\(\int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx\) [58]

Optimal result

Integrand size = 31, antiderivative size = 523 \[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=-\frac {g \sqrt {-1+c x} \sqrt {-\frac {1-c x}{1+c x}} (1+c x)^{3/2} (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (f+g x) \sqrt {d-c^2 d x^2}}+\frac {c^2 f \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {c^2 f \sqrt {-1+c x} \sqrt {1+c x} (a+b \text {arccosh}(c x)) \log \left (1+\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}+\frac {b c \sqrt {-1+c x} \sqrt {1+c x} \log (f+g x)}{\left (c^2 f^2-g^2\right ) \sqrt {d-c^2 d x^2}}+\frac {b c^2 f \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}}-\frac {b c^2 f \sqrt {-1+c x} \sqrt {1+c x} \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{\left (c^2 f^2-g^2\right )^{3/2} \sqrt {d-c^2 d x^2}} \] Output:

-g*(c*x-1)^(1/2)*(-(-c*x+1)/(c*x+1))^(1/2)*(c*x+1)^(3/2)*(a+b*arccosh(c*x) 
)/(c^2*f^2-g^2)/(g*x+f)/(-c^2*d*x^2+d)^(1/2)+c^2*f*(c*x-1)^(1/2)*(c*x+1)^( 
1/2)*(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^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)-c^2*f*(c*x-1)^( 
1/2)*(c*x+1)^(1/2)*(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^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^(1/2)+b 
*c*(c*x-1)^(1/2)*(c*x+1)^(1/2)*ln(g*x+f)/(c^2*f^2-g^2)/(-c^2*d*x^2+d)^(1/2 
)+b*c^2*f*(c*x-1)^(1/2)*(c*x+1)^(1/2)*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^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d)^( 
1/2)-b*c^2*f*(c*x-1)^(1/2)*(c*x+1)^(1/2)*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^2*f^2-g^2)^(3/2)/(-c^2*d*x^2+d 
)^(1/2)
 

Mathematica [C] (warning: unable to verify)

Result contains complex when optimal does not.

Time = 3.98 (sec) , antiderivative size = 1115, normalized size of antiderivative = 2.13 \[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx =\text {Too large to display} \] Input:

Integrate[(a + b*ArcCosh[c*x])/((f + g*x)^2*Sqrt[d - c^2*d*x^2]),x]
 

Output:

-((a*g*Sqrt[d - c^2*d*x^2])/(d*(-(c^2*f^2) + g^2)*(f + g*x))) - (a*c^2*f*L 
og[f + g*x])/(Sqrt[d]*(-(c^2*f^2) + g^2)^(3/2)) - (a*c^2*f*Log[d*(g + c^2* 
f*x) + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[d - c^2*d*x^2]])/(Sqrt[d]*(c*f 
- g)*(c*f + g)*Sqrt[-(c^2*f^2) + g^2]) + (b*c*Sqrt[(-1 + c*x)/(1 + c*x)]*( 
1 + c*x)*(-((g*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x])/((c*f - 
g)*(c*f + g)*(c*f + c*g*x))) + Log[1 + (g*x)/f]/(c^2*f^2 - g^2) + (c*f*(2* 
ArcCosh[c*x]*ArcTan[((c*f + g)*Coth[ArcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2 
]] - (2*I)*ArcCos[-((c*f)/g)]*ArcTan[((-(c*f) + g)*Tanh[ArcCosh[c*x]/2])/S 
qrt[-(c^2*f^2) + g^2]] + (ArcCos[-((c*f)/g)] + 2*(ArcTan[((c*f + g)*Coth[A 
rcCosh[c*x]/2])/Sqrt[-(c^2*f^2) + g^2]] + ArcTan[((-(c*f) + g)*Tanh[ArcCos 
h[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 + g*x)])] + (ArcCos[-((c*f)/g)] - 2*(A 
rcTan[((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^(ArcCos 
h[c*x]/2)*Sqrt[-(c^2*f^2) + g^2])/(Sqrt[2]*Sqrt[g]*Sqrt[c*(f + 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[ArcCos 
h[c*x]/2]))] - (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[-(...
 

