Integrand size = 12, antiderivative size = 238 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^5} \, dx=-\frac {a}{4 x^4}-\frac {b}{3 x^3}-\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x}}{4 x^4}+\frac {a b \sqrt {-1+a+b x} \sqrt {1+a+b x}}{12 \left (1-a^2\right ) x^3}+\frac {\left (3+2 a^2\right ) b^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{24 \left (1-a^2\right )^2 x^2}+\frac {a \left (13+2 a^2\right ) b^3 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{24 \left (1-a^2\right )^3 x}-\frac {\left (1+4 a^2\right ) b^4 \arctan \left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )}{4 \left (1-a^2\right )^{7/2}} \] Output:
-1/4*a/x^4-1/3*b/x^3-1/4*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)/x^4+1/12*a*b*(b*x +a-1)^(1/2)*(b*x+a+1)^(1/2)/(-a^2+1)/x^3+1/24*(2*a^2+3)*b^2*(b*x+a-1)^(1/2 )*(b*x+a+1)^(1/2)/(-a^2+1)^2/x^2+1/24*a*(2*a^2+13)*b^3*(b*x+a-1)^(1/2)*(b* x+a+1)^(1/2)/(-a^2+1)^3/x-1/4*(4*a^2+1)*b^4*arctan((1-a)^(1/2)*(b*x+a+1)^( 1/2)/(1+a)^(1/2)/(b*x+a-1)^(1/2))/(-a^2+1)^(7/2)
Result contains complex when optimal does not.
Time = 0.32 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.83 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^5} \, dx=\frac {1}{24} \left (-\frac {6 a}{x^4}-\frac {8 b}{x^3}-\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x} \left (6+\frac {2 a b x}{-1+a^2}-\frac {\left (3+2 a^2\right ) b^2 x^2}{\left (-1+a^2\right )^2}+\frac {a \left (13+2 a^2\right ) b^3 x^3}{\left (-1+a^2\right )^3}\right )}{x^4}-\frac {3 i \left (1+4 a^2\right ) b^4 \log \left (\frac {16 i \left (1-a^2\right )^{5/2} \left (-1+a^2+a b x-i \sqrt {1-a^2} \sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{b^4 \left (x+4 a^2 x\right )}\right )}{\left (1-a^2\right )^{7/2}}\right ) \] Input:
Integrate[E^ArcCosh[a + b*x]/x^5,x]
Output:
((-6*a)/x^4 - (8*b)/x^3 - (Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(6 + (2*a* b*x)/(-1 + a^2) - ((3 + 2*a^2)*b^2*x^2)/(-1 + a^2)^2 + (a*(13 + 2*a^2)*b^3 *x^3)/(-1 + a^2)^3))/x^4 - ((3*I)*(1 + 4*a^2)*b^4*Log[((16*I)*(1 - a^2)^(5 /2)*(-1 + a^2 + a*b*x - I*Sqrt[1 - a^2]*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b* x]))/(b^4*(x + 4*a^2*x))])/(1 - a^2)^(7/2))/24
Time = 0.84 (sec) , antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {6435, 2010, 2009}
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.
