3.61 \(\int \frac{(d-c^2 d x^2)^{3/2} (a+b \cosh ^{-1}(c x))}{f+g x} \, dx\)

Optimal. Leaf size=1270 \[ \text{result too large to display} \]

[Out]

-((a*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2])/g^3) + (b*c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2])/(g^3*
Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^2*d*(c*f - g)*x^2*Sqrt[d - c^2*d*x^2])/(4*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x
]) + (a*d*(2 + 3*c*x - 2*c^2*x^2)*Sqrt[d - c^2*d*x^2])/(6*g) + (b*c*d*x*(-12 - 9*c*x + 4*c^2*x^2)*Sqrt[d - c^2
*d*x^2])/(36*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/g^3
- (a*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/(2*g*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (b*d*(2 + 3*c*x - 2*c^2*x^2)*Sqr
t[d - c^2*d*x^2]*ArcCosh[c*x])/(6*g) - (b*d*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]^2)/(4*g*Sqrt[-1 + c*x]*Sqrt[1 + c
*x]) + (c*d*(c*f - g)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*g^2) - (d*(c*f - g)*Sqrt[d - c^2*d*x^2]*(
a + b*ArcCosh[c*x])^2)/(4*b*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2]
*(a + b*ArcCosh[c*x])^2)/(2*b*g^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d*(c*f - g)^2*(c*f + g)^2*Sqrt[d - c^2*d*x^
2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) + (d*(c*f - g)*(c*f + g)*(1 - c^
2*x^2)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) - (2*a*d
*(c*f - g)^(3/2)*(c*f + g)^(3/2)*Sqrt[d - c^2*d*x^2]*ArcTanh[(Sqrt[c*f + g]*Sqrt[1 + c*x])/(Sqrt[c*f - g]*Sqrt
[-1 + c*x])])/(g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d*(c*f - g)*(c*f + g)*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d
*x^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])
+ (b*d*(c*f - g)*(c*f + g)*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c
*f + Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d*(c*f - g)*(c*f + g)*Sqrt[c^2*f^2 - g^2]*
Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^4*Sqrt[-1 + c*x]*Sqrt[1
+ c*x]) + (b*d*(c*f - g)*(c*f + g)*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*
f + Sqrt[c^2*f^2 - g^2]))])/(g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

________________________________________________________________________________________

Rubi [F]  time = 3.8533, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{f+g x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(f + g*x),x]

[Out]

(b*c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2])/(g^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*c^2*d*(c*f - g)*x^2*
Sqrt[d - c^2*d*x^2])/(4*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (a*d*(c*f - g)*(c*f + g)*(1 - c^2*x^2)*Sqrt[d - c^
2*d*x^2])/(g^3*(1 - c*x)*(1 + c*x)) - (b*d*(c*f - g)*(c*f + g)*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x])/g^3 + (c*d*(c
*f - g)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x]))/(2*g^2) - (d*(c*f - g)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[
c*x])^2)/(4*b*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (c*d*(c*f - g)*(c*f + g)*x*Sqrt[d - c^2*d*x^2]*(a + b*ArcCos
h[c*x])^2)/(2*b*g^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (d*(c*f - g)^2*(c*f + g)^2*Sqrt[d - c^2*d*x^2]*(a + b*ArcC
osh[c*x])^2)/(2*b*c*g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) + (d*(c*f - g)*(c*f + g)*(1 - c^2*x^2)*Sqrt[d
- c^2*d*x^2]*(a + b*ArcCosh[c*x])^2)/(2*b*c*g^2*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(f + g*x)) + (a*d*(c*f - g)*(c*f
+ g)*Sqrt[c^2*f^2 - g^2]*Sqrt[-1 + c^2*x^2]*Sqrt[d - c^2*d*x^2]*ArcTanh[(g + c^2*f*x)/(Sqrt[c^2*f^2 - g^2]*Sqr
t[-1 + c^2*x^2])])/(g^4*(1 - c*x)*(1 + c*x)) - (b*d*(c*f - g)*(c*f + g)*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2
]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) + (
b*d*(c*f - g)*(c*f + g)*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*ArcCosh[c*x]*Log[1 + (E^ArcCosh[c*x]*g)/(c*f +
Sqrt[c^2*f^2 - g^2])])/(g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (b*d*(c*f - g)*(c*f + g)*Sqrt[c^2*f^2 - g^2]*Sqrt
[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f - Sqrt[c^2*f^2 - g^2]))])/(g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*
x]) + (b*d*(c*f - g)*(c*f + g)*Sqrt[c^2*f^2 - g^2]*Sqrt[d - c^2*d*x^2]*PolyLog[2, -((E^ArcCosh[c*x]*g)/(c*f +
Sqrt[c^2*f^2 - g^2]))])/(g^4*Sqrt[-1 + c*x]*Sqrt[1 + c*x]) - (c*d*Sqrt[d - c^2*d*x^2]*Defer[Int][(-1 + c*x)^(3
/2)*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x]), x])/(g*Sqrt[-1 + c*x]*Sqrt[1 + c*x])

