Integrand size = 31, antiderivative size = 773 \[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=-\frac {(1-c x) (a+b \text {arccosh}(c x))}{2 d (c f-g) \sqrt {d-c^2 d x^2}}+\frac {(1+c x) (a+b \text {arccosh}(c x))}{2 d (c f+g) \sqrt {d-c^2 d x^2}}-\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {g^2 \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\sqrt {-\frac {1-c x}{1+c x}}\right )}{d (c f+g) \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{1+c x}\right )}{2 d (c f-g) \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b \sqrt {(1-c x) (1+c x)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{1+c x}\right )}{2 d (c f+g) \sqrt {-\frac {1-c x}{1+c x}} (1+c x) \sqrt {d-c^2 d x^2}}-\frac {b g^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b g^2 \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{d \left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}} \]
-1/2*(-c*x+1)*(a+b*arccosh(c*x))/d/(c*f-g)/(-c^2*d*x^2+d)^(1/2)+1/2*(c*x+1 )*(a+b*arccosh(c*x))/d/(c*f+g)/(-c^2*d*x^2+d)^(1/2)-g^2*(a+b*arccosh(c*x)) *ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g^2)^(1/2)))*(c*x- 1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)+g^2*(a+b *arccosh(c*x))*ln(1+(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f+(c^2*f^2-g^2) ^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^ (1/2)-b*g^2*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f-(c^2*f^2-g ^2)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+ d)^(1/2)+b*g^2*polylog(2,-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g/(c*f+(c^2*f^ 2-g^2)^(1/2)))*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(c^2*f^2-g^2)^(3/2)/(-c^2*d*x ^2+d)^(1/2)-1/2*b*ln(2/(c*x+1))*((-c*x+1)*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2 )/d/(c*f-g)/(c*x+1)/((c*x-1)/(c*x+1))^(1/2)/(-c^2*d*x^2+d)^(1/2)-1/2*b*ln( 2/(c*x+1))*((-c*x+1)*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c*f+g)/(c*x+1)/( (c*x-1)/(c*x+1))^(1/2)/(-c^2*d*x^2+d)^(1/2)+1/2*b*ln((c*x-1)/(c*x+1))*((-c *x+1)*(c*x+1))^(1/2)*(-c^2*x^2+1)^(1/2)/d/(c*f+g)/(c*x+1)/((c*x-1)/(c*x+1) )^(1/2)/(-c^2*d*x^2+d)^(1/2)
Result contains complex when optimal does not.
Time = 9.28 (sec) , antiderivative size = 1173, normalized size of antiderivative = 1.52 \[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx =\text {Too large to display} \]
((-(a*g) + a*c^2*f*x)*Sqrt[-(d*(-1 + c^2*x^2))])/(d^2*(-(c^2*f^2) + g^2)*( -1 + c^2*x^2)) + (a*g^2*Log[f + g*x])/(d^(3/2)*(-(c*f) + g)*(c*f + g)*Sqrt [-(c^2*f^2) + g^2]) - (a*g^2*Log[d*g + c^2*d*f*x + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[-(d*(-1 + c^2*x^2))]])/(d^(3/2)*(-(c*f) + g)*(c*f + g)*Sqrt[- (c^2*f^2) + g^2]) - (b*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(-((ArcCosh[c* x]*Coth[ArcCosh[c*x]/2])/(c*f + g)) + (2*Log[Cosh[ArcCosh[c*x]/2]])/(c*f - g) + (2*Log[Sinh[ArcCosh[c*x]/2]])/(c*f + g) + (2*g^2*(2*ArcCosh[c*x]*Arc Tan[((c*f + g)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] - (2*I)*ArcCo s[-((c*f)/g)]*ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] + 2*(ArcTan[((c*f + g)*Coth[ArcCosh[c*x]/2])/ Sqrt[-(c^2*f^2) + g^2]] + ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[ -(c^2*f^2) + g^2]]))*Log[Sqrt[-(c^2*f^2) + g^2]/(Sqrt[2]*E^(ArcCosh[c*x]/2 )*Sqrt[g]*Sqrt[c*f + c*g*x])] + (ArcCos[-((c*f)/g)] - 2*(ArcTan[((c*f + g) *Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + ArcTan[((-(c*f) + g)*Tanh [ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]))*Log[(E^(ArcCosh[c*x]/2)*Sqrt[- (c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*f + c*g*x])] - (ArcCos[-((c*f)/g )] + 2*ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]]) *Log[((c*f + g)*(c*f - g + I*Sqrt[-(c^2*f^2) + g^2])*(-1 + Tanh[ArcCosh[c* x]/2]))/(g*(c*f + g + I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))] - ( ArcCos[-((c*f)/g)] - 2*ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/Sqrt[...
