b^2 + 11*b^4)*e^(-x) + 120*(a^3*b + 2*a*b^3)*e^(-2*x) + 10*(4*a^2*b^2 + 7*b^4)*e^(-3*x))/b^5 - 1/8*(8*a^5 + 2
0*a^3*b^2 + 15*a*b^4)*x/b^6 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*log((b*e^(-x) - a - sqrt(a^2 + b^2))/(b*e^(-
x) - a + sqrt(a^2 + b^2)))/(sqrt(a^2 + b^2)*b^6)

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (133) = 266\).

Time = 0.28 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.99 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\frac {6 \, b^{4} e^{\left (5 \, x\right )} - 15 \, a b^{3} e^{\left (4 \, x\right )} + 40 \, a^{2} b^{2} e^{\left (3 \, x\right )} + 70 \, b^{4} e^{\left (3 \, x\right )} - 120 \, a^{3} b e^{\left (2 \, x\right )} - 240 \, a b^{3} e^{\left (2 \, x\right )} + 480 \, a^{4} e^{x} + 1080 \, a^{2} b^{2} e^{x} + 660 \, b^{4} e^{x}}{960 \, b^{5}} - \frac {{\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x}{8 \, b^{6}} + \frac {{\left (15 \, a b^{4} e^{x} + 6 \, b^{5} + 60 \, {\left (8 \, a^{4} b + 18 \, a^{2} b^{3} + 11 \, b^{5}\right )} e^{\left (4 \, x\right )} + 120 \, {\left (a^{3} b^{2} + 2 \, a b^{4}\right )} e^{\left (3 \, x\right )} + 10 \, {\left (4 \, a^{2} b^{3} + 7 \, b^{5}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-5 \, x\right )}}{960 \, b^{6}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{6}} \]

[In]

integrate(cosh(x)^6/(a+b*sinh(x)),x, algorithm="giac")

[Out]

1/960*(6*b^4*e^(5*x) - 15*a*b^3*e^(4*x) + 40*a^2*b^2*e^(3*x) + 70*b^4*e^(3*x) - 120*a^3*b*e^(2*x) - 240*a*b^3*
e^(2*x) + 480*a^4*e^x + 1080*a^2*b^2*e^x + 660*b^4*e^x)/b^5 - 1/8*(8*a^5 + 20*a^3*b^2 + 15*a*b^4)*x/b^6 + 1/96
0*(15*a*b^4*e^x + 6*b^5 + 60*(8*a^4*b + 18*a^2*b^3 + 11*b^5)*e^(4*x) + 120*(a^3*b^2 + 2*a*b^4)*e^(3*x) + 10*(4
*a^2*b^3 + 7*b^5)*e^(2*x))*e^(-5*x)/b^6 + (a^6 + 3*a^4*b^2 + 3*a^2*b^4 + b^6)*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)))/(sqrt(a^2 + b^2)*b^6)

Mupad [B] (verification not implemented)

Time = 2.01 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.08 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\frac {{\mathrm {e}}^{-5\,x}}{160\,b}+\frac {{\mathrm {e}}^{5\,x}}{160\,b}-\frac {\ln \left (-\frac {2\,{\mathrm {e}}^x\,{\left (a^2+b^2\right )}^3}{b^7}-\frac {2\,\left (b-a\,{\mathrm {e}}^x\right )\,{\left (a^2+b^2\right )}^{5/2}}{b^7}\right )\,{\left (a^2+b^2\right )}^{5/2}}{b^6}+\frac {\ln \left (\frac {2\,\left (b-a\,{\mathrm {e}}^x\right )\,{\left (a^2+b^2\right )}^{5/2}}{b^7}-\frac {2\,{\mathrm {e}}^x\,{\left (a^2+b^2\right )}^3}{b^7}\right )\,{\left (a^2+b^2\right )}^{5/2}}{b^6}-\frac {x\,\left (8\,a^5+20\,a^3\,b^2+15\,a\,b^4\right )}{8\,b^6}+\frac {{\mathrm {e}}^x\,\left (8\,a^4+18\,a^2\,b^2+11\,b^4\right )}{16\,b^5}+\frac {a\,{\mathrm {e}}^{-4\,x}}{64\,b^2}-\frac {a\,{\mathrm {e}}^{4\,x}}{64\,b^2}+\frac {{\mathrm {e}}^{-x}\,\left (8\,a^4+18\,a^2\,b^2+11\,b^4\right )}{16\,b^5}+\frac {{\mathrm {e}}^{-3\,x}\,\left (4\,a^2+7\,b^2\right )}{96\,b^3}+\frac {{\mathrm {e}}^{3\,x}\,\left (4\,a^2+7\,b^2\right )}{96\,b^3}+\frac {{\mathrm {e}}^{-2\,x}\,\left (a^3+2\,a\,b^2\right )}{8\,b^4}-\frac {{\mathrm {e}}^{2\,x}\,\left (a^3+2\,a\,b^2\right )}{8\,b^4} \]

[In]

int(cosh(x)^6/(a + b*sinh(x)),x)

[Out]

exp(-5*x)/(160*b) + exp(5*x)/(160*b) - (log(- (2*exp(x)*(a^2 + b^2)^3)/b^7 - (2*(b - a*exp(x))*(a^2 + b^2)^(5/
2))/b^7)*(a^2 + b^2)^(5/2))/b^6 + (log((2*(b - a*exp(x))*(a^2 + b^2)^(5/2))/b^7 - (2*exp(x)*(a^2 + b^2)^3)/b^7
)*(a^2 + b^2)^(5/2))/b^6 - (x*(15*a*b^4 + 8*a^5 + 20*a^3*b^2))/(8*b^6) + (exp(x)*(8*a^4 + 11*b^4 + 18*a^2*b^2)
)/(16*b^5) + (a*exp(-4*x))/(64*b^2) - (a*exp(4*x))/(64*b^2) + (exp(-x)*(8*a^4 + 11*b^4 + 18*a^2*b^2))/(16*b^5)
+ (exp(-3*x)*(4*a^2 + 7*b^2))/(96*b^3) + (exp(3*x)*(4*a^2 + 7*b^2))/(96*b^3) + (exp(-2*x)*(2*a*b^2 + a^3))/(8
*b^4) - (exp(2*x)*(2*a*b^2 + a^3))/(8*b^4)