Integrand size = 31, antiderivative size = 549 \[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\frac {b g^3 x \sqrt {-1+c x} \sqrt {1+c x}}{c^3 d \sqrt {d-c^2 d x^2}}-\frac {(c f-g)^3 (1-c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {(c f+g)^3 (1+c x) (a+b \text {arccosh}(c x))}{2 c^4 d \sqrt {d-c^2 d x^2}}+\frac {g^3 (1-c x) (1+c x) (a+b \text {arccosh}(c x))}{c^4 d \sqrt {d-c^2 d x^2}}-\frac {3 f g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x))^2}{2 b c^3 d \sqrt {d-c^2 d x^2}}+\frac {b (c f+g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\sqrt {-\frac {1-c x}{1+c x}}\right )}{c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b (c f-g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{1+c x}\right )}{2 c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b (c f+g)^3 \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{1+c x}\right )}{2 c^4 d \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}} \]
-1/2*(c*f-g)^3*(-c*x+1)*(a+b*arccosh(c*x))/c^4/d/(-c^2*d*x^2+d)^(1/2)+1/2* (c*f+g)^3*(c*x+1)*(a+b*arccosh(c*x))/c^4/d/(-c^2*d*x^2+d)^(1/2)+g^3*(-c*x+ 1)*(c*x+1)*(a+b*arccosh(c*x))/c^4/d/(-c^2*d*x^2+d)^(1/2)+b*g^3*x*(c*x-1)^( 1/2)*(c*x+1)^(1/2)/c^3/d/(-c^2*d*x^2+d)^(1/2)-3/2*f*g^2*(a+b*arccosh(c*x)) ^2*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c^3/d/(-c^2*d*x^2+d)^(1/2)-1/2*b*(c*f-g)^ 3*ln(2/(c*x+1))*((-c*x+1)*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^4/d/(c*x+1)/ ((c*x-1)/(c*x+1))^(1/2)/(-c^2*d*x^2+d)^(1/2)-1/2*b*(c*f+g)^3*ln(2/(c*x+1)) *((-c*x+1)*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^4/d/(c*x+1)/((c*x-1)/(c*x+1 ))^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/2*b*(c*f+g)^3*ln((c*x-1)/(c*x+1))*((-c*x+1 )*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/c^4/d/(c*x+1)/((c*x-1)/(c*x+1))^(1/2)/ (-c^2*d*x^2+d)^(1/2)
Time = 2.11 (sec) , antiderivative size = 501, normalized size of antiderivative = 0.91 \[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\sqrt {-d \left (-1+c^2 x^2\right )} \left (\frac {a g^3}{c^4 d^2}-\frac {a \left (3 c^2 f^2 g+g^3+c^4 f^3 x+3 c^2 f g^2 x\right )}{c^4 d^2 \left (-1+c^2 x^2\right )}\right )+\frac {3 a f g^2 \arctan \left (\frac {c x \sqrt {-d \left (-1+c^2 x^2\right )}}{\sqrt {d} \left (-1+c^2 x^2\right )}\right )}{c^3 d^{3/2}}-\frac {b f^3 \left (-c x \text {arccosh}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )}{c d \sqrt {-d (-1+c x) (1+c x)}}-\frac {3 b f g^2 \left (-2 c x \text {arccosh}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\text {arccosh}(c x)^2+2 \log \left (\sqrt {\frac {-1+c x}{1+c x}} (1+c x)\right )\right )\right )}{2 c^3 d \sqrt {-d (-1+c x) (1+c x)}}+\frac {3 b f^2 g \left (\text {arccosh}(c x)+\sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )\right )\right )}{c^2 d \sqrt {-d (-1+c x) (1+c x)}}-\frac {b g^3 \left (-3 \text {arccosh}(c x)+\text {arccosh}(c x) \cosh (2 \text {arccosh}(c x))-2 \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \left (\log \left (\cosh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )-\log \left (\sinh \left (\frac {1}{2} \text {arccosh}(c x)\right )\right )\right )-\sinh (2 \text {arccosh}(c x))\right )}{2 c^4 d \sqrt {-d (-1+c x) (1+c x)}} \]
Sqrt[-(d*(-1 + c^2*x^2))]*((a*g^3)/(c^4*d^2) - (a*(3*c^2*f^2*g + g^3 + c^4 *f^3*x + 3*c^2*f*g^2*x))/(c^4*d^2*(-1 + c^2*x^2))) + (3*a*f*g^2*ArcTan[(c* x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(c^3*d^(3/2)) - (b *f^3*(-(c*x*ArcCosh[c*x]) + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*Log[Sqrt[ (-1 + c*x)/(1 + c*x)]*(1 + c*x)]))/(c*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]) - (3*b*f*g^2*(-2*c*x*ArcCosh[c*x] + Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(A rcCosh[c*x]^2 + 2*Log[Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)])))/(2*c^3*d*Sq rt[-(d*(-1 + c*x)*(1 + c*x))]) + (3*b*f^2*g*(ArcCosh[c*x] + Sqrt[(-1 + c*x )/(1 + c*x)]*(1 + c*x)*(Log[Cosh[ArcCosh[c*x]/2]] - Log[Sinh[ArcCosh[c*x]/ 2]])))/(c^2*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]) - (b*g^3*(-3*ArcCosh[c*x] + ArcCosh[c*x]*Cosh[2*ArcCosh[c*x]] - 2*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x )*(Log[Cosh[ArcCosh[c*x]/2]] - Log[Sinh[ArcCosh[c*x]/2]]) - Sinh[2*ArcCosh [c*x]]))/(2*c^4*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))])
Time = 1.83 (sec) , antiderivative size = 474, normalized size of antiderivative = 0.86, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.097, Rules used = {6387, 6396, 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.