| \(\displaystyle \int \frac {e^{\text {arccosh}(a+b x)}}{x^5} \, dx\) |
| \(\Big \downarrow \) 6435 |
| \(\displaystyle \int \frac {\sqrt {a+b x-1} \sqrt {a+b x+1}+a+b x}{x^5}dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (\frac {\sqrt {a+b x-1} \sqrt {a+b x+1}}{x^5}+\frac {a}{x^5}+\frac {b}{x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\left (4 a^2+1\right ) b^4 \arctan \left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {a+b x-1}}\right )}{4 \left (1-a^2\right )^{7/2}}+\frac {a \left (2 a^2+13\right ) b^3 \sqrt {a+b x-1} \sqrt {a+b x+1}}{24 \left (1-a^2\right )^3 x}+\frac {\left (2 a^2+3\right ) b^2 \sqrt {a+b x-1} \sqrt {a+b x+1}}{24 \left (1-a^2\right )^2 x^2}+\frac {a b \sqrt {a+b x-1} \sqrt {a+b x+1}}{12 \left (1-a^2\right ) x^3}-\frac {\sqrt {a+b x-1} \sqrt {a+b x+1}}{4 x^4}-\frac {a}{4 x^4}-\frac {b}{3 x^3}\) |
Input:
Int[E^ArcCosh[a + b*x]/x^5,x]
Output:
-1/4*a/x^4 - b/(3*x^3) - (Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(4*x^4) + (a*b*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(12*(1 - a^2)*x^3) + ((3 + 2*a^ 2)*b^2*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(24*(1 - a^2)^2*x^2) + (a*(13 + 2*a^2)*b^3*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(24*(1 - a^2)^3*x) - ( (1 + 4*a^2)*b^4*ArcTan[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[- 1 + a + b*x])])/(4*(1 - a^2)^(7/2))
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]]
Int[E^(ArcCosh[u_]*(n_.))*(x_)^(m_.), x_Symbol] :> Int[x^m*(u + Sqrt[-1 + u ]*Sqrt[1 + u])^n, x] /; RationalQ[m] && IntegerQ[n] && PolyQ[u, x]
Leaf count of result is larger than twice the leaf count of optimal. \(602\) vs. \(2(198)=396\).
Time = 0.20 (sec) , antiderivative size = 603, normalized size of antiderivative = 2.53
| method | result | size |
| default | \(\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (12 \sqrt {a^{2}-1}\, \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a^{2} b^{4} x^{4}-2 a^{5} b^{3} x^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+2 a^{6} b^{2} x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+3 \sqrt {a^{2}-1}\, \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) b^{4} x^{4}-2 a^{7} b x \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-11 a^{3} b^{3} x^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-6 a^{8} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-a^{4} b^{2} x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+6 a^{5} b x \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+13 a \,b^{3} x^{3} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+24 a^{6} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-4 a^{2} b^{2} x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-6 a^{3} b x \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-36 a^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+3 b^{2} x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+2 a b x \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+24 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2}-6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\right )}{24 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{4} x^{4}}-\frac {a}{4 x^{4}}-\frac {b}{3 x^{3}}\) | \(603\) |
Input:
int((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^5,x,method=_RETURNVERBOSE)
Output:
1/24*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)*(12*(a^2-1)^(1/2)*ln(2*(a*b*x+(a^2-1) ^(1/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2-1)/x)*a^2*b^4*x^4-2*a^5*b^3*x^3*( b^2*x^2+2*a*b*x+a^2-1)^(1/2)+2*a^6*b^2*x^2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+3 *(a^2-1)^(1/2)*ln(2*(a*b*x+(a^2-1)^(1/2)*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a^2 -1)/x)*b^4*x^4-2*a^7*b*x*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)-11*a^3*b^3*x^3*(b^2 *x^2+2*a*b*x+a^2-1)^(1/2)-6*a^8*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)-a^4*b^2*x^2* (b^2*x^2+2*a*b*x+a^2-1)^(1/2)+6*a^5*b*x*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+13*a *b^3*x^3*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+24*a^6*(b^2*x^2+2*a*b*x+a^2-1)^(1/2 )-4*a^2*b^2*x^2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)-6*a^3*b*x*(b^2*x^2+2*a*b*x+a ^2-1)^(1/2)-36*a^4*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+3*b^2*x^2*(b^2*x^2+2*a*b* x+a^2-1)^(1/2)+2*a*b*x*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+24*(b^2*x^2+2*a*b*x+a ^2-1)^(1/2)*a^2-6*(b^2*x^2+2*a*b*x+a^2-1)^(1/2))/(b^2*x^2+2*a*b*x+a^2-1)^( 1/2)/(a^2-1)^4/x^4-1/4*a/x^4-1/3*b/x^3
