Integrand size = 16, antiderivative size = 88 \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^2} \, dx=-\frac {a+b \text {arccosh}(c x)}{e (d+e x)}+\frac {2 b c \text {arctanh}\left (\frac {\sqrt {c d+e} \sqrt {1+c x}}{\sqrt {c d-e} \sqrt {-1+c x}}\right )}{\sqrt {c d-e} e \sqrt {c d+e}} \] Output:
-(a+b*arccosh(c*x))/e/(e*x+d)+2*b*c*arctanh((c*d+e)^(1/2)*(c*x+1)^(1/2)/(c *d-e)^(1/2)/(c*x-1)^(1/2))/(c*d-e)^(1/2)/e/(c*d+e)^(1/2)
Time = 0.13 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.38 \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^2} \, dx=-\frac {\frac {a}{d+e x}+\frac {b \text {arccosh}(c x)}{d+e x}-\frac {b c \log (d+e x)}{\sqrt {c^2 d^2-e^2}}+\frac {b c \log \left (e+c^2 d x-\sqrt {c^2 d^2-e^2} \sqrt {-1+c x} \sqrt {1+c x}\right )}{\sqrt {c^2 d^2-e^2}}}{e} \] Input:
Integrate[(a + b*ArcCosh[c*x])/(d + e*x)^2,x]
Output:
-((a/(d + e*x) + (b*ArcCosh[c*x])/(d + e*x) - (b*c*Log[d + e*x])/Sqrt[c^2* d^2 - e^2] + (b*c*Log[e + c^2*d*x - Sqrt[c^2*d^2 - e^2]*Sqrt[-1 + c*x]*Sqr t[1 + c*x]])/Sqrt[c^2*d^2 - e^2])/e)
Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.188, Rules used = {6378, 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.
\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^2} \, dx\) |
\(\Big \downarrow \) 6378 |
\(\displaystyle \frac {b c \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1} (d+e x)}dx}{e}-\frac {a+b \text {arccosh}(c x)}{e (d+e x)}\) |
\(\Big \downarrow \) 104 |
\(\displaystyle \frac {2 b c \int \frac {1}{c d-e-\frac {(c d+e) (c x+1)}{c x-1}}d\frac {\sqrt {c x+1}}{\sqrt {c x-1}}}{e}-\frac {a+b \text {arccosh}(c x)}{e (d+e x)}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 b c \text {arctanh}\left (\frac {\sqrt {c x+1} \sqrt {c d+e}}{\sqrt {c x-1} \sqrt {c d-e}}\right )}{e \sqrt {c d-e} \sqrt {c d+e}}-\frac {a+b \text {arccosh}(c x)}{e (d+e x)}\) |
Input:
Int[(a + b*ArcCosh[c*x])/(d + e*x)^2,x]
Output:
-((a + b*ArcCosh[c*x])/(e*(d + e*x))) + (2*b*c*ArcTanh[(Sqrt[c*d + e]*Sqrt [1 + c*x])/(Sqrt[c*d - e]*Sqrt[-1 + c*x])])/(Sqrt[c*d - e]*e*Sqrt[c*d + 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_)^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]
Time = 1.01 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.60
method | result | size |
parts | \(-\frac {a}{\left (e x +d \right ) e}-\frac {b c \,\operatorname {arccosh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {b c \sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}}\) | \(141\) |
derivativedivides | \(\frac {-\frac {a \,c^{2}}{\left (e c x +c d \right ) e}+b \,c^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {\sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}}\right )}{c}\) | \(153\) |
default | \(\frac {-\frac {a \,c^{2}}{\left (e c x +c d \right ) e}+b \,c^{2} \left (-\frac {\operatorname {arccosh}\left (c x \right )}{\left (e c x +c d \right ) e}-\frac {\sqrt {c x +1}\, \sqrt {c x -1}\, \ln \left (-\frac {2 \left (c^{2} d x -\sqrt {c^{2} x^{2}-1}\, \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, e +e \right )}{e c x +c d}\right )}{e^{2} \sqrt {\frac {c^{2} d^{2}-e^{2}}{e^{2}}}\, \sqrt {c^{2} x^{2}-1}}\right )}{c}\) | \(153\) |
Input:
int((a+b*arccosh(c*x))/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
-a/(e*x+d)/e-b*c/(c*e*x+c*d)/e*arccosh(c*x)-b*c/e^2*(c*x+1)^(1/2)*(c*x-1)^ (1/2)*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)/(c*e *x+c*d))/((c^2*d^2-e^2)/e^2)^(1/2)/(c^2*x^2-1)^(1/2)
Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (76) = 152\).