| \(\displaystyle \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 6387 |
| \(\displaystyle -\frac {\sqrt {c x-1} \sqrt {c x+1} \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{(c x-1)^{3/2} (c x+1)^{3/2}}dx}{d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6396 |
| \(\displaystyle -\frac {\sqrt {c x-1} \sqrt {c x+1} \int \left (-\frac {(a+b \text {arccosh}(c x)) (c f-g)^3}{2 c^3 \sqrt {c x-1} (c x+1)^{3/2}}+\frac {3 f g^2 (a+b \text {arccosh}(c x))}{c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {g^3 x (a+b \text {arccosh}(c x))}{c^2 \sqrt {c x-1} \sqrt {c x+1}}+\frac {(c f+g)^3 (a+b \text {arccosh}(c x))}{2 c^3 (c x-1)^{3/2} \sqrt {c x+1}}\right )dx}{d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {c x-1} \sqrt {c x+1} \left (-\frac {\sqrt {c x-1} (c f-g)^3 (a+b \text {arccosh}(c x))}{2 c^4 \sqrt {c x+1}}-\frac {\sqrt {c x+1} (c f+g)^3 (a+b \text {arccosh}(c x))}{2 c^4 \sqrt {c x-1}}+\frac {g^3 \sqrt {c x-1} \sqrt {c x+1} (a+b \text {arccosh}(c x))}{c^4}+\frac {3 f g^2 (a+b \text {arccosh}(c x))^2}{2 b c^3}-\frac {b g^3 x}{c^3}+\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f-g)^3 \log \left (\frac {2}{c x+1}\right )}{2 c^4 \sqrt {c x-1} \sqrt {-\frac {1-c x}{c x+1}} (c x+1)^{3/2}}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f+g)^3 \log \left (\sqrt {-\frac {1-c x}{c x+1}}\right )}{c^4 \sqrt {c x-1} \sqrt {-\frac {1-c x}{c x+1}} (c x+1)^{3/2}}+\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} (c f+g)^3 \log \left (\frac {2}{c x+1}\right )}{2 c^4 \sqrt {c x-1} \sqrt {-\frac {1-c x}{c x+1}} (c x+1)^{3/2}}\right )}{d \sqrt {d-c^2 d x^2}}\) |
-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((b*g^3*x)/c^3) - ((c*f - g)^3*Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c^4*Sqrt[1 + c*x]) - ((c*f + g)^3*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*c^4*Sqrt[-1 + c*x]) + (g^3*Sqrt[-1 + c*x]*Sq rt[1 + c*x]*(a + b*ArcCosh[c*x]))/c^4 + (3*f*g^2*(a + b*ArcCosh[c*x])^2)/( 2*b*c^3) - (b*(c*f + g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[ Sqrt[-((1 - c*x)/(1 + c*x))]])/(c^4*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c *x))]*(1 + c*x)^(3/2)) + (b*(c*f - g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(2*c^4*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c* x))]*(1 + c*x)^(3/2)) + (b*(c*f + g)^3*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(2*c^4*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x ))]*(1 + c*x)^(3/2))))/(d*Sqrt[d - c^2*d*x^2]))
3.1.71.3.1 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[(a + b*ArcCosh[c*x])^n/(Sqrt[d1 + e1*x]*Sqrt[d2 + e2*x]), (f + g* x)^m*(d1 + e1*x)^(p + 1/2)*(d2 + e2*x)^(p + 1/2), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && Int egerQ[m] && ILtQ[p + 1/2, 0] && GtQ[d1, 0] && LtQ[d2, 0] && IGtQ[n, 0]
Leaf count of result is larger than twice the leaf count of optimal. \(1231\) vs. \(2(486)=972\).
