Integrand size = 12, antiderivative size = 189 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^4} \, dx=-\frac {a}{3 x^3}-\frac {b}{2 x^2}+\frac {a b^2 \sqrt {-1+a+b x} \sqrt {1+a+b x}}{2 \left (1-a^2\right )^2 x}-\frac {a b \sqrt {-1+a+b x} (1+a+b x)^{3/2}}{2 (1-a) (1+a)^2 x^2}+\frac {(-1+a+b x)^{3/2} (1+a+b x)^{3/2}}{3 \left (1-a^2\right ) x^3}-\frac {a b^3 \arctan \left (\frac {\sqrt {1-a} \sqrt {1+a+b x}}{\sqrt {1+a} \sqrt {-1+a+b x}}\right )}{\left (1-a^2\right )^{5/2}} \] Output:
-1/3*a/x^3-1/2*b/x^2+1/2*a*b^2*(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2)/(-a^2+1)^2/ x-1/2*a*b*(b*x+a-1)^(1/2)*(b*x+a+1)^(3/2)/(1-a)/(1+a)^2/x^2+1/3*(b*x+a-1)^ (3/2)*(b*x+a+1)^(3/2)/(-a^2+1)/x^3-a*b^3*arctan((1-a)^(1/2)*(b*x+a+1)^(1/2 )/(1+a)^(1/2)/(b*x+a-1)^(1/2))/(-a^2+1)^(5/2)
Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.95 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^4} \, dx=\frac {1}{6} \left (-\frac {2 a}{x^3}-\frac {3 b}{x^2}+\frac {\sqrt {-1+a+b x} \sqrt {1+a+b x} \left (-2-2 a^4+a b x-a^3 b x+2 b^2 x^2+a^2 \left (4+b^2 x^2\right )\right )}{\left (-1+a^2\right )^2 x^3}-\frac {3 i a b^3 \log \left (\frac {4 \left (1-a^2\right )^{3/2} \left (-i+i a^2+i a b x+\sqrt {1-a^2} \sqrt {-1+a+b x} \sqrt {1+a+b x}\right )}{a b^3 x}\right )}{\left (1-a^2\right )^{5/2}}\right ) \] Input:
Integrate[E^ArcCosh[a + b*x]/x^4,x]
Output:
((-2*a)/x^3 - (3*b)/x^2 + (Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]*(-2 - 2*a^ 4 + a*b*x - a^3*b*x + 2*b^2*x^2 + a^2*(4 + b^2*x^2)))/((-1 + a^2)^2*x^3) - ((3*I)*a*b^3*Log[(4*(1 - a^2)^(3/2)*(-I + I*a^2 + I*a*b*x + Sqrt[1 - a^2] *Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x]))/(a*b^3*x)])/(1 - a^2)^(5/2))/6
Time = 0.73 (sec) , antiderivative size = 189, 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.
| \(\displaystyle \int \frac {e^{\text {arccosh}(a+b x)}}{x^4} \, dx\) |
| \(\Big \downarrow \) 6435 |
| \(\displaystyle \int \frac {\sqrt {a+b x-1} \sqrt {a+b x+1}+a+b x}{x^4}dx\) |
| \(\Big \downarrow \) 2010 |
| \(\displaystyle \int \left (\frac {\sqrt {a+b x-1} \sqrt {a+b x+1}}{x^4}+\frac {a}{x^4}+\frac {b}{x^3}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {a b^3 \arctan \left (\frac {\sqrt {1-a} \sqrt {a+b x+1}}{\sqrt {a+1} \sqrt {a+b x-1}}\right )}{\left (1-a^2\right )^{5/2}}+\frac {a b^2 \sqrt {a+b x-1} \sqrt {a+b x+1}}{2 \left (1-a^2\right )^2 x}+\frac {(a+b x-1)^{3/2} (a+b x+1)^{3/2}}{3 \left (1-a^2\right ) x^3}-\frac {a b \sqrt {a+b x-1} (a+b x+1)^{3/2}}{2 (1-a) (a+1)^2 x^2}-\frac {a}{3 x^3}-\frac {b}{2 x^2}\) |
Input:
Int[E^ArcCosh[a + b*x]/x^4,x]
Output:
-1/3*a/x^3 - b/(2*x^2) + (a*b^2*Sqrt[-1 + a + b*x]*Sqrt[1 + a + b*x])/(2*( 1 - a^2)^2*x) - (a*b*Sqrt[-1 + a + b*x]*(1 + a + b*x)^(3/2))/(2*(1 - a)*(1 + a)^2*x^2) + ((-1 + a + b*x)^(3/2)*(1 + a + b*x)^(3/2))/(3*(1 - a^2)*x^3 ) - (a*b^3*ArcTan[(Sqrt[1 - a]*Sqrt[1 + a + b*x])/(Sqrt[1 + a]*Sqrt[-1 + a + b*x])])/(1 - a^2)^(5/2)
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. \(373\) vs. \(2(157)=314\).
