Integrand size = 25, antiderivative size = 74 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {b x \sqrt {d-c^2 d x^2}}{c d \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^2 d} \] Output:
b*x*(-c^2*d*x^2+d)^(1/2)/c/d/(c*x-1)^(1/2)/(c*x+1)^(1/2)-(-c^2*d*x^2+d)^(1 /2)*(a+b*arccosh(c*x))/c^2/d
Time = 0.19 (sec) , antiderivative size = 85, normalized size of antiderivative = 1.15 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {d-c^2 d x^2} \left (a-a c^2 x^2+b c x \sqrt {-1+c x} \sqrt {1+c x}+\left (b-b c^2 x^2\right ) \text {arccosh}(c x)\right )}{c^2 d (-1+c x) (1+c x)} \] Input:
Integrate[(x*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]
Output:
(Sqrt[d - c^2*d*x^2]*(a - a*c^2*x^2 + b*c*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x] + (b - b*c^2*x^2)*ArcCosh[c*x]))/(c^2*d*(-1 + c*x)*(1 + c*x))
Time = 0.44 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.97, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {6329, 24}
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.
\(\displaystyle \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx\) |
\(\Big \downarrow \) 6329 |
\(\displaystyle -\frac {b \sqrt {c x-1} \sqrt {c x+1} \int 1dx}{c \sqrt {d-c^2 d x^2}}-\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^2 d}\) |
\(\Big \downarrow \) 24 |
\(\displaystyle -\frac {\sqrt {d-c^2 d x^2} (a+b \text {arccosh}(c x))}{c^2 d}-\frac {b x \sqrt {c x-1} \sqrt {c x+1}}{c \sqrt {d-c^2 d x^2}}\) |
Input:
Int[(x*(a + b*ArcCosh[c*x]))/Sqrt[d - c^2*d*x^2],x]
Output:
-((b*x*Sqrt[-1 + c*x]*Sqrt[1 + c*x])/(c*Sqrt[d - c^2*d*x^2])) - (Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(c^2*d)
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*(x_)*((d_) + (e_.)*(x_)^2)^(p _.), x_Symbol] :> Simp[(d + e*x^2)^(p + 1)*((a + b*ArcCosh[c*x])^n/(2*e*(p + 1))), x] - Simp[b*(n/(2*c*(p + 1)))*Simp[(d + e*x^2)^p/((1 + c*x)^p*(-1 + c*x)^p)] Int[(1 + c*x)^(p + 1/2)*(-1 + c*x)^(p + 1/2)*(a + b*ArcCosh[c*x ])^(n - 1), x], x] /; FreeQ[{a, b, c, d, e, p}, x] && EqQ[c^2*d + e, 0] && GtQ[n, 0] && NeQ[p, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(133\) vs. \(2(66)=132\).
Time = 0.34 (sec) , antiderivative size = 134, normalized size of antiderivative = 1.81
method | result | size |
orering | \(\frac {\left (c^{2} x^{2}-2\right ) \left (a +b \,\operatorname {arccosh}\left (c x \right )\right )}{c^{2} \sqrt {-c^{2} d \,x^{2}+d}}-\frac {\left (c x -1\right ) \left (c x +1\right ) \left (\frac {a +b \,\operatorname {arccosh}\left (c x \right )}{\sqrt {-c^{2} d \,x^{2}+d}}+\frac {x b c}{\sqrt {c x -1}\, \sqrt {c x +1}\, \sqrt {-c^{2} d \,x^{2}+d}}+\frac {x^{2} \left (a +b \,\operatorname {arccosh}\left (c x \right )\right ) c^{2} d}{\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}}}\right )}{c^{2}}\) | \(134\) |
default | \(-\frac {a \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\) | \(158\) |
parts | \(-\frac {a \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) \left (1+\operatorname {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\) | \(158\) |
Input:
int(x*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERBOSE)
Output:
(c^2*x^2-2)/c^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2)-1/c^2*(c*x-1)*(c*x +1)*((a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2)+x*b*c/(c*x-1)^(1/2)/(c*x+1)^( 1/2)/(-c^2*d*x^2+d)^(1/2)+x^2*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(3/2)*c^2* d)
Time = 0.09 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.58 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {\sqrt {-c^{2} d x^{2} + d} \sqrt {c^{2} x^{2} - 1} b c x - {\left (b c^{2} x^{2} - b\right )} \sqrt {-c^{2} d x^{2} + d} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) - {\left (a c^{2} x^{2} - a\right )} \sqrt {-c^{2} d x^{2} + d}}{c^{4} d x^{2} - c^{2} d} \] Input:
integrate(x*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="fricas")
Output:
(sqrt(-c^2*d*x^2 + d)*sqrt(c^2*x^2 - 1)*b*c*x - (b*c^2*x^2 - b)*sqrt(-c^2* d*x^2 + d)*log(c*x + sqrt(c^2*x^2 - 1)) - (a*c^2*x^2 - a)*sqrt(-c^2*d*x^2 + d))/(c^4*d*x^2 - c^2*d)
\[ \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x \left (a + b \operatorname {acosh}{\left (c x \right )}\right )}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \] Input:
integrate(x*(a+b*acosh(c*x))/(-c**2*d*x**2+d)**(1/2),x)
Output:
Integral(x*(a + b*acosh(c*x))/sqrt(-d*(c*x - 1)*(c*x + 1)), x)
Time = 0.04 (sec) , antiderivative size = 63, normalized size of antiderivative = 0.85 \[ \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {b \sqrt {-d} x}{c d} - \frac {\sqrt {-c^{2} d x^{2} + d} b \operatorname {arcosh}\left (c x\right )}{c^{2} d} - \frac {\sqrt {-c^{2} d x^{2} + d} a}{c^{2} d} \] Input:
integrate(x*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="maxima")
Output:
b*sqrt(-d)*x/(c*d) - sqrt(-c^2*d*x^2 + d)*b*arccosh(c*x)/(c^2*d) - sqrt(-c ^2*d*x^2 + d)*a/(c^2*d)
\[ \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (b \operatorname {arcosh}\left (c x\right ) + a\right )} x}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \] Input:
integrate(x*(a+b*arccosh(c*x))/(-c^2*d*x^2+d)^(1/2),x, algorithm="giac")
Output:
integrate((b*arccosh(c*x) + a)*x/sqrt(-c^2*d*x^2 + d), x)
Timed out. \[ \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {x\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:
int((x*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(1/2),x)
Output:
int((x*(a + b*acosh(c*x)))/(d - c^2*d*x^2)^(1/2), x)
\[ \int \frac {x (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-\sqrt {-c^{2} x^{2}+1}\, a +\left (\int \frac {\mathit {acosh} \left (c x \right ) x}{\sqrt {-c^{2} x^{2}+1}}d x \right ) b \,c^{2}}{\sqrt {d}\, c^{2}} \] Input:
int(x*(a+b*acosh(c*x))/(-c^2*d*x^2+d)^(1/2),x)
Output:
( - sqrt( - c**2*x**2 + 1)*a + int((acosh(c*x)*x)/sqrt( - c**2*x**2 + 1),x )*b*c**2)/(sqrt(d)*c**2)