Time = 0.09 (sec) , antiderivative size = 569, normalized size of antiderivative = 2.39 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^5} \, dx=\left [\frac {3 \, {\left (4 \, a^{2} + 1\right )} \sqrt {a^{2} - 1} b^{4} x^{4} \log \left (\frac {a^{2} b x + a^{3} + {\left (a^{2} + \sqrt {a^{2} - 1} a - 1\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - 6 \, a^{9} - {\left (2 \, a^{5} + 11 \, a^{3} - 13 \, a\right )} b^{4} x^{4} + 24 \, a^{7} - 36 \, a^{5} + 24 \, a^{3} - 8 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} b x - {\left (6 \, a^{8} + {\left (2 \, a^{5} + 11 \, a^{3} - 13 \, a\right )} b^{3} x^{3} - 24 \, a^{6} - {\left (2 \, a^{6} - a^{4} - 4 \, a^{2} + 3\right )} b^{2} x^{2} + 36 \, a^{4} + 2 \, {\left (a^{7} - 3 \, a^{5} + 3 \, a^{3} - a\right )} b x - 24 \, a^{2} + 6\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - 6 \, a}{24 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} x^{4}}, -\frac {6 \, {\left (4 \, a^{2} + 1\right )} \sqrt {-a^{2} + 1} b^{4} x^{4} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {-a^{2} + 1} \sqrt {b x + a + 1} \sqrt {b x + a - 1}}{a^{2} - 1}\right ) + 6 \, a^{9} + {\left (2 \, a^{5} + 11 \, a^{3} - 13 \, a\right )} b^{4} x^{4} - 24 \, a^{7} + 36 \, a^{5} - 24 \, a^{3} + 8 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} b x + {\left (6 \, a^{8} + {\left (2 \, a^{5} + 11 \, a^{3} - 13 \, a\right )} b^{3} x^{3} - 24 \, a^{6} - {\left (2 \, a^{6} - a^{4} - 4 \, a^{2} + 3\right )} b^{2} x^{2} + 36 \, a^{4} + 2 \, {\left (a^{7} - 3 \, a^{5} + 3 \, a^{3} - a\right )} b x - 24 \, a^{2} + 6\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + 6 \, a}{24 \, {\left (a^{8} - 4 \, a^{6} + 6 \, a^{4} - 4 \, a^{2} + 1\right )} x^{4}}\right ] \] Input:
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^5,x, algorithm="fricas ")
Output:
[1/24*(3*(4*a^2 + 1)*sqrt(a^2 - 1)*b^4*x^4*log((a^2*b*x + a^3 + (a^2 + sqr t(a^2 - 1)*a - 1)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + (a*b*x + a^2 - 1)* sqrt(a^2 - 1) - a)/x) - 6*a^9 - (2*a^5 + 11*a^3 - 13*a)*b^4*x^4 + 24*a^7 - 36*a^5 + 24*a^3 - 8*(a^8 - 4*a^6 + 6*a^4 - 4*a^2 + 1)*b*x - (6*a^8 + (2*a ^5 + 11*a^3 - 13*a)*b^3*x^3 - 24*a^6 - (2*a^6 - a^4 - 4*a^2 + 3)*b^2*x^2 + 36*a^4 + 2*(a^7 - 3*a^5 + 3*a^3 - a)*b*x - 24*a^2 + 6)*sqrt(b*x + a + 1)* sqrt(b*x + a - 1) - 6*a)/((a^8 - 4*a^6 + 6*a^4 - 4*a^2 + 1)*x^4), -1/24*(6 *(4*a^2 + 1)*sqrt(-a^2 + 1)*b^4*x^4*arctan(-(sqrt(-a^2 + 1)*b*x - sqrt(-a^ 2 + 1)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1))/(a^2 - 1)) + 6*a^9 + (2*a^5 + 11*a^3 - 13*a)*b^4*x^4 - 24*a^7 + 36*a^5 - 24*a^3 + 8*(a^8 - 4*a^6 + 6*a^4 - 4*a^2 + 1)*b*x + (6*a^8 + (2*a^5 + 11*a^3 - 13*a)*b^3*x^3 - 24*a^6 - (2 *a^6 - a^4 - 4*a^2 + 3)*b^2*x^2 + 36*a^4 + 2*(a^7 - 3*a^5 + 3*a^3 - a)*b*x - 24*a^2 + 6)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + 6*a)/((a^8 - 4*a^6 + 6*a^4 - 4*a^2 + 1)*x^4)]
\[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^5} \, dx=\int \frac {a + b x + \sqrt {a + b x - 1} \sqrt {a + b x + 1}}{x^{5}}\, dx \] Input:
integrate((b*x+a+(b*x+a-1)**(1/2)*(b*x+a+1)**(1/2))/x**5,x)
Output:
Integral((a + b*x + sqrt(a + b*x - 1)*sqrt(a + b*x + 1))/x**5, x)
Exception generated. \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^5} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^5,x, algorithm="maxima ")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(a-1>0)', see `assume?` for more details)Is
Leaf count of result is larger than twice the leaf count of optimal. 817 vs. \(2 (192) = 384\).