Time = 0.19 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.98
| method | result | size |
| default | \(-\frac {\sqrt {b x +a -1}\, \sqrt {b x +a +1}\, \left (3 \sqrt {a^{2}-1}\, \ln \left (\frac {2 a^{2}-2+2 a b x +2 \sqrt {a^{2}-1}\, \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}}{x}\right ) a \,b^{3} x^{3}-a^{4} b^{2} x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+a^{5} b x \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+2 a^{6} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-a^{2} b^{2} x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-2 a^{3} b x \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}-6 a^{4} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+2 b^{2} x^{2} \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+a b x \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}+6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, a^{2}-2 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\right )}{6 \sqrt {b^{2} x^{2}+2 a b x +a^{2}-1}\, \left (a^{2}-1\right )^{3} x^{3}}-\frac {b}{2 x^{2}}-\frac {a}{3 x^{3}}\) | \(374\) |
Input:
int((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^4,x,method=_RETURNVERBOSE)
Output:
-1/6*(b*x+a-1)^(1/2)*(b*x+a+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)*a*b^3*x^3-a^4*b^2*x^2*(b^2*x ^2+2*a*b*x+a^2-1)^(1/2)+a^5*b*x*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+2*a^6*(b^2*x ^2+2*a*b*x+a^2-1)^(1/2)-a^2*b^2*x^2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)-2*a^3*b* x*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)-6*a^4*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+2*b^2* x^2*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+a*b*x*(b^2*x^2+2*a*b*x+a^2-1)^(1/2)+6*(b ^2*x^2+2*a*b*x+a^2-1)^(1/2)*a^2-2*(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)^3/x^3-1/2*b/x^2-1/3*a/x^3
Time = 0.08 (sec) , antiderivative size = 431, normalized size of antiderivative = 2.28 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^4} \, dx=\left [\frac {3 \, \sqrt {a^{2} - 1} a b^{3} x^{3} \log \left (\frac {a^{2} b x + a^{3} + {\left (a^{2} - \sqrt {a^{2} - 1} a - 1\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} - {\left (a b x + a^{2} - 1\right )} \sqrt {a^{2} - 1} - a}{x}\right ) - 2 \, a^{7} + {\left (a^{4} + a^{2} - 2\right )} b^{3} x^{3} + 6 \, a^{5} - 6 \, a^{3} - 3 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b x - {\left (2 \, a^{6} - {\left (a^{4} + a^{2} - 2\right )} b^{2} x^{2} - 6 \, a^{4} + {\left (a^{5} - 2 \, a^{3} + a\right )} b x + 6 \, a^{2} - 2\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + 2 \, a}{6 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3}}, \frac {6 \, \sqrt {-a^{2} + 1} a b^{3} x^{3} \arctan \left (-\frac {\sqrt {-a^{2} + 1} b x - \sqrt {-a^{2} + 1} \sqrt {b x + a + 1} \sqrt {b x + a - 1}}{a^{2} - 1}\right ) - 2 \, a^{7} + {\left (a^{4} + a^{2} - 2\right )} b^{3} x^{3} + 6 \, a^{5} - 6 \, a^{3} - 3 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} b x - {\left (2 \, a^{6} - {\left (a^{4} + a^{2} - 2\right )} b^{2} x^{2} - 6 \, a^{4} + {\left (a^{5} - 2 \, a^{3} + a\right )} b x + 6 \, a^{2} - 2\right )} \sqrt {b x + a + 1} \sqrt {b x + a - 1} + 2 \, a}{6 \, {\left (a^{6} - 3 \, a^{4} + 3 \, a^{2} - 1\right )} x^{3}}\right ] \] Input:
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^4,x, algorithm="fricas ")
Output:
[1/6*(3*sqrt(a^2 - 1)*a*b^3*x^3*log((a^2*b*x + a^3 + (a^2 - sqrt(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) - 2*a^7 + (a^4 + a^2 - 2)*b^3*x^3 + 6*a^5 - 6*a^3 - 3*(a^6 - 3* a^4 + 3*a^2 - 1)*b*x - (2*a^6 - (a^4 + a^2 - 2)*b^2*x^2 - 6*a^4 + (a^5 - 2 *a^3 + a)*b*x + 6*a^2 - 2)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + 2*a)/((a^ 6 - 3*a^4 + 3*a^2 - 1)*x^3), 1/6*(6*sqrt(-a^2 + 1)*a*b^3*x^3*arctan(-(sqrt (-a^2 + 1)*b*x - sqrt(-a^2 + 1)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1))/(a^2 - 1)) - 2*a^7 + (a^4 + a^2 - 2)*b^3*x^3 + 6*a^5 - 6*a^3 - 3*(a^6 - 3*a^4 + 3*a^2 - 1)*b*x - (2*a^6 - (a^4 + a^2 - 2)*b^2*x^2 - 6*a^4 + (a^5 - 2*a^3 + a)*b*x + 6*a^2 - 2)*sqrt(b*x + a + 1)*sqrt(b*x + a - 1) + 2*a)/((a^6 - 3 *a^4 + 3*a^2 - 1)*x^3)]
\[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^4} \, dx=\int \frac {a + b x + \sqrt {a + b x - 1} \sqrt {a + b x + 1}}{x^{4}}\, dx \] Input:
integrate((b*x+a+(b*x+a-1)**(1/2)*(b*x+a+1)**(1/2))/x**4,x)
Output:
Integral((a + b*x + sqrt(a + b*x - 1)*sqrt(a + b*x + 1))/x**4, x)
Exception generated. \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^4} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^4,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. 429 vs. \(2 (151) = 302\).