Time = 1.94 (sec) , antiderivative size = 1232, normalized size of antiderivative = 2.24
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1232\) |
| parts | \(\text {Expression too large to display}\) | \(1232\) |
a*(f^3/d*x/(-c^2*d*x^2+d)^(1/2)+g^3*(-x^2/c^2/d/(-c^2*d*x^2+d)^(1/2)+2/d/c ^4/(-c^2*d*x^2+d)^(1/2))+3*f*g^2*(x/c^2/d/(-c^2*d*x^2+d)^(1/2)-1/c^2/d/(c^ 2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-c^2*d*x^2+d)^(1/2)))+3*f^2*g/c^2/d/(-c ^2*d*x^2+d)^(1/2))-b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^ 2/c/(c^2*x^2-1)*f^3*arccosh(c*x)-3*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d ^2/(c^2*x^2-1)*x^2*f^2*g+b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/c^4/(c^ 2*x^2-1)*(c*x-1)*(c*x+1)*g^3-3*b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/c ^2/(c^2*x^2-1)*x*f*g^2+3/2*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^ (1/2)/d^2/c^3/(c^2*x^2-1)*arccosh(c*x)^2*f*g^2-b*(-d*(c^2*x^2-1))^(1/2)*ar ccosh(c*x)/d^2/(c^2*x^2-1)*x*f^3-b*(-d*(c^2*x^2-1))^(1/2)*g^3/d^2/c^3/(c^2 *x^2-1)*(c*x+1)^(1/2)*(c*x-1)^(1/2)*x+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/ 2)*(c*x+1)^(1/2)/d^2/c/(c^2*x^2-1)*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*f ^3+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c^4/(c^2*x^2-1 )*ln((c*x-1)^(1/2)*(c*x+1)^(1/2)+c*x-1)*g^3+b*(-d*(c^2*x^2-1))^(1/2)*(c*x- 1)^(1/2)*(c*x+1)^(1/2)*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))/d^2/c/(c^2*x^ 2-1)*f^3-b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(1+c*x+(c* x-1)^(1/2)*(c*x+1)^(1/2))/d^2/c^4/(c^2*x^2-1)*g^3-3*b*(-d*(c^2*x^2-1))^(1/ 2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c^3/(c^2*x^2-1)*f*arccosh(c*x)*g^2+3*b* (-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d^2/c^2/(c^2*x^2-1)*(c*x-1)*(c*x+1)*f^ 2*g+3*b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d^2/c^2/(c^2...
\[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
integral((a*g^3*x^3 + 3*a*f*g^2*x^2 + 3*a*f^2*g*x + a*f^3 + (b*g^3*x^3 + 3 *b*f*g^2*x^2 + 3*b*f^2*g*x + b*f^3)*arccosh(c*x))*sqrt(-c^2*d*x^2 + d)/(c^ 4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2), x)
\[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right ) \left (f + g x\right )^{3}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
-1/2*b*c*f^3*sqrt(-1/(c^4*d))*log(x^2 - 1/c^2)/d - a*g^3*(x^2/(sqrt(-c^2*d *x^2 + d)*c^2*d) - 2/(sqrt(-c^2*d*x^2 + d)*c^4*d)) + 3*a*f*g^2*(x/(sqrt(-c ^2*d*x^2 + d)*c^2*d) - arcsin(c*x)/(c^3*d^(3/2))) + b*f^3*x*arccosh(c*x)/( sqrt(-c^2*d*x^2 + d)*d) + a*f^3*x/(sqrt(-c^2*d*x^2 + d)*d) + 3*a*f^2*g/(sq rt(-c^2*d*x^2 + d)*c^2*d) + integrate(b*g^3*x^3*log(c*x + sqrt(c*x + 1)*sq rt(c*x - 1))/(-c^2*d*x^2 + d)^(3/2) + 3*b*f*g^2*x^2*log(c*x + sqrt(c*x + 1 )*sqrt(c*x - 1))/(-c^2*d*x^2 + d)^(3/2) + 3*b*f^2*g*x*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))/(-c^2*d*x^2 + d)^(3/2), x)
\[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {{\left (g x + f\right )}^{3} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {(f+g x)^3 (a+b \text {arccosh}(c x))}{\left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {{\left (f+g\,x\right )}^3\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}{{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]