Integrand size = 10, antiderivative size = 97 \[ \int (d+e x) \text {arccosh}(c x) \, dx=-\frac {3 d \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {\sqrt {-1+c x} \sqrt {1+c x} (d+e x)}{4 c}-\frac {1}{4} \left (\frac {2 d^2}{e}+\frac {e}{c^2}\right ) \text {arccosh}(c x)+\frac {(d+e x)^2 \text {arccosh}(c x)}{2 e} \] Output:
-3/4*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/4*(c*x-1)^(1/2)*(c*x+1)^(1/2)*(e*x+ d)/c-1/4*(2*d^2/e+e/c^2)*arccosh(c*x)+1/2*(e*x+d)^2*arccosh(c*x)/e
Time = 0.05 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.75 \[ \int (d+e x) \text {arccosh}(c x) \, dx=-\frac {c \sqrt {-1+c x} \sqrt {1+c x} (4 d+e x)-2 c^2 x (2 d+e x) \text {arccosh}(c x)+2 e \text {arctanh}\left (\sqrt {\frac {-1+c x}{1+c x}}\right )}{4 c^2} \] Input:
Integrate[(d + e*x)*ArcCosh[c*x],x]
Output:
-1/4*(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(4*d + e*x) - 2*c^2*x*(2*d + e*x)*Arc Cosh[c*x] + 2*e*ArcTanh[Sqrt[(-1 + c*x)/(1 + c*x)]])/c^2
Time = 0.24 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.12, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {6378, 101, 90, 43}
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.
\(\displaystyle \int \text {arccosh}(c x) (d+e x) \, dx\) |
\(\Big \downarrow \) 6378 |
\(\displaystyle \frac {\text {arccosh}(c x) (d+e x)^2}{2 e}-\frac {c \int \frac {(d+e x)^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 e}\) |
\(\Big \downarrow \) 101 |
\(\displaystyle \frac {\text {arccosh}(c x) (d+e x)^2}{2 e}-\frac {c \left (\frac {\int \frac {2 d^2 c^2+3 d e x c^2+e^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)}{2 c^2}\right )}{2 e}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {\text {arccosh}(c x) (d+e x)^2}{2 e}-\frac {c \left (\frac {\left (2 c^2 d^2+e^2\right ) \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx+3 d e \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)}{2 c^2}\right )}{2 e}\) |
\(\Big \downarrow \) 43 |
\(\displaystyle \frac {\text {arccosh}(c x) (d+e x)^2}{2 e}-\frac {c \left (\frac {\frac {\text {arccosh}(c x) \left (2 c^2 d^2+e^2\right )}{c}+3 d e \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)}{2 c^2}\right )}{2 e}\) |
Input:
Int[(d + e*x)*ArcCosh[c*x],x]
Output:
((d + e*x)^2*ArcCosh[c*x])/(2*e) - (c*((e*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(d + e*x))/(2*c^2) + (3*d*e*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + ((2*c^2*d^2 + e^2) *ArcCosh[c*x])/c)/(2*c^2)))/(2*e)
Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ ArcCosh[b*(x/a)]/(b*Sqrt[d/b]), x] /; FreeQ[{a, b, c, d}, x] && EqQ[b*c + a *d, 0] && GtQ[a, 0] && GtQ[d/b, 0]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
Int[((a_.) + (b_.)*(x_))^2*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^( p_), x_] :> Simp[b*(a + b*x)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 3))), x] + Simp[1/(d*f*(n + p + 3)) Int[(c + d*x)^n*(e + f*x)^p*Simp [a^2*d*f*(n + p + 3) - b*(b*c*e + a*(d*e*(n + 1) + c*f*(p + 1))) + b*(a*d*f *(n + p + 4) - b*(d*e*(n + 2) + c*f*(p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 3, 0]
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 = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(\frac {\operatorname {arccosh}\left (c x \right ) c x d +\frac {c \,\operatorname {arccosh}\left (c x \right ) e \,x^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 d c \sqrt {c^{2} x^{2}-1}+\sqrt {c^{2} x^{2}-1}\, e c x +\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) e \right )}{4 c \sqrt {c^{2} x^{2}-1}}}{c}\) | \(104\) |
default | \(\frac {\operatorname {arccosh}\left (c x \right ) c x d +\frac {c \,\operatorname {arccosh}\left (c x \right ) e \,x^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 d c \sqrt {c^{2} x^{2}-1}+\sqrt {c^{2} x^{2}-1}\, e c x +\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) e \right )}{4 c \sqrt {c^{2} x^{2}-1}}}{c}\) | \(104\) |