Time = 0.29 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.27 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^4} \, dx=-\frac {1}{6} \, b^{3} {\left (\frac {6 \, a \arctan \left (\frac {{\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} - 2 \, a}{2 \, \sqrt {-a^{2} + 1}}\right )}{{\left (a^{4} - 2 \, a^{2} + 1\right )} \sqrt {-a^{2} + 1}} - \frac {4 \, {\left (12 \, a^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{8} - 16 \, a^{5} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} - 3 \, a {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{10} + 6 \, a^{2} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{8} - 56 \, a^{3} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} + 48 \, a^{4} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} + 12 \, {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{8} - 48 \, a {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{6} + 192 \, a^{2} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} - 96 \, a^{3} {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} - 144 \, a {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} + 32 \, a^{2} + 64\right )}}{{\left (a^{4} - 2 \, a^{2} + 1\right )} {\left ({\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{4} - 4 \, a {\left (\sqrt {b x + a + 1} - \sqrt {b x + a - 1}\right )}^{2} + 4\right )}^{3}} + \frac {3 \, b x + 2 \, a}{b^{3} x^{3}}\right )} \] Input:
integrate((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^4,x, algorithm="giac")
Output:
-1/6*b^3*(6*a*arctan(1/2*((sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 - 2*a) /sqrt(-a^2 + 1))/((a^4 - 2*a^2 + 1)*sqrt(-a^2 + 1)) - 4*(12*a^4*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^8 - 16*a^5*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 - 3*a*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^10 + 6*a^2*(sqrt(b* x + a + 1) - sqrt(b*x + a - 1))^8 - 56*a^3*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 + 48*a^4*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^4 + 12*(sqrt(b *x + a + 1) - sqrt(b*x + a - 1))^8 - 48*a*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^6 + 192*a^2*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^4 - 96*a^3*(sq rt(b*x + a + 1) - sqrt(b*x + a - 1))^2 - 144*a*(sqrt(b*x + a + 1) - sqrt(b *x + a - 1))^2 + 32*a^2 + 64)/((a^4 - 2*a^2 + 1)*((sqrt(b*x + a + 1) - sqr t(b*x + a - 1))^4 - 4*a*(sqrt(b*x + a + 1) - sqrt(b*x + a - 1))^2 + 4)^3) + (3*b*x + 2*a)/(b^3*x^3))
Time = 47.24 (sec) , antiderivative size = 1537, normalized size of antiderivative = 8.13 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^4} \, dx=\text {Too large to display} \] Input:
int((a + (a + b*x - 1)^(1/2)*(a + b*x + 1)^(1/2) + b*x)/x^4,x)
Output:
((((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^2*((3*b^3)/32 - (a^2*b^3)/32))/((( a + 1)^(1/2) - (a + b*x + 1)^(1/2))^2*(a^4 - 2*a^2 + 1)) - b^3/(192*(a^2 - 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^4*((9*a^2*b^3)/8 - b^3/2 + ( 5*a^4*b^3)/8))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^4*(3*a^2 - 3*a^4 + a ^6 - 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^8*((a^2*b^3)/32 - (21*b^ 3)/64 + (3*a^4*b^3)/64))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^8*(3*a^2 - 3*a^4 + a^6 - 1)) + (((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^6*((103*b^3)/9 6 - (121*a^2*b^3)/32 + (11*a^4*b^3)/32 + (67*a^6*b^3)/96))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/2))^6*(6*a^4 - 4*a^2 - 4*a^6 + a^8 + 1)) - (((a - 1)^( 1/2) - (a + b*x - 1)^(1/2))^3*(a - 1)^(1/2)*(a + 1)^(1/2)*((17*a*b^3)/32 + (17*a^3*b^3)/96))/(((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))^7*(a - 1)^(1/2)*(a + 1)^(1/2)*((3*a^3*b^3)/16 - (63*a*b^3)/32 + (9*a^5*b^3)/32))/(((a + 1)^(1/ 2) - (a + b*x + 1)^(1/2))^7*(6*a^4 - 4*a^2 - 4*a^6 + a^8 + 1)) - (((a - 1) ^(1/2) - (a + b*x - 1)^(1/2))^5*(a - 1)^(1/2)*(a + 1)^(1/2)*((17*a^3*b^3)/ 16 - (79*a*b^3)/32 + (29*a^5*b^3)/32))/(((a + 1)^(1/2) - (a + b*x + 1)^(1/ 2))^5*(6*a^4 - 4*a^2 - 4*a^6 + a^8 + 1)) + (a*b^3*((a - 1)^(1/2) - (a + b* x - 1)^(1/2))*(a - 1)^(1/2)*(a + 1)^(1/2))/(32*((a + 1)^(1/2) - (a + b*x + 1)^(1/2))*(a^4 - 2*a^2 + 1)))/(((a - 1)^(1/2) - (a + b*x - 1)^(1/2))^3/(( a + 1)^(1/2) - (a + b*x + 1)^(1/2))^3 + ((a - 1)^(1/2) - (a + b*x - 1)^...