Time = 2.27 (sec) , antiderivative size = 636, normalized size of antiderivative = 0.82, 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.
| \(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{\left (d-c^2 d x^2\right )^{3/2} (f+g x)} \, dx\) |
| \(\Big \downarrow \) 6387 |
| \(\displaystyle -\frac {\sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{(c x-1)^{3/2} (c x+1)^{3/2} (f+g x)}dx}{d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 6396 |
| \(\displaystyle -\frac {\sqrt {c x-1} \sqrt {c x+1} \int \left (\frac {(a+b \text {arccosh}(c x)) g^2}{(c f-g) (c f+g) \sqrt {c x-1} \sqrt {c x+1} (f+g x)}+\frac {c (a+b \text {arccosh}(c x))}{2 (c f+g) (c x-1)^{3/2} \sqrt {c x+1}}-\frac {c (a+b \text {arccosh}(c x))}{2 (c f-g) \sqrt {c x-1} (c x+1)^{3/2}}\right )dx}{d \sqrt {d-c^2 d x^2}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt {c x-1} \sqrt {c x+1} \left (\frac {g^2 (a+b \text {arccosh}(c x)) \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{\left (c^2 f^2-g^2\right )^{3/2}}-\frac {g^2 (a+b \text {arccosh}(c x)) \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{\left (c^2 f^2-g^2\right )^{3/2}}-\frac {\sqrt {c x+1} (a+b \text {arccosh}(c x))}{2 \sqrt {c x-1} (c f+g)}-\frac {\sqrt {c x-1} (a+b \text {arccosh}(c x))}{2 \sqrt {c x+1} (c f-g)}+\frac {b g^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2}}-\frac {b g^2 \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2}}-\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} \log \left (\sqrt {-\frac {1-c x}{c x+1}}\right )}{\sqrt {c x-1} \sqrt {-\frac {1-c x}{c x+1}} (c x+1)^{3/2} (c f+g)}+\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{c x+1}\right )}{2 \sqrt {c x-1} \sqrt {-\frac {1-c x}{c x+1}} (c x+1)^{3/2} (c f-g)}+\frac {b \sqrt {(1-c x) (c x+1)} \sqrt {1-c^2 x^2} \log \left (\frac {2}{c x+1}\right )}{2 \sqrt {c x-1} \sqrt {-\frac {1-c x}{c x+1}} (c x+1)^{3/2} (c f+g)}\right )}{d \sqrt {d-c^2 d x^2}}\) |
-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-1/2*(Sqrt[-1 + c*x]*(a + b*ArcCosh[c*x]) )/((c*f - g)*Sqrt[1 + c*x]) - (Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]))/(2*(c*f + g)*Sqrt[-1 + c*x]) + (g^2*(a + b*ArcCosh[c*x])*Log[1 + (E^ArcCosh[c*x]* g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(c^2*f^2 - g^2)^(3/2) - (g^2*(a + b*ArcCo sh[c*x])*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])])/(c^2*f^2 - g^2)^(3/2) - (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[Sqrt[-( (1 - c*x)/(1 + c*x))]])/((c*f + g)*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c* x))]*(1 + c*x)^(3/2)) + (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log [2/(1 + c*x)])/(2*(c*f - g)*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x)^(3/2)) + (b*Sqrt[(1 - c*x)*(1 + c*x)]*Sqrt[1 - c^2*x^2]*Log[2/(1 + c*x)])/(2*(c*f + g)*Sqrt[-1 + c*x]*Sqrt[-((1 - c*x)/(1 + c*x))]*(1 + c*x) ^(3/2)) + (b*g^2*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2 ]))])/(c^2*f^2 - g^2)^(3/2) - (b*g^2*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/(c^2*f^2 - g^2)^(3/2)))/(d*Sqrt[d - c^2*d*x^2]))
3.1.74.3.1 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. \(1925\) vs. \(2(738)=1476\).