parts | \(\frac {\operatorname {arccosh}\left (c x \right ) e \,x^{2}}{2}+\operatorname {arccosh}\left (c x \right ) d x -\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (\operatorname {csgn}\left (c \right ) c \sqrt {c^{2} x^{2}-1}\, e x +4 \,\operatorname {csgn}\left (c \right ) c \sqrt {c^{2} x^{2}-1}\, d +\ln \left (\left (\sqrt {c^{2} x^{2}-1}\, \operatorname {csgn}\left (c \right )+c x \right ) \operatorname {csgn}\left (c \right )\right ) e \right ) \operatorname {csgn}\left (c \right )}{4 c^{2} \sqrt {c^{2} x^{2}-1}}\) | \(110\) |
orering | \(\frac {\left (3 e^{2} c^{2} x^{3}+10 e \,c^{2} x^{2} d +4 c^{2} d^{2} x -2 e^{2} x -5 d e \right ) \operatorname {arccosh}\left (c x \right )}{4 \left (e x +d \right ) c^{2}}-\frac {\left (e x +4 d \right ) \left (c x -1\right ) \left (c x +1\right ) \left (e \,\operatorname {arccosh}\left (c x \right )+\frac {\left (e x +d \right ) c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{4 c^{2} \left (e x +d \right )}\) | \(116\) |
Input:
int((e*x+d)*arccosh(c*x),x,method=_RETURNVERBOSE)
Output:
1/c*(arccosh(c*x)*c*x*d+1/2*c*arccosh(c*x)*e*x^2-1/4/c*(c*x-1)^(1/2)*(c*x+ 1)^(1/2)*(4*d*c*(c^2*x^2-1)^(1/2)+(c^2*x^2-1)^(1/2)*e*c*x+ln(c*x+(c^2*x^2- 1)^(1/2))*e)/(c^2*x^2-1)^(1/2))
Time = 0.10 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.67 \[ \int (d+e x) \text {arccosh}(c x) \, dx=\frac {{\left (2 \, c^{2} e x^{2} + 4 \, c^{2} d x - e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - \sqrt {c^{2} x^{2} - 1} {\left (c e x + 4 \, c d\right )}}{4 \, c^{2}} \] Input:
integrate((e*x+d)*arccosh(c*x),x, algorithm="fricas")
Output:
1/4*((2*c^2*e*x^2 + 4*c^2*d*x - e)*log(c*x + sqrt(c^2*x^2 - 1)) - sqrt(c^2 *x^2 - 1)*(c*e*x + 4*c*d))/c^2
\[ \int (d+e x) \text {arccosh}(c x) \, dx=\int \left (d + e x\right ) \operatorname {acosh}{\left (c x \right )}\, dx \] Input:
integrate((e*x+d)*acosh(c*x),x)
Output:
Integral((d + e*x)*acosh(c*x), x)
Time = 0.02 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.85 \[ \int (d+e x) \text {arccosh}(c x) \, dx=-\frac {1}{4} \, c {\left (\frac {\sqrt {c^{2} x^{2} - 1} e x}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} d}{c^{2}} + \frac {e \log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )} + \frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )} \operatorname {arcosh}\left (c x\right ) \] Input:
integrate((e*x+d)*arccosh(c*x),x, algorithm="maxima")
Output:
-1/4*c*(sqrt(c^2*x^2 - 1)*e*x/c^2 + 4*sqrt(c^2*x^2 - 1)*d/c^2 + e*log(2*c^ 2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^3) + 1/2*(e*x^2 + 2*d*x)*arccosh(c*x)
Time = 0.13 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.87 \[ \int (d+e x) \text {arccosh}(c x) \, dx=\frac {1}{2} \, {\left (e x^{2} + 2 \, d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - \frac {1}{4} \, \sqrt {c^{2} x^{2} - 1} {\left (\frac {e x}{c} + \frac {4 \, d}{c}\right )} + \frac {e \log \left ({\left | -x {\left | c \right |} + \sqrt {c^{2} x^{2} - 1} \right |}\right )}{4 \, c {\left | c \right |}} \] Input:
integrate((e*x+d)*arccosh(c*x),x, algorithm="giac")
Output:
1/2*(e*x^2 + 2*d*x)*log(c*x + sqrt(c^2*x^2 - 1)) - 1/4*sqrt(c^2*x^2 - 1)*( e*x/c + 4*d/c) + 1/4*e*log(abs(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c*abs(c))
Time = 3.61 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.70 \[ \int (d+e x) \text {arccosh}(c x) \, dx=d\,x\,\mathrm {acosh}\left (c\,x\right )+e\,x\,\mathrm {acosh}\left (c\,x\right )\,\left (\frac {x}{2}-\frac {1}{4\,c^2\,x}\right )-\frac {d\,\sqrt {c\,x-1}\,\sqrt {c\,x+1}}{c}-\frac {e\,x\,\sqrt {c\,x-1}\,\sqrt {c\,x+1}}{4\,c} \] Input:
int(acosh(c*x)*(d + e*x),x)
Output:
d*x*acosh(c*x) + e*x*acosh(c*x)*(x/2 - 1/(4*c^2*x)) - (d*(c*x - 1)^(1/2)*( c*x + 1)^(1/2))/c - (e*x*(c*x - 1)^(1/2)*(c*x + 1)^(1/2))/(4*c)
Time = 0.17 (sec) , antiderivative size = 79, normalized size of antiderivative = 0.81 \[ \int (d+e x) \text {arccosh}(c x) \, dx=\frac {4 \mathit {acosh} \left (c x \right ) c^{2} d x +2 \mathit {acosh} \left (c x \right ) c^{2} e \,x^{2}-\sqrt {c^{2} x^{2}-1}\, c e x -4 \sqrt {c x +1}\, \sqrt {c x -1}\, c d -\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) e}{4 c^{2}} \] Input:
int((e*x+d)*acosh(c*x),x)
Output:
(4*acosh(c*x)*c**2*d*x + 2*acosh(c*x)*c**2*e*x**2 - sqrt(c**2*x**2 - 1)*c* e*x - 4*sqrt(c*x + 1)*sqrt(c*x - 1)*c*d - log(sqrt(c**2*x**2 - 1) + c*x)*e )/(4*c**2)