Integrand size = 14, antiderivative size = 106 \[ \int (d+e x) (a+b \text {arccosh}(c x)) \, dx=-\frac {3 b d \sqrt {-1+c x} \sqrt {1+c x}}{4 c}-\frac {b \sqrt {-1+c x} \sqrt {1+c x} (d+e x)}{4 c}-\frac {b \left (2 d^2+\frac {e^2}{c^2}\right ) \text {arccosh}(c x)}{4 e}+\frac {(d+e x)^2 (a+b \text {arccosh}(c x))}{2 e} \] Output:
-3/4*b*d*(c*x-1)^(1/2)*(c*x+1)^(1/2)/c-1/4*b*(c*x-1)^(1/2)*(c*x+1)^(1/2)*( e*x+d)/c-1/4*b*(2*d^2+e^2/c^2)*arccosh(c*x)/e+1/2*(e*x+d)^2*(a+b*arccosh(c *x))/e
Time = 0.05 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.10 \[ \int (d+e x) (a+b \text {arccosh}(c x)) \, dx=a d x+\frac {1}{2} a e x^2-\frac {b d \sqrt {-1+c x} \sqrt {1+c x}}{c}-\frac {b e x \sqrt {-1+c x} \sqrt {1+c x}}{4 c}+b d x \text {arccosh}(c x)+\frac {1}{2} b e x^2 \text {arccosh}(c x)-\frac {b e \text {arctanh}\left (\frac {\sqrt {-1+c x}}{\sqrt {1+c x}}\right )}{2 c^2} \] Input:
Integrate[(d + e*x)*(a + b*ArcCosh[c*x]),x]
Output:
a*d*x + (a*e*x^2)/2 - (b*d*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/c - (b*e*x*Sqrt[- 1 + c*x]*Sqrt[1 + c*x])/(4*c) + b*d*x*ArcCosh[c*x] + (b*e*x^2*ArcCosh[c*x] )/2 - (b*e*ArcTanh[Sqrt[-1 + c*x]/Sqrt[1 + c*x]])/(2*c^2)
Time = 0.26 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.08, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, 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.
\(\displaystyle \int (d+e x) (a+b \text {arccosh}(c x)) \, dx\) |
\(\Big \downarrow \) 6378 |
\(\displaystyle \frac {(d+e x)^2 (a+b \text {arccosh}(c x))}{2 e}-\frac {b c \int \frac {(d+e x)^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 e}\) |
\(\Big \downarrow \) 101 |
\(\displaystyle \frac {(d+e x)^2 (a+b \text {arccosh}(c x))}{2 e}-\frac {b c \left (\frac {\int \frac {2 d^2 c^2+3 d e x c^2+e^2}{\sqrt {c x-1} \sqrt {c x+1}}dx}{2 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)}{2 c^2}\right )}{2 e}\) |
\(\Big \downarrow \) 90 |
\(\displaystyle \frac {(d+e x)^2 (a+b \text {arccosh}(c x))}{2 e}-\frac {b c \left (\frac {\left (2 c^2 d^2+e^2\right ) \int \frac {1}{\sqrt {c x-1} \sqrt {c x+1}}dx+3 d e \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)}{2 c^2}\right )}{2 e}\) |
\(\Big \downarrow \) 43 |
\(\displaystyle \frac {(d+e x)^2 (a+b \text {arccosh}(c x))}{2 e}-\frac {b c \left (\frac {\frac {\text {arccosh}(c x) \left (2 c^2 d^2+e^2\right )}{c}+3 d e \sqrt {c x-1} \sqrt {c x+1}}{2 c^2}+\frac {e \sqrt {c x-1} \sqrt {c x+1} (d+e x)}{2 c^2}\right )}{2 e}\) |
Input:
Int[(d + e*x)*(a + b*ArcCosh[c*x]),x]
Output:
((d + e*x)^2*(a + b*ArcCosh[c*x]))/(2*e) - (b*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 = 0.24 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.11
method | result | size |
