Integrand size = 12, antiderivative size = 132 \[ \int \frac {\text {arccosh}(c x)}{(d+e x)^3} \, dx=-\frac {c \sqrt {-1+c x} \sqrt {1+c x}}{2 \left (c^2 d^2-e^2\right ) (d+e x)}-\frac {\text {arccosh}(c x)}{2 e (d+e x)^2}+\frac {c^3 d \text {arctanh}\left (\frac {\sqrt {c d+e} \sqrt {1+c x}}{\sqrt {c d-e} \sqrt {-1+c x}}\right )}{(c d-e)^{3/2} e (c d+e)^{3/2}} \] Output:
-1/2*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^2*d^2-e^2)/(e*x+d)-1/2*arccosh(c*x)/ e/(e*x+d)^2+c^3*d*arctanh((c*d+e)^(1/2)*(c*x+1)^(1/2)/(c*d-e)^(1/2)/(c*x-1 )^(1/2))/(c*d-e)^(3/2)/e/(c*d+e)^(3/2)
Time = 0.13 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.44 \[ \int \frac {\text {arccosh}(c x)}{(d+e x)^3} \, dx=\frac {-\left (c^2 d^2-e^2\right )^{3/2} \text {arccosh}(c x)+c (d+e x) \left (-e \sqrt {c^2 d^2-e^2} \sqrt {-1+c x} \sqrt {1+c x}+c^2 d (d+e x) \log (d+e x)-c^2 d (d+e x) \log \left (e+c^2 d x-\sqrt {c^2 d^2-e^2} \sqrt {-1+c x} \sqrt {1+c x}\right )\right )}{2 (c d-e) e (c d+e) \sqrt {c^2 d^2-e^2} (d+e x)^2} \] Input:
Integrate[ArcCosh[c*x]/(d + e*x)^3,x]
Output:
(-((c^2*d^2 - e^2)^(3/2)*ArcCosh[c*x]) + c*(d + e*x)*(-(e*Sqrt[c^2*d^2 - e ^2]*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + c^2*d*(d + e*x)*Log[d + e*x] - c^2*d*( d + e*x)*Log[e + c^2*d*x - Sqrt[c^2*d^2 - e^2]*Sqrt[-1 + c*x]*Sqrt[1 + c*x ]]))/(2*(c*d - e)*e*(c*d + e)*Sqrt[c^2*d^2 - e^2]*(d + e*x)^2)
Time = 0.30 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.15, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {6378, 107, 104, 221}
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.
\(\displaystyle \int \frac {\text {arccosh}(c x)}{(d+e x)^3} \, dx\) |
\(\Big \downarrow \) 6378 |
\(\displaystyle \frac {c \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1} (d+e x)^2}dx}{2 e}-\frac {\text {arccosh}(c x)}{2 e (d+e x)^2}\) |
\(\Big \downarrow \) 107 |
\(\displaystyle \frac {c \left (\frac {c^2 d \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1} (d+e x)}dx}{c^2 d^2-e^2}-\frac {e \sqrt {c x-1} \sqrt {c x+1}}{\left (c^2 d^2-e^2\right ) (d+e x)}\right )}{2 e}-\frac {\text {arccosh}(c x)}{2 e (d+e x)^2}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {c \left (\frac {2 c^2 d \int \frac {1}{c d-e-\frac {(c d+e) (c x+1)}{c x-1}}d\frac {\sqrt {c x+1}}{\sqrt {c x-1}}}{c^2 d^2-e^2}-\frac {e \sqrt {c x-1} \sqrt {c x+1}}{\left (c^2 d^2-e^2\right ) (d+e x)}\right )}{2 e}-\frac {\text {arccosh}(c x)}{2 e (d+e x)^2}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {c \left (\frac {2 c^2 d \text {arctanh}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{\sqrt {c d-e} \sqrt {c d+e} \left (c^2 d^2-e^2\right )}-\frac {e \sqrt {c x-1} \sqrt {c x+1}}{\left (c^2 d^2-e^2\right ) (d+e x)}\right )}{2 e}-\frac {\text {arccosh}(c x)}{2 e (d+e x)^2}\) |
Input:
Int[ArcCosh[c*x]/(d + e*x)^3,x]
Output:
-1/2*ArcCosh[c*x]/(e*(d + e*x)^2) + (c*(-((e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) /((c^2*d^2 - e^2)*(d + e*x))) + (2*c^2*d*ArcTanh[(Sqrt[c*d + e]*Sqrt[1 + c *x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(Sqrt[c*d - e]*Sqrt[c*d + e]*(c^2*d^ 2 - e^2))))/(2*e)
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[(a*d*f*(m + 1) + b*c*f*(n + 1) + b*d*e*(p + 1))/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x ] && EqQ[Simplify[m + n + p + 3], 0] && (LtQ[m, -1] || SumSimplerQ[m, 1])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d_.) + (e_.)*(x_))^(m_.), x _Symbol] :> Simp[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^n/(e*(m + 1))), x] - Simp[b*c*(n/(e*(m + 1))) Int[(d + e*x)^(m + 1)*((a + b*ArcCosh[c*x])^( n - 1)/(Sqrt[-1 + c*x]*Sqrt[1 + c*x])), x], x] /; FreeQ[{a, b, c, d, e, m}, x] && IGtQ[n, 0] && NeQ[m, -1]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.16 (sec) , antiderivative size = 249, normalized size of antiderivative = 1.89
method | result | size |
parts | \(-\frac {\operatorname {arccosh}\left (c x \right )}{2 e \left (e x +d \right )^{2}}+\frac {c \left (-\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e x +d}\right ) c^{2} d e x -\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e x +d}\right ) c^{2} d^{2}-e^{2} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\right ) \operatorname {csgn}\left (c \right )^{2} \sqrt {c x +1}\, \sqrt {c x -1}}{2 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (e x +d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}\) | \(249\) |
derivativedivides | \(\frac {-\frac {c^{3} \operatorname {arccosh}\left (c x \right )}{2 \left (e c x +c d \right )^{2} e}+\frac {c^{3} \left (-\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d^{2}-\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d e x -e^{2} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{2 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\) | \(266\) |
default | \(\frac {-\frac {c^{3} \operatorname {arccosh}\left (c x \right )}{2 \left (e c x +c d \right )^{2} e}+\frac {c^{3} \left (-\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d^{2}-\ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right ) c^{2} d e x -e^{2} \sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\right ) \sqrt {c x -1}\, \sqrt {c x +1}}{2 e^{2} \sqrt {c^{2} x^{2}-1}\, \left (c d +e \right ) \left (c d -e \right ) \left (e c x +c d \right ) \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}}}{c}\) | \(266\) |
Input:
int(arccosh(c*x)/(e*x+d)^3,x,method=_RETURNVERBOSE)
Output:
-1/2*arccosh(c*x)/e/(e*x+d)^2+1/2/e^2*c*(-ln(-2*(c^2*d*x-(c^2*x^2-1)^(1/2) *((c^2*d^2-e^2)/e^2)^(1/2)*e+e)/(e*x+d))*c^2*d*e*x-ln(-2*(c^2*d*x-(c^2*x^2 -1)^(1/2)*((c^2*d^2-e^2)/e^2)^(1/2)*e+e)/(e*x+d))*c^2*d^2-e^2*(c^2*x^2-1)^ (1/2)*((c^2*d^2-e^2)/e^2)^(1/2))*csgn(c)^2*(c*x+1)^(1/2)*(c*x-1)^(1/2)/(c^ 2*x^2-1)^(1/2)/(c*d+e)/(c*d-e)/(e*x+d)/((c^2*d^2-e^2)/e^2)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 511 vs. \(2 (112) = 224\).