Time = 0.37 (sec) , antiderivative size = 817, normalized size of antiderivative = 3.43 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^5} \, dx =\text {Too large to display} \] Input:
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^5,x, algorithm="giac")
Output:
1/12*(3*(4*a^2*b^5 + b^5)*arctan(1/2*((sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 - 2*a)/sqrt(-a^2 + 1))/((a^6 - 3*a^4 + 3*a^2 - 1)*sqrt(-a^2 + 1)) + 2*(128*a^6*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^10 + 12*a^2*b^5*(sq rt(b*x + a + 1) - sqrt(b*x + a - 1))^14 - 128*a^7*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^8 - 168*a^3*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1) )^12 + 448*a^4*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^10 + 3*b^5*(sqr t(b*x + a + 1) - sqrt(b*x + a - 1))^14 - 1216*a^5*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^8 - 42*a*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^1 2 + 512*a^6*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 + 768*a^2*b^5*(s qrt(b*x + a + 1) - sqrt(b*x + a - 1))^10 - 2544*a^3*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^8 + 5632*a^4*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 - 84*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^10 - 1536*a^5*b^5* (sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^4 - 312*a*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^8 + 1920*a^2*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1 ))^6 - 7552*a^3*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^4 + 1024*a^4*b ^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 + 336*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 - 992*a*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1)) ^4 + 5888*a^2*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 - 256*a^3*b^5 - 192*b^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 - 1664*a*b^5)/((a^6 - 3*a^4 + 3*a^2 - 1)*((sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^4 - 4*a*(sq...
Time = 68.21 (sec) , antiderivative size = 2347, normalized size of antiderivative = 9.86 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^5} \, dx=\text {Too large to display} \] Input:
int((a + (a + b*x - 1)^(1/2)*(a + b*x + 1)^(1/2) + b*x)/x^5,x)
Output:
(log(((a - 1)^(1/2) - (a + b*x - 1)^(1/2))/((a + 1)^(1/2) - (a + b*x + 1)^ (1/2)))*(b^4*(a - 1)^(1/2)*(a + 1)^(1/2) + 4*a^2*b^4*(a - 1)^(1/2)*(a + 1) ^(1/2)))/(48*a^4 - 32*a^2 - 32*a^6 + 8*a^8 + 8) - (a/4 + (b*x)/3)/x^4 - (l og(((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^2/((a + 1)^(1/2) - (a + b*x + 1)^ (1/2))^2 - a^2 - (a^2*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^2)/((a + 1)^(1 /2) - (a + b*x + 1)^(1/2))^2 + (2*a*((a - 1)^(1/2) - (a + b*x - 1)^(1/2))* (a - 1)^(1/2)*(a + 1)^(1/2))/((a + 1)^(1/2) - (a + b*x + 1)^(1/2)) + 1)*(b ^4*(a - 1)^(1/2)*(a + 1)^(1/2) + 4*a^2*b^4*(a - 1)^(1/2)*(a + 1)^(1/2)))/( 