Rubi steps

\begin{align*} \int \frac{\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{f+g x} \, dx &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \frac{(-1+c x)^{3/2} (1+c x)^{3/2} \left (a+b \cosh ^{-1}(c x)\right )}{f+g x} \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{\left (d \sqrt{d-c^2 d x^2}\right ) \int \left (-\frac{c (c f-g) \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{g^2}+\frac{c (-1+c x)^{3/2} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{g}+\frac{(c f-g) (c f+g) \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{g^2 (f+g x)}\right ) \, dx}{\sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{\left (c d (c f-g) \sqrt{d-c^2 d x^2}\right ) \int \sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (c d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \int \frac{\sqrt{-1+c x} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right )}{f+g x} \, dx}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{c d (c f-g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}+\frac{d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{\left (c d (c f-g) \sqrt{d-c^2 d x^2}\right ) \int \frac{a+b \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{2 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c^2 d (c f-g) \sqrt{d-c^2 d x^2}\right ) \int x \, dx}{2 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (c d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (g+2 c^2 f x+c^2 g x^2\right ) \left (a+b \cosh ^{-1}(c x)\right )^2}{(f+g x)^2} \, dx}{2 b c g^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c^2 d (c f-g) x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac{d (c f-g) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{\left (c d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (\frac{1}{f+g x}-\frac{c^2 \left (g x+\frac{f^2}{f+g x}\right )}{g^2}\right ) \left (-a-b \cosh ^{-1}(c x)\right )}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c^2 d (c f-g) x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac{d (c f-g) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{\left (c d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{a \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right )}{g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{b \left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \cosh ^{-1}(c x)}{g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}\right ) \, dx}{g^2 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c^2 d (c f-g) x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac{d (c f-g) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{\left (c d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (a d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \int \frac{c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)} \, dx}{g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \int \frac{\left (c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2\right ) \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)} \, dx}{g^4 \sqrt{-1+c x} \sqrt{1+c x}}\\ &=-\frac{b c^2 d (c f-g) x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac{d (c f-g) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{\left (c d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \int \left (\frac{c^2 g x \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (c^2 f^2-g^2\right ) \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)}\right ) \, dx}{g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (a d (c f-g) (c f+g) \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \int \frac{c^2 f^2-g^2+c^2 f g x+c^2 g^2 x^2}{(f+g x) \sqrt{-1+c^2 x^2}} \, dx}{g^4 (-1+c x) (1+c x)}\\ &=-\frac{b c^2 d (c f-g) x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g^3 (1-c x) (1+c x)}+\frac{c d (c f-g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac{d (c f-g) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{\left (c d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b c^2 d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \int \frac{x \cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x}} \, dx}{g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2}\right ) \int \frac{\cosh ^{-1}(c x)}{\sqrt{-1+c x} \sqrt{1+c x} (f+g x)} \, dx}{g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (a d (c f-g) (c f+g) \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \int \frac{c^4 f^2 g^2-c^2 g^4}{(f+g x) \sqrt{-1+c^2 x^2}} \, dx}{c^2 g^6 (-1+c x) (1+c x)}\\ &=-\frac{b c^2 d (c f-g) x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g^3 (1-c x) (1+c x)}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^3}+\frac{c d (c f-g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac{d (c f-g) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{\left (c d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b c d (c f-g) (c f+g) \sqrt{d-c^2 d x^2}\right ) \int 1 \, dx}{g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{x}{c f+g \cosh (x)} \, dx,x,\cosh ^{-1}(c x)\right )}{g^4 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (a d (c f-g)^2 (c f+g)^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \int \frac{1}{(f+g x) \sqrt{-1+c^2 x^2}} \, dx}{g^4 (-1+c x) (1+c x)}\\ &=\frac{b