Integrand size = 13, antiderivative size = 145 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=-\frac {a \left (8 a^4+20 a^2 b^2+15 b^4\right ) x}{8 b^6}-\frac {2 \left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {b-a \tanh \left (\frac {x}{2}\right )}{\sqrt {a^2+b^2}}\right )}{b^6}+\frac {\cosh ^5(x)}{5 b}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^3}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{8 b^5} \] Output:
-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*cosh(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
Result contains complex when optimal does not.
Time = 4.41 (sec) , antiderivative size = 463, normalized size of antiderivative = 3.19 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\frac {\cosh (x) \left (8 \left (15 a^4+35 a^2 b^2+23 b^4\right )-15 a b \left (4 a^2+9 b^2\right ) \sinh (x)+8 b^2 \left (5 a^2+11 b^2\right ) \sinh ^2(x)-30 a b^3 \sinh ^3(x)+24 b^4 \sinh ^4(x)-\frac {30 (-1)^{3/4} \sqrt {b} \left (8 a^4-4 i a^3 b+16 a^2 b^2-7 i a b^3+8 b^4\right ) \arcsin \left (\frac {\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt {a-i b} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {b}}\right )}{\sqrt {a-i b} \sqrt {1+i \sinh (x)} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}-\frac {240 \left (a^2+b^2\right )^2 \text {arctanh}\left (\frac {\sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {-\frac {b (-i+\sinh (x))}{a+i b}}}\right )}{\sqrt {-\frac {b (-i+\sinh (x))}{a+i b}} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}+\frac {240 (a-i b)^{5/2} (a+i b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a-i b} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}{\sqrt {a+i b} \sqrt {-\frac {b (-i+\sinh (x))}{a+i b}}}\right )}{\sqrt {-\frac {b (-i+\sinh (x))}{a+i b}} \sqrt {-\frac {b (i+\sinh (x))}{a-i b}}}\right )}{120 b^5} \] Input:
Integrate[Cosh[x]^6/(a + b*Sinh[x]),x]
Output:
(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 + Sinh[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 )
Time = 0.98 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.17, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.154, Rules used = {3042, 3174, 26, 3042, 3344, 25, 3042, 3344, 25, 3042, 3214, 3042, 3139, 1083, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\cos (i x)^6}{a-i b \sin (i x)}dx\) |
| \(\Big \downarrow \) 3174 |
| \(\displaystyle \frac {\cosh ^5(x)}{5 b}+\frac {i \int -\frac {i \cosh ^4(x) (b-a \sinh (x))}{a+b \sinh (x)}dx}{b}\) |
| \(\Big \downarrow \) 26 |
| \(\displaystyle \frac {\int \frac {\cosh ^4(x) (b-a \sinh (x))}{a+b \sinh (x)}dx}{b}+\frac {\cosh ^5(x)}{5 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cosh ^5(x)}{5 b}+\frac {\int \frac {\cos (i x)^4 (b+i a \sin (i x))}{a-i b \sin (i x)}dx}{b}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^2}-\frac {\int -\frac {\cosh ^2(x) \left (b \left (a^2+4 b^2\right )-a \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)}dx}{4 b^2}}{b}+\frac {\cosh ^5(x)}{5 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\cosh ^2(x) \left (b \left (a^2+4 b^2\right )-a \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{a+b \sinh (x)}dx}{4 b^2}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^2}}{b}+\frac {\cosh ^5(x)}{5 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cosh ^5(x)}{5 b}+\frac {\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^2}+\frac {\int \frac {\cos (i x)^2 \left (b \left (a^2+4 b^2\right )+i a \left (4 a^2+7 b^2\right ) \sin (i x)\right )}{a-i b \sin (i x)}dx}{4 b^2}}{b}\) |
| \(\Big \downarrow \) 3344 |