48*a^4 - 32*a^2 - 32*a^6 + 8*a^8 + 8) - ((((a - 1)^(1/2) - (a + b*x - 1)^( 1/2))^3*((17*a*b^4)/192 - (5*a^3*b^4)/192))/(((a + 1)^(1/2) - (a + b*x + 1 )^(1/2))^3*(3*a^2 - 3*a^4 + a^6 - 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1 /2))^11*((7*a^3*b^4)/64 - (81*a*b^4)/128 + (3*a^5*b^4)/128))/(((a + 1)^(1/ 2) - (a + b*x + 1)^(1/2))^11*(6*a^4 - 4*a^2 - 4*a^6 + a^8 + 1)) + (((a - 1 )^(1/2) - (a + b*x - 1)^(1/2))^5*((229*a^3*b^4)/64 - (119*a*b^4)/128 + (11 9*a^5*b^4)/384))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^5*(6*a^4 - 4*a^2 - 4*a^6 + a^8 + 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^9*((1025*a*b^4 )/384 - (1745*a^3*b^4)/128 + (385*a^5*b^4)/128 + (239*a^7*b^4)/384))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^9*(5*a^2 - 10*a^4 + 10*a^6 - 5*a^8 + a^1 0 - 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^7*((1103*a*b^4)/384 - (21 99*a^3*b^4)/128 + (1039*a^5*b^4)/128 + (521*a^7*b^4)/384))/(((a + 1)^(1...
Time = 2.12 (sec) , antiderivative size = 828, normalized size of antiderivative = 3.48 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^5} \, dx =\text {Too large to display} \] Input:
int((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^5,x)
Output:
(48*sqrt(a + 1)*sqrt( - a + 1)*log(sqrt(a + 1)*sqrt( - a + 1)*i + sqrt(a + b*x + 1)*sqrt(a + b*x - 1) + b*x)*a**3*b**4*i*x**4 + 12*sqrt(a + 1)*sqrt( - a + 1)*log(sqrt(a + 1)*sqrt( - a + 1)*i + sqrt(a + b*x + 1)*sqrt(a + b* x - 1) + b*x)*a*b**4*i*x**4 - 48*sqrt(a + 1)*sqrt( - a + 1)*log(( - sqrt(s qrt(a + 1)*sqrt( - a + 1)*i + a)*sqrt(2) + sqrt(a + b*x - 1) + sqrt(a + b* x + 1))/sqrt(2))*a**3*b**4*i*x**4 - 12*sqrt(a + 1)*sqrt( - a + 1)*log(( - sqrt(sqrt(a + 1)*sqrt( - a + 1)*i + a)*sqrt(2) + sqrt(a + b*x - 1) + sqrt( a + b*x + 1))/sqrt(2))*a*b**4*i*x**4 - 48*sqrt(a + 1)*sqrt( - a + 1)*log(( sqrt(sqrt(a + 1)*sqrt( - a + 1)*i + a)*sqrt(2) + sqrt(a + b*x - 1) + sqrt( a + b*x + 1))/sqrt(2))*a**3*b**4*i*x**4 - 12*sqrt(a + 1)*sqrt( - a + 1)*lo g((sqrt(sqrt(a + 1)*sqrt( - a + 1)*i + a)*sqrt(2) + sqrt(a + b*x - 1) + sq rt(a + b*x + 1))/sqrt(2))*a*b**4*i*x**4 - 24*sqrt(a + b*x + 1)*sqrt(a + b* x - 1)*a**9 - 8*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**8*b*x + 8*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**7*b**2*x**2 + 96*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**7 - 8*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**6*b**3*x**3 + 24* sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**6*b*x - 4*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**5*b**2*x**2 - 144*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**5 - 44*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**4*b**3*x**3 - 24*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**4*b*x - 16*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a* *3*b**2*x**2 + 96*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**3 + 52*sqrt(a ...