Time = 0.22 (sec) , antiderivative size = 459, normalized size of antiderivative = 2.43 \[ \int \frac {e^{\text {arccosh}(a+b x)}}{x^4} \, dx=\frac {-3 \sqrt {a +1}\, \sqrt {-a +1}\, \mathrm {log}\left (\sqrt {a +1}\, \sqrt {-a +1}\, i +\sqrt {b x +a +1}\, \sqrt {b x +a -1}+b x \right ) a \,b^{3} i \,x^{3}+3 \sqrt {a +1}\, \sqrt {-a +1}\, \mathrm {log}\left (\frac {-\sqrt {\sqrt {a +1}\, \sqrt {-a +1}\, i +a}\, \sqrt {2}+\sqrt {b x +a -1}+\sqrt {b x +a +1}}{\sqrt {2}}\right ) a \,b^{3} i \,x^{3}+3 \sqrt {a +1}\, \sqrt {-a +1}\, \mathrm {log}\left (\frac {\sqrt {\sqrt {a +1}\, \sqrt {-a +1}\, i +a}\, \sqrt {2}+\sqrt {b x +a -1}+\sqrt {b x +a +1}}{\sqrt {2}}\right ) a \,b^{3} i \,x^{3}-2 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a^{6}-\sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a^{5} b x +\sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a^{4} b^{2} x^{2}+6 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a^{4}+2 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a^{3} b x +\sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a^{2} b^{2} x^{2}-6 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a^{2}-\sqrt {b x +a +1}\, \sqrt {b x +a -1}\, a b x -2 \sqrt {b x +a +1}\, \sqrt {b x +a -1}\, b^{2} x^{2}+2 \sqrt {b x +a +1}\, \sqrt {b x +a -1}-2 a^{7}-3 a^{6} b x +6 a^{5}-a^{4} b^{3} x^{3}+9 a^{4} b x -6 a^{3}-9 a^{2} b x +2 a +b^{3} x^{3}+3 b x}{6 x^{3} \left (a^{6}-3 a^{4}+3 a^{2}-1\right )} \] Input:
int((b*x+a+(b*x+a-1)^(1/2)*(b*x+a+1)^(1/2))/x^4,x)
Output:
( - 3*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**3*i*x**3 + 3*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**3*i*x**3 + 3*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**3*i*x**3 - 2*sqrt(a + b*x + 1)* sqrt(a + b*x - 1)*a**6 - sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**5*b*x + sq rt(a + b*x + 1)*sqrt(a + b*x - 1)*a**4*b**2*x**2 + 6*sqrt(a + b*x + 1)*sqr t(a + b*x - 1)*a**4 + 2*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a**3*b*x + sqr t(a + b*x + 1)*sqrt(a + b*x - 1)*a**2*b**2*x**2 - 6*sqrt(a + b*x + 1)*sqrt (a + b*x - 1)*a**2 - sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*a*b*x - 2*sqrt(a + b*x + 1)*sqrt(a + b*x - 1)*b**2*x**2 + 2*sqrt(a + b*x + 1)*sqrt(a + b*x - 1) - 2*a**7 - 3*a**6*b*x + 6*a**5 - a**4*b**3*x**3 + 9*a**4*b*x - 6*a**3 - 9*a**2*b*x + 2*a + b**3*x**3 + 3*b*x)/(6*x**3*(a**6 - 3*a**4 + 3*a**2 - 1))