| \(\displaystyle \frac {\frac {\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{2 b^2}-\frac {\int -\frac {b \left (4 a^4+9 b^2 a^2+8 b^4\right )-a \left (8 a^4+20 b^2 a^2+15 b^4\right ) \sinh (x)}{a+b \sinh (x)}dx}{2 b^2}}{4 b^2}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^2}}{b}+\frac {\cosh ^5(x)}{5 b}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {b \left (4 a^4+9 b^2 a^2+8 b^4\right )-a \left (8 a^4+20 b^2 a^2+15 b^4\right ) \sinh (x)}{a+b \sinh (x)}dx}{2 b^2}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{2 b^2}}{4 b^2}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^2}}{b}+\frac {\cosh ^5(x)}{5 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cosh ^5(x)}{5 b}+\frac {\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^2}+\frac {\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{2 b^2}+\frac {\int \frac {b \left (4 a^4+9 b^2 a^2+8 b^4\right )+i a \left (8 a^4+20 b^2 a^2+15 b^4\right ) \sin (i x)}{a-i b \sin (i x)}dx}{2 b^2}}{4 b^2}}{b}\) |
| \(\Big \downarrow \) 3214 |
| \(\displaystyle \frac {\frac {\frac {\frac {8 \left (a^2+b^2\right )^3 \int \frac {1}{a+b \sinh (x)}dx}{b}-\frac {a x \left (8 a^4+20 a^2 b^2+15 b^4\right )}{b}}{2 b^2}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{2 b^2}}{4 b^2}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^2}}{b}+\frac {\cosh ^5(x)}{5 b}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\cosh ^5(x)}{5 b}+\frac {\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^2}+\frac {\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{2 b^2}+\frac {-\frac {a x \left (8 a^4+20 a^2 b^2+15 b^4\right )}{b}+\frac {8 \left (a^2+b^2\right )^3 \int \frac {1}{a-i b \sin (i x)}dx}{b}}{2 b^2}}{4 b^2}}{b}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {\frac {\frac {16 \left (a^2+b^2\right )^3 \int \frac {1}{-a \tanh ^2\left (\frac {x}{2}\right )+2 b \tanh \left (\frac {x}{2}\right )+a}d\tanh \left (\frac {x}{2}\right )}{b}-\frac {a x \left (8 a^4+20 a^2 b^2+15 b^4\right )}{b}}{2 b^2}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{2 b^2}}{4 b^2}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^2}}{b}+\frac {\cosh ^5(x)}{5 b}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {-\frac {32 \left (a^2+b^2\right )^3 \int \frac {1}{4 \left (a^2+b^2\right )-\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )^2}d\left (2 b-2 a \tanh \left (\frac {x}{2}\right )\right )}{b}-\frac {a x \left (8 a^4+20 a^2 b^2+15 b^4\right )}{b}}{2 b^2}+\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{2 b^2}}{4 b^2}+\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^2}}{b}+\frac {\cosh ^5(x)}{5 b}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {\frac {\cosh ^3(x) \left (4 \left (a^2+b^2\right )-3 a b \sinh (x)\right )}{12 b^2}+\frac {\frac {\cosh (x) \left (8 \left (a^2+b^2\right )^2-a b \left (4 a^2+7 b^2\right ) \sinh (x)\right )}{2 b^2}+\frac {-\frac {16 \left (a^2+b^2\right )^{5/2} \text {arctanh}\left (\frac {2 b-2 a \tanh \left (\frac {x}{2}\right )}{2 \sqrt {a^2+b^2}}\right )}{b}-\frac {a x \left (8 a^4+20 a^2 b^2+15 b^4\right )}{b}}{2 b^2}}{4 b^2}}{b}+\frac {\cosh ^5(x)}{5 b}\) |
Input:
Int[Cosh[x]^6/(a + b*Sinh[x]),x]
Output:
Cosh[x]^5/(5*b) + ((Cosh[x]^3*(4*(a^2 + b^2) - 3*a*b*Sinh[x]))/(12*b^2) + ((-((a*(8*a^4 + 20*a^2*b^2 + 15*b^4)*x)/b) - (16*(a^2 + b^2)^(5/2)*ArcTanh [(2*b - 2*a*Tanh[x/2])/(2*Sqrt[a^2 + b^2])])/b)/(2*b^2) + (Cosh[x]*(8*(a^2 + b^2)^2 - a*b*(4*a^2 + 7*b^2)*Sinh[x]))/(2*b^2))/(4*b^2))/b
Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a]) I nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
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] && (Gt Q[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[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]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[g*(g*Cos[e + f*x])^(p - 1)*((a + b*Sin[e + f*x ])^(m + 1)/(b*f*(m + p))), x] + Simp[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] /; F reeQ[{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]
Int[((a_.) + (b_.)*sin[(e_.) + (f_.)*(x_)])/((c_.) + (d_.)*sin[(e_.) + (f_. )*(x_)]), x_Symbol] :> Simp[b*(x/d), x] - Simp[(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]
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] + Simp[g^2*( (p - 1)/(b^2*(m + p)*(m + p + 1))) Int[(g*Cos[e + f*x])^(p - 2)*(a + b*Si n[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]
Leaf count of result is larger than twice the leaf count of optimal. \(343\) vs. \(2(132)=264\).