parts | \(a \left (\frac {1}{2} e \,x^{2}+d x \right )+\frac {b \left (\operatorname {arccosh}\left (c x \right ) c x d +\frac {c \,\operatorname {arccosh}\left (c x \right ) e \,x^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 d c \sqrt {c^{2} x^{2}-1}+\sqrt {c^{2} x^{2}-1}\, e c x +\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) e \right )}{4 c \sqrt {c^{2} x^{2}-1}}\right )}{c}\) | \(118\) |
orering | \(\frac {\left (3 e^{2} c^{2} x^{3}+10 c^{2} d e \,x^{2}+4 c^{2} d^{2} x -2 e^{2} x -5 d e \right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{4 \left (e x +d \right ) c^{2}}-\frac {\left (e x +4 d \right ) \left (c x -1\right ) \left (c x +1\right ) \left (e \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )+\frac {\left (e x +d \right ) b c}{\sqrt {c x -1}\, \sqrt {c x +1}}\right )}{4 c^{2} \left (e x +d \right )}\) | \(125\) |
derivativedivides | \(\frac {\frac {a \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arccosh}\left (c x \right ) c^{2} e \,x^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 d c \sqrt {c^{2} x^{2}-1}+\sqrt {c^{2} x^{2}-1}\, e c x +\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) e \right )}{4 \sqrt {c^{2} x^{2}-1}}\right )}{c}}{c}\) | \(132\) |
default | \(\frac {\frac {a \left (c^{2} d x +\frac {1}{2} c^{2} e \,x^{2}\right )}{c}+\frac {b \left (\operatorname {arccosh}\left (c x \right ) d \,c^{2} x +\frac {\operatorname {arccosh}\left (c x \right ) c^{2} e \,x^{2}}{2}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (4 d c \sqrt {c^{2} x^{2}-1}+\sqrt {c^{2} x^{2}-1}\, e c x +\ln \left (c x +\sqrt {c^{2} x^{2}-1}\right ) e \right )}{4 \sqrt {c^{2} x^{2}-1}}\right )}{c}}{c}\) | \(132\) |
Input:
int((e*x+d)*(a+b*arccosh(c*x)),x,method=_RETURNVERBOSE)
Output:
a*(1/2*e*x^2+d*x)+b/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 = 0.08 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.83 \[ \int (d+e x) (a+b \text {arccosh}(c x)) \, dx=\frac {2 \, a c^{2} e x^{2} + 4 \, a c^{2} d x + {\left (2 \, b c^{2} e x^{2} + 4 \, b c^{2} d x - b e\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (b c e x + 4 \, b c d\right )} \sqrt {c^{2} x^{2} - 1}}{4 \, c^{2}} \] Input:
integrate((e*x+d)*(a+b*arccosh(c*x)),x, algorithm="fricas")
Output:
1/4*(2*a*c^2*e*x^2 + 4*a*c^2*d*x + (2*b*c^2*e*x^2 + 4*b*c^2*d*x - b*e)*log (c*x + sqrt(c^2*x^2 - 1)) - (b*c*e*x + 4*b*c*d)*sqrt(c^2*x^2 - 1))/c^2
\[ \int (d+e x) (a+b \text {arccosh}(c x)) \, dx=\int \left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (d + e x\right )\, dx \] Input:
integrate((e*x+d)*(a+b*acosh(c*x)),x)
Output:
Integral((a + b*acosh(c*x))*(d + e*x), x)