Rubi [A] (verified)

Time = 1.54 (sec) , antiderivative size = 360, normalized size of antiderivative = 0.69, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.484, Rules used = {6387, 6395, 3042, 3805, 26, 3042, 26, 3147, 16, 3801, 2694, 27, 2620, 2715, 2838}

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.

\(\displaystyle \int \frac {a+b \text {arccosh}(c x)}{\sqrt {d-c^2 d x^2} (f+g x)^2} \, dx\)

\(\Big \downarrow \) 6387

\(\displaystyle \frac {\sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{\sqrt {c x-1} \sqrt {c x+1} (f+g x)^2}dx}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 6395

\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{(c f+c g x)^2}d\text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \int \frac {a+b \text {arccosh}(c x)}{\left (c f+g \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )\right )^2}d\text {arccosh}(c x)}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3805

\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {c f \int \frac {a+b \text {arccosh}(c x)}{c f+c g x}d\text {arccosh}(c x)}{c^2 f^2-g^2}+\frac {i b g \int -\frac {i \sqrt {\frac {c x-1}{c x+1}} (c x+1)}{c f+c g x}d\text {arccosh}(c x)}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {c f \int \frac {a+b \text {arccosh}(c x)}{c f+c g x}d\text {arccosh}(c x)}{c^2 f^2-g^2}+\frac {b g \int \frac {\sqrt {\frac {c x-1}{c x+1}} (c x+1)}{c f+c g x}d\text {arccosh}(c x)}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {c f \int \frac {a+b \text {arccosh}(c x)}{c f+g \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 f^2-g^2}+\frac {b g \int -\frac {i \cos \left (i \text {arccosh}(c x)-\frac {\pi }{2}\right )}{c f-g \sin \left (i \text {arccosh}(c x)-\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 26

\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {c f \int \frac {a+b \text {arccosh}(c x)}{c f+g \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 f^2-g^2}-\frac {i b g \int \frac {\cos \left (i \text {arccosh}(c x)-\frac {\pi }{2}\right )}{c f-g \sin \left (i \text {arccosh}(c x)-\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3147

\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {c f \int \frac {a+b \text {arccosh}(c x)}{c f+g \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 f^2-g^2}+\frac {b \int \frac {1}{c f+c g x}d(c g x)}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 16

\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {c f \int \frac {a+b \text {arccosh}(c x)}{c f+g \sin \left (i \text {arccosh}(c x)+\frac {\pi }{2}\right )}d\text {arccosh}(c x)}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 3801

\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {2 c f \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{2 c e^{\text {arccosh}(c x)} f+e^{2 \text {arccosh}(c x)} g+g}d\text {arccosh}(c x)}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2694

\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {2 c f \left (\frac {g \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{2 \left (c f+e^{\text {arccosh}(c x)} g-\sqrt {c^2 f^2-g^2}\right )}d\text {arccosh}(c x)}{\sqrt {c^2 f^2-g^2}}-\frac {g \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{2 \left (c f+e^{\text {arccosh}(c x)} g+\sqrt {c^2 f^2-g^2}\right )}d\text {arccosh}(c x)}{\sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {2 c f \left (\frac {g \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{c f+e^{\text {arccosh}(c x)} g-\sqrt {c^2 f^2-g^2}}d\text {arccosh}(c x)}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \int \frac {e^{\text {arccosh}(c x)} (a+b \text {arccosh}(c x))}{c f+e^{\text {arccosh}(c x)} g+\sqrt {c^2 f^2-g^2}}d\text {arccosh}(c x)}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2620