c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2}}{g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^2 d (c f-g) x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g^3 (1-c x) (1+c x)}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^3}+\frac{c d (c f-g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac{d (c f-g) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}-\frac{\left (c d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c e^x f+g+e^{2 x} g} \, dx,x,\cosh ^{-1}(c x)\right )}{g^4 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (a d (c f-g)^2 (c f+g)^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{1}{c^2 f^2-g^2-x^2} \, dx,x,\frac{-g-c^2 f x}{\sqrt{-1+c^2 x^2}}\right )}{g^4 (-1+c x) (1+c x)}\\ &=\frac{b c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2}}{g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^2 d (c f-g) x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g^3 (1-c x) (1+c x)}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^3}+\frac{c d (c f-g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac{d (c f-g) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{a d (c f-g)^2 (c f+g)^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{-1+c^2 x^2}}\right )}{g^4 \sqrt{c^2 f^2-g^2} (1-c x) (1+c x)}-\frac{\left (c d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (2 b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g-2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{g^3 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (2 b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{e^x x}{2 c f+2 e^x g+2 \sqrt{c^2 f^2-g^2}} \, dx,x,\cosh ^{-1}(c x)\right )}{g^3 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2}}{g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^2 d (c f-g) x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g^3 (1-c x) (1+c x)}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^3}+\frac{c d (c f-g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac{d (c f-g) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{a d (c f-g)^2 (c f+g)^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{-1+c^2 x^2}}\right )}{g^4 \sqrt{c^2 f^2-g^2} (1-c x) (1+c x)}-\frac{b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (c d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x g}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{g^4 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \log \left (1+\frac{2 e^x g}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{g^4 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2}}{g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^2 d (c f-g) x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g^3 (1-c x) (1+c x)}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^3}+\frac{c d (c f-g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac{d (c f-g) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{a d (c f-g)^2 (c f+g)^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{-1+c^2 x^2}}\right )}{g^4 \sqrt{c^2 f^2-g^2} (1-c x) (1+c x)}-\frac{b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (c d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}+\frac{\left (b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 g x}{2 c f-2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{g^4 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2}\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{2 g x}{2 c f+2 \sqrt{c^2 f^2-g^2}}\right )}{x} \, dx,x,e^{\cosh ^{-1}(c x)}\right )}{g^4 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}\\ &=\frac{b c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2}}{g^3 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b c^2 d (c f-g) x^2 \sqrt{d-c^2 d x^2}}{4 g^2 \sqrt{-1+c x} \sqrt{1+c x}}-\frac{a d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2}}{g^3 (1-c x) (1+c x)}-\frac{b d (c f-g) (c f+g) \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x)}{g^3}+\frac{c d (c f-g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )}{2 g^2}-\frac{d (c f-g) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{4 b g^2 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{c d (c f-g) (c f+g) x \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b g^3 \sqrt{-1+c x} \sqrt{1+c x}}+\frac{d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^4 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{d (c f-g) (c f+g) \left (1-c^2 x^2\right ) \sqrt{d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^2}{2 b c g^2 \sqrt{-1+c x} \sqrt{1+c x} (f+g x)}+\frac{a d (c f-g)^2 (c f+g)^2 \sqrt{-1+c^2 x^2} \sqrt{d-c^2 d x^2} \tanh ^{-1}\left (\frac{g+c^2 f x}{\sqrt{c^2 f^2-g^2} \sqrt{-1+c^2 x^2}}\right )}{g^4 \sqrt{c^2 f^2-g^2} (1-c x) (1+c x)}-\frac{b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \cosh ^{-1}(c x) \log \left (1+\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f-\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}+\frac{b d (c f-g)^2 (c f+g)^2 \sqrt{d-c^2 d x^2} \text{Li}_2\left (-\frac{e^{\cosh ^{-1}(c x)} g}{c f+\sqrt{c^2 f^2-g^2}}\right )}{g^4 \sqrt{c^2 f^2-g^2} \sqrt{-1+c x} \sqrt{1+c x}}-\frac{\left (c d \sqrt{d-c^2 d x^2}\right ) \int (-1+c x)^{3/2} \sqrt{1+c x} \left (a+b \cosh ^{-1}(c x)\right ) \, dx}{g \sqrt{-1+c x} \sqrt{1+c x}}\\ \end{align*}