Time = 0.03 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.93 \[ \int (d+e x) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{2} \, a e x^{2} + \frac {1}{4} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} + \frac {\log \left (2 \, c^{2} x + 2 \, \sqrt {c^{2} x^{2} - 1} c\right )}{c^{3}}\right )}\right )} b e + a d x + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d}{c} \] Input:
integrate((e*x+d)*(a+b*arccosh(c*x)),x, algorithm="maxima")
Output:
1/2*a*e*x^2 + 1/4*(2*x^2*arccosh(c*x) - c*(sqrt(c^2*x^2 - 1)*x/c^2 + log(2 *c^2*x + 2*sqrt(c^2*x^2 - 1)*c)/c^3))*b*e + a*d*x + (c*x*arccosh(c*x) - sq rt(c^2*x^2 - 1))*b*d/c
Time = 0.19 (sec) , antiderivative size = 124, normalized size of antiderivative = 1.17 \[ \int (d+e x) (a+b \text {arccosh}(c x)) \, dx=\frac {1}{2} \, a e x^{2} + {\left (x \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - \frac {\sqrt {c^{2} x^{2} - 1}}{c}\right )} b d + \frac {1}{4} \, {\left (2 \, x^{2} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x}{c^{2}} - \frac {\log \left ({\left | -x {\left | c \right |} + \sqrt {c^{2} x^{2} - 1} \right |}\right )}{c^{2} {\left | c \right |}}\right )}\right )} b e + a d x \] Input:
integrate((e*x+d)*(a+b*arccosh(c*x)),x, algorithm="giac")
Output:
1/2*a*e*x^2 + (x*log(c*x + sqrt(c^2*x^2 - 1)) - sqrt(c^2*x^2 - 1)/c)*b*d + 1/4*(2*x^2*log(c*x + sqrt(c^2*x^2 - 1)) - c*(sqrt(c^2*x^2 - 1)*x/c^2 - lo g(abs(-x*abs(c) + sqrt(c^2*x^2 - 1)))/(c^2*abs(c))))*b*e + a*d*x
Time = 3.64 (sec) , antiderivative size = 83, normalized size of antiderivative = 0.78 \[ \int (d+e x) (a+b \text {arccosh}(c x)) \, dx=\frac {a\,x\,\left (2\,d+e\,x\right )}{2}+b\,d\,x\,\mathrm {acosh}\left (c\,x\right )+b\,e\,x\,\mathrm {acosh}\left (c\,x\right )\,\left (\frac {x}{2}-\frac {1}{4\,c^2\,x}\right )-\frac {b\,d\,\sqrt {c\,x-1}\,\sqrt {c\,x+1}}{c}-\frac {b\,e\,x\,\sqrt {c\,x-1}\,\sqrt {c\,x+1}}{4\,c} \] Input:
int((a + b*acosh(c*x))*(d + e*x),x)
Output:
(a*x*(2*d + e*x))/2 + b*d*x*acosh(c*x) + b*e*x*acosh(c*x)*(x/2 - 1/(4*c^2* x)) - (b*d*(c*x - 1)^(1/2)*(c*x + 1)^(1/2))/c - (b*e*x*(c*x - 1)^(1/2)*(c* x + 1)^(1/2))/(4*c)
Time = 0.17 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.96 \[ \int (d+e x) (a+b \text {arccosh}(c x)) \, dx=\frac {4 \mathit {acosh} \left (c x \right ) b \,c^{2} d x +2 \mathit {acosh} \left (c x \right ) b \,c^{2} e \,x^{2}-\sqrt {c^{2} x^{2}-1}\, b c e x -4 \sqrt {c x +1}\, \sqrt {c x -1}\, b c d -\mathrm {log}\left (\sqrt {c^{2} x^{2}-1}+c x \right ) b e +4 a \,c^{2} d x +2 a \,c^{2} e \,x^{2}}{4 c^{2}} \] Input:
int((e*x+d)*(a+b*acosh(c*x)),x)
Output:
(4*acosh(c*x)*b*c**2*d*x + 2*acosh(c*x)*b*c**2*e*x**2 - sqrt(c**2*x**2 - 1 )*b*c*e*x - 4*sqrt(c*x + 1)*sqrt(c*x - 1)*b*c*d - log(sqrt(c**2*x**2 - 1) + c*x)*b*e + 4*a*c**2*d*x + 2*a*c**2*e*x**2)/(4*c**2)