\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {2 c f \left (\frac {g \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g}-\frac {b \int \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )d\text {arccosh}(c x)}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{g}-\frac {b \int \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )d\text {arccosh}(c x)}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2715

\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {2 c f \left (\frac {g \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g}-\frac {b \int e^{-\text {arccosh}(c x)} \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}+1\right )de^{\text {arccosh}(c x)}}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{g}-\frac {b \int e^{-\text {arccosh}(c x)} \log \left (\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}+1\right )de^{\text {arccosh}(c x)}}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

\(\Big \downarrow \) 2838

\(\displaystyle \frac {c \sqrt {c x-1} \sqrt {c x+1} \left (\frac {2 c f \left (\frac {g \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {g e^{\text {arccosh}(c x)}}{c f-\sqrt {c^2 f^2-g^2}}+1\right )}{g}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f-\sqrt {c^2 f^2-g^2}}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}-\frac {g \left (\frac {(a+b \text {arccosh}(c x)) \log \left (\frac {g e^{\text {arccosh}(c x)}}{\sqrt {c^2 f^2-g^2}+c f}+1\right )}{g}+\frac {b \operatorname {PolyLog}\left (2,-\frac {e^{\text {arccosh}(c x)} g}{c f+\sqrt {c^2 f^2-g^2}}\right )}{g}\right )}{2 \sqrt {c^2 f^2-g^2}}\right )}{c^2 f^2-g^2}-\frac {g \sqrt {\frac {c x-1}{c x+1}} (c x+1) (a+b \text {arccosh}(c x))}{\left (c^2 f^2-g^2\right ) (c f+c g x)}+\frac {b \log (c f+c g x)}{c^2 f^2-g^2}\right )}{\sqrt {d-c^2 d x^2}}\)

Input:

Int[(a + b*ArcCosh[c*x])/((f + g*x)^2*Sqrt[d - c^2*d*x^2]),x]
 

Output:

(c*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((g*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x) 
*(a + b*ArcCosh[c*x]))/((c^2*f^2 - g^2)*(c*f + c*g*x))) + (b*Log[c*f + c*g 
*x])/(c^2*f^2 - g^2) + (2*c*f*((g*(((a + b*ArcCosh[c*x])*Log[1 + (E^ArcCos 
h[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/g + (b*PolyLog[2, -((E^ArcCosh[c*x 
]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/g))/(2*Sqrt[c^2*f^2 - g^2]) - (g*(((a 
+ b*ArcCosh[c*x])*Log[1 + (E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2])]) 
/g + (b*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f + Sqrt[c^2*f^2 - g^2]))])/g)) 
/(2*Sqrt[c^2*f^2 - g^2])))/(c^2*f^2 - g^2)))/Sqrt[d - c^2*d*x^2]
 

Defintions of rubi rules used

rule 16

Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + 
b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
 

rule 26

Int[(Complex[0, a_])*(Fx_), x_Symbol] :> Simp[(Complex[Identity[0], a])   I 
nt[Fx, x], x] /; FreeQ[a, x] && EqQ[a^2, 1]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2620

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/ 
((a_) + (b_.)*((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)), x_Symbol] :> Simp 
[((c + d*x)^m/(b*f*g*n*Log[F]))*Log[1 + b*((F^(g*(e + f*x)))^n/a)], x] - Si 
mp[d*(m/(b*f*g*n*Log[F]))   Int[(c + d*x)^(m - 1)*Log[1 + b*((F^(g*(e + f*x 
)))^n/a)], x], x] /; FreeQ[{F, a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]
 

rule 2694

Int[((F_)^(u_)*((f_.) + (g_.)*(x_))^(m_.))/((a_.) + (b_.)*(F_)^(u_) + (c_.) 
*(F_)^(v_)), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[2*(c/q)   Int 
[(f + g*x)^m*(F^u/(b - q + 2*c*F^u)), x], x] - Simp[2*(c/q)   Int[(f + g*x) 
^m*(F^u/(b + q + 2*c*F^u)), x], x]] /; FreeQ[{F, a, b, c, f, g}, x] && EqQ[ 
v, 2*u] && LinearQ[u, x] && NeQ[b^2 - 4*a*c, 0] && IGtQ[m, 0]
 