Time = 55.25 (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 | \(\frac {1}{5 b \left (\tanh \left (\frac {x}{2}\right )+1\right )^{5}}-\frac {2 b -a}{4 b^{2} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{4}}-\frac {-4 a^{2}+6 a b -13 b^{2}}{12 b^{3} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{3}}-\frac {-4 a^{3}+4 a^{2} b -11 a \,b^{2}+9 b^{3}}{8 b^{4} \left (\tanh \left (\frac {x}{2}\right )+1\right )^{2}}-\frac {-8 a^{4}+4 a^{3} b -20 a^{2} b^{2}+9 a \,b^{3}-15 b^{4}}{8 b^{5} \left (\tanh \left (\frac {x}{2}\right )+1\right )}-\frac {a \left (8 a^{4}+20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )+1\right )}{8 b^{6}}-\frac {2 \left (-a^{6}-3 a^{4} b^{2}-3 a^{2} b^{4}-b^{6}\right ) \operatorname {arctanh}\left (\frac {2 a \tanh \left (\frac {x}{2}\right )-2 b}{2 \sqrt {a^{2}+b^{2}}}\right )}{b^{6} \sqrt {a^{2}+b^{2}}}-\frac {1}{5 b \left (\tanh \left (\frac {x}{2}\right )-1\right )^{5}}-\frac {2 b +a}{4 b^{2} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{4}}-\frac {4 a^{2}+6 a b +13 b^{2}}{12 b^{3} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{3}}-\frac {4 a^{3}+4 a^{2} b +11 a \,b^{2}+9 b^{3}}{8 b^{4} \left (\tanh \left (\frac {x}{2}\right )-1\right )^{2}}-\frac {8 a^{4}+4 a^{3} b +20 a^{2} b^{2}+9 a \,b^{3}+15 b^{4}}{8 b^{5} \left (\tanh \left (\frac {x}{2}\right )-1\right )}+\frac {a \left (8 a^{4}+20 a^{2} b^{2}+15 b^{4}\right ) \ln \left (\tanh \left (\frac {x}{2}\right )-1\right )}{8 b^{6}}\) | \(410\) |
Input:
int(cosh(x)^6/(a+b*sinh(x)),x,method=_RETURNVERBOSE)
Output:
-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/exp(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))
Leaf count of result is larger than twice the leaf count of optimal. 1486 vs. \(2 (133) = 266\).
Time = 0.11 (sec) , antiderivative size = 1486, normalized size of antiderivative = 10.25 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\text {Too large to display} \] Input:
integrate(cosh(x)^6/(a+b*sinh(x)),x, algorithm="fricas")
Output:
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)*c osh(x))*sinh(x)^6 + 15*a*b^4*cosh(x) + 2*(756*b^5*cosh(x)^5 - 945*a*b^4*co sh(x)^4 + 280*(4*a^2*b^3 + 7*b^5)*cosh(x)^3 - 1260*(a^3*b^2 + 2*a*b^4)*cos h(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)*cosh(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)...
Timed out. \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\text {Timed out} \] Input:
integrate(cosh(x)**6/(a+b*sinh(x)),x)
Output:
Timed out
Leaf count of result is larger than twice the leaf count of optimal. 283 vs. \(2 (133) = 266\).
