Integrand size = 29, antiderivative size = 136 \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=-\frac {b g x \sqrt {-1+c x} \sqrt {1+c x}}{c \sqrt {d-c^2 d x^2}}-\frac {g (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{c^2 \sqrt {d-c^2 d x^2}}+\frac {f \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c \sqrt {d-c^2 d x^2}} \]
-g*(-c*x+1)*(c*x+1)*(a+b*arccosh(c*x))/c^2/(-c^2*d*x^2+d)^(1/2)-b*g*x*(c*x -1)^(1/2)*(c*x+1)^(1/2)/c/(-c^2*d*x^2+d)^(1/2)+1/2*f*(a+b*arccosh(c*x))^2* (c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/(-c^2*d*x^2+d)^(1/2)
Time = 0.73 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.36 \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\frac {-\frac {2 a g \sqrt {d-c^2 d x^2}}{d}+\frac {b c f \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \text {arccosh}(c x)^2}{\sqrt {d-c^2 d x^2}}-\frac {2 b g \sqrt {d-c^2 d x^2} \left (-1+\sqrt {\frac {-1+c x}{1+c x}} \text {arccosh}(c x)\right )}{d \sqrt {\frac {-1+c x}{1+c x}}}-\frac {2 a c f \arctan \left (\frac {c x \sqrt {d-c^2 d x^2}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{\sqrt {d}}}{2 c^2} \]
((-2*a*g*Sqrt[d - c^2*d*x^2])/d + (b*c*f*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c *x)*ArcCosh[c*x]^2)/Sqrt[d - c^2*d*x^2] - (2*b*g*Sqrt[d - c^2*d*x^2]*(-1 + Sqrt[(-1 + c*x)/(1 + c*x)]*ArcCosh[c*x]))/(d*Sqrt[(-1 + c*x)/(1 + c*x)]) - (2*a*c*f*ArcTan[(c*x*Sqrt[d - c^2*d*x^2])/(Sqrt[d]*(-1 + c^2*x^2))])/Sqr t[d])/(2*c^2)
Time = 0.71 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.70, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.103, Rules used = {6387, 6390, 2009}
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.
| \(\displaystyle \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx\) |
| \(\Big \downarrow \) 6387 |
| \(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}dx}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6390 |
| \(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int \left (\frac {f (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}+\frac {g x (a+b \text {arccosh}(c x))}{\sqrt {c x-1} \sqrt {c x+1}}\right )dx}{\sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {g \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^2}+\frac {f (a+b \text {arccosh}(c x))^2}{2 b c}-\frac {b g x}{c}\right )}{\sqrt {d-c^2 d x^2}}\) |
(Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((b*g*x)/c) + (g*Sqrt[-1 + c*x]*Sqrt[1 + c *x]*(a + b*ArcCosh[c*x]))/c^2 + (f*(a + b*ArcCosh[c*x])^2)/(2*b*c)))/Sqrt[ d - c^2*d*x^2]
3.1.68.3.1 Defintions of rubi rules used
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.)*((d _) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(-d)^IntPart[p]*((d + e*x^2)^Fra cPart[p]/((-1 + c*x)^FracPart[p]*(1 + c*x)^FracPart[p])) Int[(f + g*x)^m* (-1 + c*x)^p*(1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && EqQ[c^2*d + e, 0] && IntegerQ[p - 1/2] && IntegerQ[m]
Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*(( d2_) + (e2_.)*(x_))^(p_)*((f_) + (g_.)*(x_))^(m_.), x_Symbol] :> Int[Expand Integrand[(d1 + e1*x)^p*(d2 + e2*x)^p*(a + b*ArcCosh[c*x])^n, (f + g*x)^m, x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IGtQ[m, 0] && IntegerQ[p + 1/2] && GtQ[d1, 0] && LtQ[ d2, 0] && IGtQ[n, 0] && ((EqQ[n, 1] && GtQ[p, -1]) || GtQ[p, 0] || EqQ[m, 1 ] || (EqQ[m, 2] && LtQ[p, -2]))
Leaf count of result is larger than twice the leaf count of optimal. \(247\) vs. \(2(120)=240\).
Time = 2.08 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.82
| method | result | size |
| default | \(\frac {a f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {a g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, f \operatorname {arccosh}\left (c x \right )^{2}}{2 d c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) g \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) g \left (1+\operatorname {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\) | \(248\) |
| parts | \(\frac {a f \arctan \left (\frac {\sqrt {c^{2} d}\, x}{\sqrt {-c^{2} d \,x^{2}+d}}\right )}{\sqrt {c^{2} d}}-\frac {a g \sqrt {-c^{2} d \,x^{2}+d}}{c^{2} d}+b \left (-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \sqrt {c x -1}\, \sqrt {c x +1}\, f \operatorname {arccosh}\left (c x \right )^{2}}{2 d c \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) g \left (-1+\operatorname {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}-\frac {\sqrt {-d \left (c^{2} x^{2}-1\right )}\, \left (-\sqrt {c x -1}\, \sqrt {c x +1}\, c x +c^{2} x^{2}-1\right ) g \left (1+\operatorname {arccosh}\left (c x \right )\right )}{2 c^{2} d \left (c^{2} x^{2}-1\right )}\right )\) | \(248\) |
a*f/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2))-a*g/c^2/d*( -c^2*d*x^2+d)^(1/2)+b*(-1/2*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^( 1/2)/d/c/(c^2*x^2-1)*f*arccosh(c*x)^2-1/2*(-d*(c^2*x^2-1))^(1/2)*((c*x-1)^ (1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*g*(-1+arccosh(c*x))/c^2/d/(c^2*x^2-1)-1 /2*(-d*(c^2*x^2-1))^(1/2)*(-(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*x+c^2*x^2-1)*g*( 1+arccosh(c*x))/c^2/d/(c^2*x^2-1))
\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
integral(-sqrt(-c^2*d*x^2 + d)*(a*g*x + a*f + (b*g*x + b*f)*arccosh(c*x))/ (c^2*d*x^2 - d), x)
\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )}}\, dx \]
\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
a*f*arcsin(c*x)/(c*sqrt(d)) - sqrt(-c^2*d*x^2 + d)*a*g/(c^2*d) + integrate (b*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c^2*d*x^2 + d) + b*f*l og(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/sqrt(-c^2*d*x^2 + d), x)
\[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int { \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{\sqrt {-c^{2} d x^{2} + d}} \,d x } \]
Timed out. \[ \int \frac {(f+g x) (a+b \text {arccosh}(c x))}{\sqrt {d-c^2 d x^2}} \, dx=\int \frac {\left (f+g\,x\right )\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{\sqrt {d-c^2\,d\,x^2}} \,d x \]