rule 2715

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] 
:> Simp[1/(d*e*n*Log[F])   Subst[Int[Log[a + b*x]/x, x], x, (F^(e*(c + d*x) 
))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]
 

rule 2838

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> Simp[-PolyLog[2 
, (-c)*e*x^n]/n, x] /; FreeQ[{c, d, e, n}, x] && EqQ[c*d, 1]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 3147

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m 
_.), x_Symbol] :> Simp[1/(b^p*f)   Subst[Int[(a + x)^m*(b^2 - x^2)^((p - 1) 
/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && IntegerQ[(p 
 - 1)/2] && NeQ[a^2 - b^2, 0]
 

rule 3801

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + Pi*(k_.) + (Comple 
x[0, fz_])*(f_.)*(x_)]), x_Symbol] :> Simp[2   Int[((c + d*x)^m*(E^((-I)*e 
+ f*fz*x)/(b + (2*a*E^((-I)*e + f*fz*x))/E^(I*Pi*(k - 1/2)) - (b*E^(2*((-I) 
*e + f*fz*x)))/E^(2*I*k*Pi))))/E^(I*Pi*(k - 1/2)), x], x] /; FreeQ[{a, b, c 
, d, e, f, fz}, x] && IntegerQ[2*k] && NeQ[a^2 - b^2, 0] && IGtQ[m, 0]
 

rule 3805

Int[((c_.) + (d_.)*(x_))^(m_.)/((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^2, x_ 
Symbol] :> Simp[b*(c + d*x)^m*(Cos[e + f*x]/(f*(a^2 - b^2)*(a + b*Sin[e + f 
*x]))), x] + (Simp[a/(a^2 - b^2)   Int[(c + d*x)^m/(a + b*Sin[e + f*x]), x] 
, x] - Simp[b*d*(m/(f*(a^2 - b^2)))   Int[(c + d*x)^(m - 1)*(Cos[e + f*x]/( 
a + b*Sin[e + f*x])), x], x]) /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[a^2 - 
b^2, 0] && IGtQ[m, 0]
 

rule 6387

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]
 

rule 6395

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/( 
Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_) + (e2_.)*(x_)]), x_Symbol] :> Simp[1/( 
c^(m + 1)*Sqrt[(-d1)*d2])   Subst[Int[(a + b*x)^n*(c*f + g*Cosh[x])^m, x], 
x, ArcCosh[c*x]], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, g, n}, x] && EqQ 
[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && GtQ[d1, 0] && LtQ[d2, 
 0] && (GtQ[m, 0] || IGtQ[n, 0])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(1977\) vs. \(2(521)=1042\).

Time = 0.58 (sec) , antiderivative size = 1978, normalized size of antiderivative = 3.78

method	result	size
default	\(\text {Expression too large to display}\)	\(1978\)
parts	\(\text {Expression too large to display}\)	\(1978\)

Input:

int((a+b*arccosh(c*x))/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x,method=_RETURNVERB 
OSE)
 

Output:

a/d/(c^2*f^2-g^2)/(x+f/g)*(-(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/g*c^2*f/(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)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f) 
*(c*x-1)*(c*x+1)*x*c^2*f+b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d/(c^2*x^2- 
1)/(c^2*f^2-g^2)/(g*x+f)*x^3*c^4*f-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d 
/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*x*c*g+b*(-d 
*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*x^2*c 
^2*g-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g* 
x+f)*(c*x-1)^(1/2)*(c*x+1)^(1/2)*c*f-b*(-d*(c^2*x^2-1))^(1/2)*arccosh(c*x) 
/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*x*c^2*f-b*(-d*(c^2*x^2-1))^(1/2)*arcc 
osh(c*x)/d/(c^2*x^2-1)/(c^2*f^2-g^2)/(g*x+f)*g-b*(-d*(c^2*x^2-1))^(1/2)*(c 
*x-1)^(1/2)*(c*x+1)^(1/2)/d/(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)*c^2*arccosh(c*x)*(c^2*f^2-g^2)^(1/2)*ln((-(c*x+(c* 
x-1)^(1/2)*(c*x+1)^(1/2))*g-f*c+(c^2*f^2-g^2)^(1/2))/(-f*c+(c^2*f^2-g^2)^( 
1/2)))*f+b*(-d*(c^2*x^2-1))^(1/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/d/(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)*c^2*arccosh(c* 
x)*(c^2*f^2-g^2)^(1/2)*ln(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+f*c+(c^2*f^ 
2-g^2)^(1/2))/(f*c+(c^2*f^2-g^2)^(1/2)))*f+2*b*(-d*(c^2*x^2-1))^(1/2)*(...
 

Fricas [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x, algorithm=" 
fricas")
 

Output:

integral(-sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)/(c^2*d*g^2*x^4 + 2*c^2 
*d*f*g*x^3 - 2*d*f*g*x - d*f^2 + (c^2*d*f^2 - d*g^2)*x^2), x)
 

Sympy [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a + b \operatorname {acosh}{\left (c x \right )}}{\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (f + g x\right )^{2}}\, dx \] Input:

integrate((a+b*acosh(c*x))/(g*x+f)**2/(-c**2*d*x**2+d)**(1/2),x)
 

Output:

Integral((a + b*acosh(c*x))/(sqrt(-d*(c*x - 1)*(c*x + 1))*(f + g*x)**2), x 
)
 

Maxima [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x, algorithm=" 
maxima")
 

Output:

integrate((b*arccosh(c*x) + a)/(sqrt(-c^2*d*x^2 + d)*(g*x + f)^2), x)
 

Giac [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int { \frac {b \operatorname {arcosh}\left (c x\right ) + a}{\sqrt {-c^{2} d x^{2} + d} {\left (g x + f\right )}^{2}} \,d x } \] Input:

integrate((a+b*arccosh(c*x))/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x, algorithm=" 
giac")
 

Output:

integrate((b*arccosh(c*x) + a)/(sqrt(-c^2*d*x^2 + d)*(g*x + f)^2), x)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\int \frac {a+b\,\mathrm {acosh}\left (c\,x\right )}{{\left (f+g\,x\right )}^2\,\sqrt {d-c^2\,d\,x^2}} \,d x \] Input:

int((a + b*acosh(c*x))/((f + g*x)^2*(d - c^2*d*x^2)^(1/2)),x)
 

Output:

int((a + b*acosh(c*x))/((f + g*x)^2*(d - c^2*d*x^2)^(1/2)), x)
 

Reduce [F]