Time = 0.17 (sec) , antiderivative size = 1044, normalized size of antiderivative = 7.91 \[ \int \frac {\text {arccosh}(c x)}{(d+e x)^3} \, dx =\text {Too large to display} \] Input:
integrate(arccosh(c*x)/(e*x+d)^3,x, algorithm="fricas")
Output:
[-1/2*(c^4*d^6 - c^2*d^4*e^2 + (c^4*d^4*e^2 - c^2*d^2*e^4)*x^2 + (c^3*d^3* e^2*x^2 + 2*c^3*d^4*e*x + c^3*d^5)*sqrt(c^2*d^2 - e^2)*log((c^3*d^2*x + c* d*e - sqrt(c^2*d^2 - e^2)*(c^2*d*x + e) + (c^2*d^2 - sqrt(c^2*d^2 - e^2)*c *d - e^2)*sqrt(c^2*x^2 - 1))/(e*x + d)) + 2*(c^4*d^5*e - c^2*d^3*e^3)*x - ((c^4*d^4*e^2 - 2*c^2*d^2*e^4 + e^6)*x^2 + 2*(c^4*d^5*e - 2*c^2*d^3*e^3 + d*e^5)*x)*log(c*x + sqrt(c^2*x^2 - 1)) - (c^4*d^6 - 2*c^2*d^4*e^2 + d^2*e^ 4 + (c^4*d^4*e^2 - 2*c^2*d^2*e^4 + e^6)*x^2 + 2*(c^4*d^5*e - 2*c^2*d^3*e^3 + d*e^5)*x)*log(-c*x + sqrt(c^2*x^2 - 1)) + (c^3*d^5*e - c*d^3*e^3 + (c^3 *d^4*e^2 - c*d^2*e^4)*x)*sqrt(c^2*x^2 - 1))/(c^4*d^8*e - 2*c^2*d^6*e^3 + d ^4*e^5 + (c^4*d^6*e^3 - 2*c^2*d^4*e^5 + d^2*e^7)*x^2 + 2*(c^4*d^7*e^2 - 2* c^2*d^5*e^4 + d^3*e^6)*x), -1/2*(c^4*d^6 - c^2*d^4*e^2 + (c^4*d^4*e^2 - c^ 2*d^2*e^4)*x^2 + 2*(c^3*d^3*e^2*x^2 + 2*c^3*d^4*e*x + c^3*d^5)*sqrt(-c^2*d ^2 + e^2)*arctan(-(sqrt(-c^2*d^2 + e^2)*sqrt(c^2*x^2 - 1)*e - sqrt(-c^2*d^ 2 + e^2)*(c*e*x + c*d))/(c^2*d^2 - e^2)) + 2*(c^4*d^5*e - c^2*d^3*e^3)*x - ((c^4*d^4*e^2 - 2*c^2*d^2*e^4 + e^6)*x^2 + 2*(c^4*d^5*e - 2*c^2*d^3*e^3 + d*e^5)*x)*log(c*x + sqrt(c^2*x^2 - 1)) - (c^4*d^6 - 2*c^2*d^4*e^2 + d^2*e ^4 + (c^4*d^4*e^2 - 2*c^2*d^2*e^4 + e^6)*x^2 + 2*(c^4*d^5*e - 2*c^2*d^3*e^ 3 + d*e^5)*x)*log(-c*x + sqrt(c^2*x^2 - 1)) + (c^3*d^5*e - c*d^3*e^3 + (c^ 3*d^4*e^2 - c*d^2*e^4)*x)*sqrt(c^2*x^2 - 1))/(c^4*d^8*e - 2*c^2*d^6*e^3 + d^4*e^5 + (c^4*d^6*e^3 - 2*c^2*d^4*e^5 + d^2*e^7)*x^2 + 2*(c^4*d^7*e^2 ...
\[ \int \frac {\text {arccosh}(c x)}{(d+e x)^3} \, dx=\int \frac {\operatorname {acosh}{\left (c x \right )}}{\left (d + e x\right )^{3}}\, dx \] Input:
integrate(acosh(c*x)/(e*x+d)**3,x)
Output:
Integral(acosh(c*x)/(d + e*x)**3, x)
Exception generated. \[ \int \frac {\text {arccosh}(c x)}{(d+e x)^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(arccosh(c*x)/(e*x+d)^3,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((e-c*d)*(e+c*d)>0)', see `assume ?` for mor
Exception generated. \[ \int \frac {\text {arccosh}(c x)}{(d+e x)^3} \, dx=\text {Exception raised: TypeError} \] Input:
integrate(arccosh(c*x)/(e*x+d)^3,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Limit: Max order reached or unable to make series expansion Error: Bad Argument Value
Timed out. \[ \int \frac {\text {arccosh}(c x)}{(d+e x)^3} \, dx=\int \frac {\mathrm {acosh}\left (c\,x\right )}{{\left (d+e\,x\right )}^3} \,d x \] Input:
int(acosh(c*x)/(d + e*x)^3,x)
Output:
int(acosh(c*x)/(d + e*x)^3, x)
\[ \int \frac {\text {arccosh}(c x)}{(d+e x)^3} \, dx=\int \frac {\mathit {acosh} \left (c x \right )}{e^{3} x^{3}+3 d \,e^{2} x^{2}+3 d^{2} e x +d^{3}}d x \] Input:
int(acosh(c*x)/(e*x+d)^3,x)
Output:
int(acosh(c*x)/(d**3 + 3*d**2*e*x + 3*d*e**2*x**2 + e**3*x**3),x)