Time = 0.12 (sec) , antiderivative size = 283, normalized size of antiderivative = 1.95 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=-\frac {{\left (15 \, a b^{3} e^{\left (-x\right )} - 6 \, b^{4} - 10 \, {\left (4 \, a^{2} b^{2} + 7 \, b^{4}\right )} e^{\left (-2 \, x\right )} + 120 \, {\left (a^{3} b + 2 \, a b^{3}\right )} e^{\left (-3 \, x\right )} - 60 \, {\left (8 \, a^{4} + 18 \, a^{2} b^{2} + 11 \, b^{4}\right )} e^{\left (-4 \, x\right )}\right )} e^{\left (5 \, x\right )}}{960 \, b^{5}} + \frac {15 \, a b^{3} e^{\left (-4 \, x\right )} + 6 \, b^{4} e^{\left (-5 \, x\right )} + 60 \, {\left (8 \, a^{4} + 18 \, a^{2} b^{2} + 11 \, b^{4}\right )} e^{\left (-x\right )} + 120 \, {\left (a^{3} b + 2 \, a b^{3}\right )} e^{\left (-2 \, x\right )} + 10 \, {\left (4 \, a^{2} b^{2} + 7 \, b^{4}\right )} e^{\left (-3 \, x\right )}}{960 \, b^{5}} - \frac {{\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x}{8 \, b^{6}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac {b e^{\left (-x\right )} - a - \sqrt {a^{2} + b^{2}}}{b e^{\left (-x\right )} - a + \sqrt {a^{2} + b^{2}}}\right )}{\sqrt {a^{2} + b^{2}} b^{6}} \] Input:
integrate(cosh(x)^6/(a+b*sinh(x)),x, algorithm="maxima")
Output:
-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 + 20*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)
Leaf count of result is larger than twice the leaf count of optimal. 288 vs. \(2 (133) = 266\).
Time = 0.15 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.99 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\frac {6 \, b^{4} e^{\left (5 \, x\right )} - 15 \, a b^{3} e^{\left (4 \, x\right )} + 40 \, a^{2} b^{2} e^{\left (3 \, x\right )} + 70 \, b^{4} e^{\left (3 \, x\right )} - 120 \, a^{3} b e^{\left (2 \, x\right )} - 240 \, a b^{3} e^{\left (2 \, x\right )} + 480 \, a^{4} e^{x} + 1080 \, a^{2} b^{2} e^{x} + 660 \, b^{4} e^{x}}{960 \, b^{5}} - \frac {{\left (8 \, a^{5} + 20 \, a^{3} b^{2} + 15 \, a b^{4}\right )} x}{8 \, b^{6}} + \frac {{\left (15 \, a b^{4} e^{x} + 6 \, b^{5} + 60 \, {\left (8 \, a^{4} b + 18 \, a^{2} b^{3} + 11 \, b^{5}\right )} e^{\left (4 \, x\right )} + 120 \, {\left (a^{3} b^{2} + 2 \, a b^{4}\right )} e^{\left (3 \, x\right )} + 10 \, {\left (4 \, a^{2} b^{3} + 7 \, b^{5}\right )} e^{\left (2 \, x\right )}\right )} e^{\left (-5 \, x\right )}}{960 \, b^{6}} + \frac {{\left (a^{6} + 3 \, a^{4} b^{2} + 3 \, a^{2} b^{4} + b^{6}\right )} \log \left (\frac {{\left | 2 \, b e^{x} + 2 \, a - 2 \, \sqrt {a^{2} + b^{2}} \right |}}{{\left | 2 \, b e^{x} + 2 \, a + 2 \, \sqrt {a^{2} + b^{2}} \right |}}\right )}{\sqrt {a^{2} + b^{2}} b^{6}} \] Input:
integrate(cosh(x)^6/(a+b*sinh(x)),x, algorithm="giac")
Output:
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 + 1/960* (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)
