Integrand size = 16, antiderivative size = 180 \[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {d \sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}-\frac {e x \sqrt {1+c^2 x^2}}{b c (a+b \text {arcsinh}(c x))}+\frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {d \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right ) \sinh \left (\frac {a}{b}\right )}{b^2 c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2} \] Output:
-d*(c^2*x^2+1)^(1/2)/b/c/(a+b*arcsinh(c*x))-e*x*(c^2*x^2+1)^(1/2)/b/c/(a+b *arcsinh(c*x))+e*cosh(2*a/b)*Chi(2*(a+b*arcsinh(c*x))/b)/b^2/c^2-d*Chi((a+ b*arcsinh(c*x))/b)*sinh(a/b)/b^2/c+d*cosh(a/b)*Shi((a+b*arcsinh(c*x))/b)/b ^2/c-e*sinh(2*a/b)*Shi(2*(a+b*arcsinh(c*x))/b)/b^2/c^2
Time = 0.80 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.83 \[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=-\frac {\frac {b c d \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}+\frac {b c e x \sqrt {1+c^2 x^2}}{a+b \text {arcsinh}(c x)}-e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )+c d \text {Chi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right ) \sinh \left (\frac {a}{b}\right )-c d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\text {arcsinh}(c x)\right )+e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (2 \left (\frac {a}{b}+\text {arcsinh}(c x)\right )\right )}{b^2 c^2} \] Input:
Integrate[(d + e*x)/(a + b*ArcSinh[c*x])^2,x]
Output:
-(((b*c*d*Sqrt[1 + c^2*x^2])/(a + b*ArcSinh[c*x]) + (b*c*e*x*Sqrt[1 + c^2* x^2])/(a + b*ArcSinh[c*x]) - e*Cosh[(2*a)/b]*CoshIntegral[2*(a/b + ArcSinh [c*x])] + c*d*CoshIntegral[a/b + ArcSinh[c*x]]*Sinh[a/b] - c*d*Cosh[a/b]*S inhIntegral[a/b + ArcSinh[c*x]] + e*Sinh[(2*a)/b]*SinhIntegral[2*(a/b + Ar cSinh[c*x])])/(b^2*c^2))
Time = 0.57 (sec) , antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {6244, 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.
\(\displaystyle \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx\) |
\(\Big \downarrow \) 6244 |
\(\displaystyle \int \left (\frac {d}{(a+b \text {arcsinh}(c x))^2}+\frac {e x}{(a+b \text {arcsinh}(c x))^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {e \cosh \left (\frac {2 a}{b}\right ) \text {Chi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {e \sinh \left (\frac {2 a}{b}\right ) \text {Shi}\left (\frac {2 (a+b \text {arcsinh}(c x))}{b}\right )}{b^2 c^2}-\frac {d \sinh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}+\frac {d \cosh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \text {arcsinh}(c x)}{b}\right )}{b^2 c}-\frac {d \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}-\frac {e x \sqrt {c^2 x^2+1}}{b c (a+b \text {arcsinh}(c x))}\) |
Input:
Int[(d + e*x)/(a + b*ArcSinh[c*x])^2,x]
Output:
-((d*Sqrt[1 + c^2*x^2])/(b*c*(a + b*ArcSinh[c*x]))) - (e*x*Sqrt[1 + c^2*x^ 2])/(b*c*(a + b*ArcSinh[c*x])) + (e*Cosh[(2*a)/b]*CoshIntegral[(2*(a + b*A rcSinh[c*x]))/b])/(b^2*c^2) - (d*CoshIntegral[(a + b*ArcSinh[c*x])/b]*Sinh [a/b])/(b^2*c) + (d*Cosh[a/b]*SinhIntegral[(a + b*ArcSinh[c*x])/b])/(b^2*c ) - (e*Sinh[(2*a)/b]*SinhIntegral[(2*(a + b*ArcSinh[c*x]))/b])/(b^2*c^2)
Int[((a_.) + ArcSinh[(c_.)*(x_)]*(b_.))^(n_)*((d_) + (e_.)*(x_))^(m_.), x_S ymbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + b*ArcSinh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e}, x] && IGtQ[m, 0] && LtQ[n, -1]
Time = 3.32 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.51
method | result | size |
derivativedivides | \(\frac {\frac {\left (x c -\sqrt {c^{2} x^{2}+1}\right ) d}{2 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}+\frac {d \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {d \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{2 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {d \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right )}{2 b^{2}}+\frac {\left (-2 \sqrt {c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}+1\right ) e}{4 c b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arcsinh}\left (x c \right )+\frac {2 a}{b}\right )}{2 c \,b^{2}}-\frac {e \left (2 c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +1\right )}{4 c b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arcsinh}\left (x c \right )-\frac {2 a}{b}\right )}{2 c \,b^{2}}}{c}\) | \(272\) |