Mathematica [C]  time = 11.717, size = 3068, normalized size = 2.42 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((d - c^2*d*x^2)^(3/2)*(a + b*ArcCosh[c*x]))/(f + g*x),x]

[Out]

Sqrt[-(d*(-1 + c^2*x^2))]*((a*d*(-3*c^2*f^2 + 4*g^2))/(3*g^3) + (a*c^2*d*f*x)/(2*g^2) - (a*c^2*d*x^2)/(3*g)) +
(a*c*d^(3/2)*f*(2*c^2*f^2 - 3*g^2)*ArcTan[(c*x*Sqrt[-(d*(-1 + c^2*x^2))])/(Sqrt[d]*(-1 + c^2*x^2))])/(2*g^4)
+ (a*d^(3/2)*(-(c^2*f^2) + g^2)^(3/2)*Log[f + g*x])/g^4 - (a*d^(3/2)*(-(c^2*f^2) + g^2)^(3/2)*Log[d*g + c^2*d*
f*x + Sqrt[d]*Sqrt[-(c^2*f^2) + g^2]*Sqrt[-(d*(-1 + c^2*x^2))]])/g^4 + (b*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*((
-2*c*g*x)/(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)) + 2*g*ArcCosh[c*x] - (c*f*ArcCosh[c*x]^2)/(Sqrt[(-1 + c*x)/(1
+ c*x)]*(1 + c*x)) + (2*(-(c*f) + g)*(c*f + g)*(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])/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)*S
qrt[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[ArcC
osh[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[-(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]))] + I*(PolyLog[2, ((
c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))/(g*(c*f + g + I*Sqr
t[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))] - PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - I*Sqrt[-
(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))/(g*(c*f + g + I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))])))/(Sqr
t[-(c^2*f^2) + g^2]*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x))))/(2*g^2) - (b*d*Sqrt[-(d*(-1 + c*x)*(1 + c*x))]*((-
9*(-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])/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[-(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]))] - I*(PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - I*
Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))/(g*(c*f + g + I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))] -
PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))/(g*(c
*f + g + I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))])))/Sqrt[-(c^2*f^2) + g^2] - (-18*c*g*(-4*c^2*f^2 + g
^2)*x + 18*g*(-4*c^2*f^2 + g^2)*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*ArcCosh[c*x] + 18*c*f*(2*c^2*f^2 - g^2)*A
rcCosh[c*x]^2 - 9*c*f*g^2*Cosh[2*ArcCosh[c*x]] + 2*g^3*Cosh[3*ArcCosh[c*x]] + (9*(8*c^4*f^4 - 8*c^2*f^2*g^2 +
g^4)*(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])/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[-(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]))] + I*(PolyLog[2, ((c*f - I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g -
I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))/(g*(c*f + g + I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))]
- PolyLog[2, ((c*f + I*Sqrt[-(c^2*f^2) + g^2])*(c*f + g - I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))/(g*
(c*f + g + I*Sqrt[-(c^2*f^2) + g^2]*Tanh[ArcCosh[c*x]/2]))])))/Sqrt[-(c^2*f^2) + g^2] + 18*c*f*g^2*ArcCosh[c*x
]*Sinh[2*ArcCosh[c*x]] - 6*g^3*ArcCosh[c*x]*Sinh[3*ArcCosh[c*x]])/g^4))/(72*Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*
x))