Time = 2.08 (sec) , antiderivative size = 302, normalized size of antiderivative = 2.08 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\frac {{\mathrm {e}}^{-5\,x}}{160\,b}+\frac {{\mathrm {e}}^{5\,x}}{160\,b}-\frac {\ln \left (-\frac {2\,{\mathrm {e}}^x\,{\left (a^2+b^2\right )}^3}{b^7}-\frac {2\,\left (b-a\,{\mathrm {e}}^x\right )\,{\left (a^2+b^2\right )}^{5/2}}{b^7}\right )\,{\left (a^2+b^2\right )}^{5/2}}{b^6}+\frac {\ln \left (\frac {2\,\left (b-a\,{\mathrm {e}}^x\right )\,{\left (a^2+b^2\right )}^{5/2}}{b^7}-\frac {2\,{\mathrm {e}}^x\,{\left (a^2+b^2\right )}^3}{b^7}\right )\,{\left (a^2+b^2\right )}^{5/2}}{b^6}-\frac {x\,\left (8\,a^5+20\,a^3\,b^2+15\,a\,b^4\right )}{8\,b^6}+\frac {{\mathrm {e}}^x\,\left (8\,a^4+18\,a^2\,b^2+11\,b^4\right )}{16\,b^5}+\frac {a\,{\mathrm {e}}^{-4\,x}}{64\,b^2}-\frac {a\,{\mathrm {e}}^{4\,x}}{64\,b^2}+\frac {{\mathrm {e}}^{-x}\,\left (8\,a^4+18\,a^2\,b^2+11\,b^4\right )}{16\,b^5}+\frac {{\mathrm {e}}^{-3\,x}\,\left (4\,a^2+7\,b^2\right )}{96\,b^3}+\frac {{\mathrm {e}}^{3\,x}\,\left (4\,a^2+7\,b^2\right )}{96\,b^3}+\frac {{\mathrm {e}}^{-2\,x}\,\left (a^3+2\,a\,b^2\right )}{8\,b^4}-\frac {{\mathrm {e}}^{2\,x}\,\left (a^3+2\,a\,b^2\right )}{8\,b^4} \] Input:
int(cosh(x)^6/(a + b*sinh(x)),x)
Output:
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) + (ex p(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)
Time = 0.16 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.57 \[ \int \frac {\cosh ^6(x)}{a+b \sinh (x)} \, dx=\frac {1920 e^{5 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{4} i +3840 e^{5 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) a^{2} b^{2} i +1920 e^{5 x} \sqrt {a^{2}+b^{2}}\, \mathit {atan} \left (\frac {e^{x} b i +a i}{\sqrt {a^{2}+b^{2}}}\right ) b^{4} i +6 e^{10 x} b^{5}-15 e^{9 x} a \,b^{4}+40 e^{8 x} a^{2} b^{3}+70 e^{8 x} b^{5}-120 e^{7 x} a^{3} b^{2}-240 e^{7 x} a \,b^{4}+480 e^{6 x} a^{4} b +1080 e^{6 x} a^{2} b^{3}+660 e^{6 x} b^{5}-960 e^{5 x} a^{5} x -2400 e^{5 x} a^{3} b^{2} x -1800 e^{5 x} a \,b^{4} x +480 e^{4 x} a^{4} b +1080 e^{4 x} a^{2} b^{3}+660 e^{4 x} b^{5}+120 e^{3 x} a^{3} b^{2}+240 e^{3 x} a \,b^{4}+40 e^{2 x} a^{2} b^{3}+70 e^{2 x} b^{5}+15 e^{x} a \,b^{4}+6 b^{5}}{960 e^{5 x} b^{6}} \] Input:
int(cosh(x)^6/(a+b*sinh(x)),x)
Output:
(1920*e**(5*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))* a**4*i + 3840*e**(5*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i)/sqrt(a**2 + b**2))*a**2*b**2*i + 1920*e**(5*x)*sqrt(a**2 + b**2)*atan((e**x*b*i + a*i )/sqrt(a**2 + b**2))*b**4*i + 6*e**(10*x)*b**5 - 15*e**(9*x)*a*b**4 + 40*e **(8*x)*a**2*b**3 + 70*e**(8*x)*b**5 - 120*e**(7*x)*a**3*b**2 - 240*e**(7* x)*a*b**4 + 480*e**(6*x)*a**4*b + 1080*e**(6*x)*a**2*b**3 + 660*e**(6*x)*b **5 - 960*e**(5*x)*a**5*x - 2400*e**(5*x)*a**3*b**2*x - 1800*e**(5*x)*a*b* *4*x + 480*e**(4*x)*a**4*b + 1080*e**(4*x)*a**2*b**3 + 660*e**(4*x)*b**5 + 120*e**(3*x)*a**3*b**2 + 240*e**(3*x)*a*b**4 + 40*e**(2*x)*a**2*b**3 + 70 *e**(2*x)*b**5 + 15*e**x*a*b**4 + 6*b**5)/(960*e**(5*x)*b**6)