Time = 0.11 (sec) , antiderivative size = 507, normalized size of antiderivative = 5.76 \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^2} \, dx=\left [-\frac {a c^{2} d^{3} - a d e^{2} - {\left (b c^{2} d^{2} e - b e^{3}\right )} x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c d e x + b c d^{2}\right )} \sqrt {c^{2} d^{2} - e^{2}} \log \left (\frac {c^{3} d^{2} x + c d e + \sqrt {c^{2} d^{2} - e^{2}} {\left (c^{2} d x + e\right )} + {\left (c^{2} d^{2} + \sqrt {c^{2} d^{2} - e^{2}} c d - e^{2}\right )} \sqrt {c^{2} x^{2} - 1}}{e x + d}\right ) - {\left (b c^{2} d^{3} - b d e^{2} + {\left (b c^{2} d^{2} e - b e^{3}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}, -\frac {a c^{2} d^{3} - a d e^{2} - {\left (b c^{2} d^{2} e - b e^{3}\right )} x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 2 \, {\left (b c d e x + b c d^{2}\right )} \sqrt {-c^{2} d^{2} + e^{2}} \arctan \left (-\frac {\sqrt {-c^{2} d^{2} + e^{2}} \sqrt {c^{2} x^{2} - 1} e - \sqrt {-c^{2} d^{2} + e^{2}} {\left (c e x + c d\right )}}{c^{2} d^{2} - e^{2}}\right ) - {\left (b c^{2} d^{3} - b d e^{2} + {\left (b c^{2} d^{2} e - b e^{3}\right )} x\right )} \log \left (-c x + \sqrt {c^{2} x^{2} - 1}\right )}{c^{2} d^{4} e - d^{2} e^{3} + {\left (c^{2} d^{3} e^{2} - d e^{4}\right )} x}\right ] \] Input:
integrate((a+b*arccosh(c*x))/(e*x+d)^2,x, algorithm="fricas")
Output:
[-(a*c^2*d^3 - a*d*e^2 - (b*c^2*d^2*e - b*e^3)*x*log(c*x + sqrt(c^2*x^2 - 1)) - (b*c*d*e*x + b*c*d^2)*sqrt(c^2*d^2 - e^2)*log((c^3*d^2*x + c*d*e + s qrt(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)) - (b*c^2*d^3 - b*d*e^2 + (b*c^2*d^2*e - b *e^3)*x)*log(-c*x + sqrt(c^2*x^2 - 1)))/(c^2*d^4*e - d^2*e^3 + (c^2*d^3*e^ 2 - d*e^4)*x), -(a*c^2*d^3 - a*d*e^2 - (b*c^2*d^2*e - b*e^3)*x*log(c*x + s qrt(c^2*x^2 - 1)) + 2*(b*c*d*e*x + b*c*d^2)*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)) - (b*c^2*d^3 - b*d*e^2 + (b*c^2*d^2*e - b*e^3)*x)*lo g(-c*x + sqrt(c^2*x^2 - 1)))/(c^2*d^4*e - d^2*e^3 + (c^2*d^3*e^2 - d*e^4)* x)]
\[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^2} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (d + e x\right )^{2}}\, dx \] Input:
integrate((a+b*acosh(c*x))/(e*x+d)**2,x)
Output:
Integral((a + b*acosh(c*x))/(d + e*x)**2, x)
Exception generated. \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^2} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*arccosh(c*x))/(e*x+d)^2,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
Leaf count of result is larger than twice the leaf count of optimal. 245 vs. \(2 (76) = 152\).
Time = 0.39 (sec) , antiderivative size = 245, normalized size of antiderivative = 2.78 \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^2} \, dx={\left (\frac {c \log \left ({\left | c^{2} d e - \sqrt {c^{2} d^{2} - e^{2}} {\left | c \right |} {\left | e \right |} \right |}\right ) \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}{\sqrt {c^{2} d^{2} - e^{2}} {\left | e \right |}} - \frac {\log \left (c x + \sqrt {c^{2} x^{2} - 1}\right )}{{\left (e x + d\right )} e} - \frac {c \log \left ({\left | c^{2} d e - \sqrt {c^{2} d^{2} - e^{2}} {\left (\sqrt {c^{2} - \frac {2 \, c^{2} d}{e x + d} + \frac {c^{2} d^{2}}{{\left (e x + d\right )}^{2}} - \frac {e^{2}}{{\left (e x + d\right )}^{2}}} + \frac {\sqrt {c^{2} d^{2} e^{2} - e^{4}}}{{\left (e x + d\right )} e}\right )} {\left | e \right |} \right |}\right )}{\sqrt {c^{2} d^{2} - e^{2}} {\left | e \right |} \mathrm {sgn}\left (\frac {1}{e x + d}\right ) \mathrm {sgn}\left (e\right )}\right )} b - \frac {a}{{\left (e x + d\right )} e} \] Input:
integrate((a+b*arccosh(c*x))/(e*x+d)^2,x, algorithm="giac")
Output:
(c*log(abs(c^2*d*e - sqrt(c^2*d^2 - e^2)*abs(c)*abs(e)))*sgn(1/(e*x + d))* sgn(e)/(sqrt(c^2*d^2 - e^2)*abs(e)) - log(c*x + sqrt(c^2*x^2 - 1))/((e*x + d)*e) - c*log(abs(c^2*d*e - sqrt(c^2*d^2 - e^2)*(sqrt(c^2 - 2*c^2*d/(e*x + d) + c^2*d^2/(e*x + d)^2 - e^2/(e*x + d)^2) + sqrt(c^2*d^2*e^2 - e^4)/(( e*x + d)*e))*abs(e)))/(sqrt(c^2*d^2 - e^2)*abs(e)*sgn(1/(e*x + d))*sgn(e)) )*b - a/((e*x + d)*e)
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^2} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (d+e\,x\right )}^2} \,d x \] Input:
int((a + b*acosh(c*x))/(d + e*x)^2,x)
Output:
int((a + b*acosh(c*x))/(d + e*x)^2, x)
\[ \int \frac {a+b \text {arccosh}(c x)}{(d+e x)^2} \, dx=\frac {\left (\int \frac {\mathit {acosh} \left (c x \right )}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b \,d^{2}+\left (\int \frac {\mathit {acosh} \left (c x \right )}{e^{2} x^{2}+2 d e x +d^{2}}d x \right ) b d e x +a x}{d \left (e x +d \right )} \] Input:
int((a+b*acosh(c*x))/(e*x+d)^2,x)
Output:
(int(acosh(c*x)/(d**2 + 2*d*e*x + e**2*x**2),x)*b*d**2 + int(acosh(c*x)/(d **2 + 2*d*e*x + e**2*x**2),x)*b*d*e*x + a*x)/(d*(d + e*x))