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Maple [A]  time = 0.27, size = 1965, normalized size = 1.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/9*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g*x^3*c^3-4/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x-1)^(1
/2)/(c*x+1)^(1/2)/g*c*x+1/8*b*(-d*(c^2*x^2-1))^(1/2)*f*c*d/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2+b*(c^2*f^2-g^2)^(3/
2)*d*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^4*dilog(((c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g+c*f+(c^
2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))-b*(c^2*f^2-g^2)^(3/2)*d*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x
+1)^(1/2)/g^4*dilog((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))
-1/4*b*(-d*(c^2*x^2-1))^(1/2)*f*c^3*d/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^2*x^2+b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x-1)^(
1/2)/(c*x+1)^(1/2)/g^3*x*c^3*f^2-b*(c^2*f^2-g^2)^(3/2)*d*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^
4*arccosh(c*x)*ln((-(c*x+(c*x-1)^(1/2)*(c*x+1)^(1/2))*g-c*f+(c^2*f^2-g^2)^(1/2))/(-c*f+(c^2*f^2-g^2)^(1/2)))+b
*(c^2*f^2-g^2)^(3/2)*d*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)/g^4*arccosh(c*x)*ln(((c*x+(c*x-1)^(1
/2)*(c*x+1)^(1/2))*g+c*f+(c^2*f^2-g^2)^(1/2))/(c*f+(c^2*f^2-g^2)^(1/2)))+1/2*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^
(1/2)/(c*x+1)^(1/2)*f^3*arccosh(c*x)^2*c^3*d/g^4-3/4*b*(-d*(c^2*x^2-1))^(1/2)/(c*x-1)^(1/2)/(c*x+1)^(1/2)*f*ar
ccosh(c*x)^2*c*d/g^2-a/g^4*d^2*c^4*f^3/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-(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^5*d^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))*c^4*f^4-4/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x-1)/(c*x+1)/g*arccosh(c*x)+2*a/g^3*d^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))*c^2*f^2+1/2*a/g^2*c^2*d*f*(-(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+3/2*a/g^2*c^2*d^2*f/(c^2*d)^(1/2)*arctan((c^2*d)^(1/2)*x/(-(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*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)+1/3*a/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)^(3/2)-1/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(
c*x-1)/(c*x+1)/g*arccosh(c*x)*x^4*c^4+5/3*b*(-d*(c^2*x^2-1))^(1/2)*d/(c*x-1)/(c*x+1)/g*arccosh(c*x)*x^2*c^2+b*
(-d*(c^2*x^2-1))^(1/2)*d/(c*x-1)/(c*x+1)/g^3*arccosh(c*x)*c^2*f^2+1/2*b*(-d*(c^2*x^2-1))^(1/2)*f*c^4*d/(c*x-1)
/(c*x+1)/g^2*arccosh(c*x)*x^3-1/2*b*(-d*(c^2*x^2-1))^(1/2)*f*c^2*d/(c*x-1)/(c*x+1)/g^2*arccosh(c*x)*x-b*(-d*(c
^2*x^2-1))^(1/2)*d/(c*x-1)/(c*x+1)/g^3*arccosh(c*x)*x^2*c^4*f^2-a/g^3*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*f^2-a/g*d^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))

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/(g*x+f),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{{\left (a c^{2} d x^{2} - a d +{\left (b c^{2} d x^{2} - b d\right )} \operatorname{arcosh}\left (c x\right )\right )} \sqrt{-c^{2} d x^{2} + d}}{g x + f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/(g*x+f),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac{3}{2}} \left (a + b \operatorname{acosh}{\left (c x \right )}\right )}{f + g x}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(3/2)*(a+b*acosh(c*x))/(g*x+f),x)

[Out]

Integral((-d*(c*x - 1)*(c*x + 1))**(3/2)*(a + b*acosh(c*x))/(f + g*x), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (-c^{2} d x^{2} + d\right )}^{\frac{3}{2}}{\left (b \operatorname{arcosh}\left (c x\right ) + a\right )}}{g x + f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*d*x^2+d)^(3/2)*(a+b*arccosh(c*x))/(g*x+f),x, algorithm="giac")

[Out]

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