Time = 2.46 (sec) , antiderivative size = 1926, normalized size of antiderivative = 2.49
| method | result | size |
| default | \(\text {Expression too large to display}\) | \(1926\) |
| parts | \(\text {Expression too large to display}\) | \(1926\) |
-a*g/d/(c^2*f^2-g^2)/(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2) /g^2)^(1/2)+a*f/(c^2*f^2-g^2)/d/(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c ^2*f^2-g^2)/g^2)^(1/2)*c^2*x+a*g/d/(c^2*f^2-g^2)/(-d*(c^2*f^2-g^2)/g^2)^(1 /2)*ln((-2*d*(c^2*f^2-g^2)/g^2+2*c^2*d*f/g*(x+f/g)+2*(-d*(c^2*f^2-g^2)/g^2 )^(1/2)*(-(x+f/g)^2*c^2*d+2*c^2*d*f/g*(x+f/g)-d*(c^2*f^2-g^2)/g^2)^(1/2))/ (x+f/g))-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/(c^2*x^2-1)/d^2/(c^2*f^2-g^ 2)*(c*x-1)*(c*x+1)*g+b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/(c^2*x^2-1)/d^2 /(c^2*f^2-g^2)*x^2*c^2*g+b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/(c^2*x^2-1) /d^2/(c^2*f^2-g^2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*f-b*(-d*(c^2*x^2-1))^(1/2 )*arccosh(c*x)/(c^2*x^2-1)/d^2/(c^2*f^2-g^2)*x*c^2*f+b*(c^2*f^2-g^2)*(-d*( c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^6*f^4*x^2-2*c^4*f^2*g^2*x ^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)/d^2*ln((c*x-1)^(1/2)*(c*x+1)^(1/ 2)+c*x-1)*c*f-2*b*(c^2*f^2-g^2)*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+ 1)^(1/2)/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2- g^4)/d^2*ln(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*c*f+b*(c^2*f^2-g^2)*(-d*(c^2* x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c ^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2-g^4)/d^2*ln(1+c*x+(c*x-1)^(1/2)*(c*x+1)^( 1/2))*c*f+b*(c^2*f^2-g^2)^(1/2)*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+ 1)^(1/2)/(c^6*f^4*x^2-2*c^4*f^2*g^2*x^2-c^4*f^4+c^2*g^4*x^2+2*c^2*f^2*g^2- g^4)/d^2*arccosh(c*x)*ln((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g-c*f+(c^2...
\[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}} \,d x } \]
integral(sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^4*d^2*g*x^5 + c^4*d^ 2*f*x^4 - 2*c^2*d^2*g*x^3 - 2*c^2*d^2*f*x^2 + d^2*g*x + d^2*f), x)
\[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (f + g x\right )}\, dx \]
\[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}} \,d x } \]
\[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (g x + f\right )}} \,d x } \]
Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x) \left (d-c^2 d x^2\right )^{3/2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{\left (f+g\,x\right )\,{\left (d-c^2\,d\,x^2\right )}^{3/2}} \,d x \]