default | \(\frac {\frac {\left (x c -\sqrt {c^{2} x^{2}+1}\right ) d}{2 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}+\frac {d \,{\mathrm e}^{\frac {a}{b}} \operatorname {expIntegral}_{1}\left (\operatorname {arcsinh}\left (x c \right )+\frac {a}{b}\right )}{2 b^{2}}-\frac {d \left (x c +\sqrt {c^{2} x^{2}+1}\right )}{2 b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {d \,{\mathrm e}^{-\frac {a}{b}} \operatorname {expIntegral}_{1}\left (-\operatorname {arcsinh}\left (x c \right )-\frac {a}{b}\right )}{2 b^{2}}+\frac {\left (-2 \sqrt {c^{2} x^{2}+1}\, x c +2 c^{2} x^{2}+1\right ) e}{4 c b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {e \,{\mathrm e}^{\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (2 \,\operatorname {arcsinh}\left (x c \right )+\frac {2 a}{b}\right )}{2 c \,b^{2}}-\frac {e \left (2 c^{2} x^{2}+2 \sqrt {c^{2} x^{2}+1}\, x c +1\right )}{4 c b \left (a +b \,\operatorname {arcsinh}\left (x c \right )\right )}-\frac {e \,{\mathrm e}^{-\frac {2 a}{b}} \operatorname {expIntegral}_{1}\left (-2 \,\operatorname {arcsinh}\left (x c \right )-\frac {2 a}{b}\right )}{2 c \,b^{2}}}{c}\) | \(272\) |
Input:
int((e*x+d)/(a+b*arcsinh(x*c))^2,x,method=_RETURNVERBOSE)
Output:
1/c*(1/2*(x*c-(c^2*x^2+1)^(1/2))*d/b/(a+b*arcsinh(x*c))+1/2*d/b^2*exp(a/b) *Ei(1,arcsinh(x*c)+a/b)-1/2/b*d*(x*c+(c^2*x^2+1)^(1/2))/(a+b*arcsinh(x*c)) -1/2/b^2*d*exp(-a/b)*Ei(1,-arcsinh(x*c)-a/b)+1/4*(-2*(c^2*x^2+1)^(1/2)*x*c +2*c^2*x^2+1)*e/c/b/(a+b*arcsinh(x*c))-1/2*e/c/b^2*exp(2*a/b)*Ei(1,2*arcsi nh(x*c)+2*a/b)-1/4*e/c/b*(2*c^2*x^2+2*(c^2*x^2+1)^(1/2)*x*c+1)/(a+b*arcsin h(x*c))-1/2*e/c/b^2*exp(-2*a/b)*Ei(1,-2*arcsinh(x*c)-2*a/b))
\[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {e x + d}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((e*x+d)/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")
Output:
integral((e*x + d)/(b^2*arcsinh(c*x)^2 + 2*a*b*arcsinh(c*x) + a^2), x)
\[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {d + e x}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}\, dx \] Input:
integrate((e*x+d)/(a+b*asinh(c*x))**2,x)
Output:
Integral((d + e*x)/(a + b*asinh(c*x))**2, x)
\[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {e x + d}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((e*x+d)/(a+b*arcsinh(c*x))^2,x, algorithm="maxima")
Output:
-(c^3*e*x^4 + c^3*d*x^3 + c*e*x^2 + c*d*x + (c^2*e*x^3 + c^2*d*x^2 + e*x + d)*sqrt(c^2*x^2 + 1))/(a*b*c^3*x^2 + sqrt(c^2*x^2 + 1)*a*b*c^2*x + a*b*c + (b^2*c^3*x^2 + sqrt(c^2*x^2 + 1)*b^2*c^2*x + b^2*c)*log(c*x + sqrt(c^2*x ^2 + 1))) + integrate((2*c^5*e*x^5 + c^5*d*x^4 + 4*c^3*e*x^3 + 2*c^3*d*x^2 + 2*c*e*x + (2*c^3*e*x^3 + c^3*d*x^2 - c*d)*(c^2*x^2 + 1) + c*d + (4*c^4* e*x^4 + 2*c^4*d*x^3 + 4*c^2*e*x^2 + c^2*d*x + e)*sqrt(c^2*x^2 + 1))/(a*b*c ^5*x^4 + (c^2*x^2 + 1)*a*b*c^3*x^2 + 2*a*b*c^3*x^2 + a*b*c + (b^2*c^5*x^4 + (c^2*x^2 + 1)*b^2*c^3*x^2 + 2*b^2*c^3*x^2 + b^2*c + 2*(b^2*c^4*x^3 + b^2 *c^2*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*x^3 + a*b*c^2*x)*sqrt(c^2*x^2 + 1)), x)
\[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {e x + d}{{\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \] Input:
integrate((e*x+d)/(a+b*arcsinh(c*x))^2,x, algorithm="giac")
Output:
integrate((e*x + d)/(b*arcsinh(c*x) + a)^2, x)
Timed out. \[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {d+e\,x}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2} \,d x \] Input:
int((d + e*x)/(a + b*asinh(c*x))^2,x)
Output:
int((d + e*x)/(a + b*asinh(c*x))^2, x)
\[ \int \frac {d+e x}{(a+b \text {arcsinh}(c x))^2} \, dx=\left (\int \frac {x}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) e +\left (\int \frac {1}{\mathit {asinh} \left (c x \right )^{2} b^{2}+2 \mathit {asinh} \left (c x \right ) a b +a^{2}}d x \right ) d \] Input:
int((e*x+d)/(a+b*asinh(c*x))^2,x)
Output:
int(x/(asinh(c*x)**2*b**2 + 2*asinh(c*x)*a*b + a**2),x)*e + int(1/(asinh(c *x)**2*b**2 + 2*asinh(c*x)*a*b + a**2),x)*d