\[ \int \frac {a+b \text {arccosh}(c x)}{(f+g x)^2 \sqrt {d-c^2 d x^2}} \, dx=\frac {2 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,c^{2} f^{2}+2 \sqrt {c^{2} f^{2}-g^{2}}\, \mathit {atan} \left (\frac {\tan \left (\frac {\mathit {asin} \left (c x \right )}{2}\right ) c f +g}{\sqrt {c^{2} f^{2}-g^{2}}}\right ) a \,c^{2} f g x +\sqrt {-c^{2} x^{2}+1}\, a \,c^{2} f^{2} g -\sqrt {-c^{2} x^{2}+1}\, a \,g^{3}+\left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, f^{2}+2 \sqrt {-c^{2} x^{2}+1}\, f g x +\sqrt {-c^{2} x^{2}+1}\, g^{2} x^{2}}d x \right ) b \,c^{4} f^{5}+\left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, f^{2}+2 \sqrt {-c^{2} x^{2}+1}\, f g x +\sqrt {-c^{2} x^{2}+1}\, g^{2} x^{2}}d x \right ) b \,c^{4} f^{4} g x -2 \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, f^{2}+2 \sqrt {-c^{2} x^{2}+1}\, f g x +\sqrt {-c^{2} x^{2}+1}\, g^{2} x^{2}}d x \right ) b \,c^{2} f^{3} g^{2}-2 \left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, f^{2}+2 \sqrt {-c^{2} x^{2}+1}\, f g x +\sqrt {-c^{2} x^{2}+1}\, g^{2} x^{2}}d x \right ) b \,c^{2} f^{2} g^{3} x +\left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, f^{2}+2 \sqrt {-c^{2} x^{2}+1}\, f g x +\sqrt {-c^{2} x^{2}+1}\, g^{2} x^{2}}d x \right ) b f \,g^{4}+\left (\int \frac {\mathit {acosh} \left (c x \right )}{\sqrt {-c^{2} x^{2}+1}\, f^{2}+2 \sqrt {-c^{2} x^{2}+1}\, f g x +\sqrt {-c^{2} x^{2}+1}\, g^{2} x^{2}}d x \right ) b \,g^{5} x}{\sqrt {d}\, \left (c^{4} f^{4} g x +c^{4} f^{5}-2 c^{2} f^{2} g^{3} x -2 c^{2} f^{3} g^{2}+g^{5} x +f \,g^{4}\right )} \] Input:

int((a+b*acosh(c*x))/(g*x+f)^2/(-c^2*d*x^2+d)^(1/2),x)
 

Output:

(2*sqrt(c**2*f**2 - g**2)*atan((tan(asin(c*x)/2)*c*f + g)/sqrt(c**2*f**2 - 
 g**2))*a*c**2*f**2 + 2*sqrt(c**2*f**2 - g**2)*atan((tan(asin(c*x)/2)*c*f 
+ g)/sqrt(c**2*f**2 - g**2))*a*c**2*f*g*x + sqrt( - c**2*x**2 + 1)*a*c**2* 
f**2*g - sqrt( - c**2*x**2 + 1)*a*g**3 + int(acosh(c*x)/(sqrt( - c**2*x**2 
 + 1)*f**2 + 2*sqrt( - c**2*x**2 + 1)*f*g*x + sqrt( - c**2*x**2 + 1)*g**2* 
x**2),x)*b*c**4*f**5 + int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*f**2 + 2*sqr 
t( - c**2*x**2 + 1)*f*g*x + sqrt( - c**2*x**2 + 1)*g**2*x**2),x)*b*c**4*f* 
*4*g*x - 2*int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*f**2 + 2*sqrt( - c**2*x* 
*2 + 1)*f*g*x + sqrt( - c**2*x**2 + 1)*g**2*x**2),x)*b*c**2*f**3*g**2 - 2* 
int(acosh(c*x)/(sqrt( - c**2*x**2 + 1)*f**2 + 2*sqrt( - c**2*x**2 + 1)*f*g 
*x + sqrt( - c**2*x**2 + 1)*g**2*x**2),x)*b*c**2*f**2*g**3*x + int(acosh(c 
*x)/(sqrt( - c**2*x**2 + 1)*f**2 + 2*sqrt( - c**2*x**2 + 1)*f*g*x + sqrt( 
- c**2*x**2 + 1)*g**2*x**2),x)*b*f*g**4 + int(acosh(c*x)/(sqrt( - c**2*x** 
2 + 1)*f**2 + 2*sqrt( - c**2*x**2 + 1)*f*g*x + sqrt( - c**2*x**2 + 1)*g**2 
*x**2),x)*b*g**5*x)/(sqrt(d)*(c**4*f**5 + c**4*f**4*g*x - 2*c**2*f**3*g**2 
 - 2*c**2*f**2*g**3*x + f*g**4 + g**5*x))