∫ (f+gx)(a+b cosh−1(cx))n ______________________________

3.76 \(\int \frac {(f+g x) (a+b \cosh ^{-1}(c x))^n}{\sqrt {1-c x} \sqrt {1+c x}} \, dx\)

Optimal. Leaf size=200 \[ \frac {g e^{-\frac {a}{b}} \sqrt {c x-1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x}}-\frac {g e^{a/b} \sqrt {c x-1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x}}+\frac {f \sqrt {c x-1} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt {1-c x}} \]

[Out]

f*(a+b*arccosh(c*x))^(1+n)*(c*x-1)^(1/2)/b/c/(1+n)/(-c*x+1)^(1/2)+1/2*g*(a+b*arccosh(c*x))^n*GAMMA(1+n,(-a-b*a

rccosh(c*x))/b)*(c*x-1)^(1/2)/c^2/exp(a/b)/(((-a-b*arccosh(c*x))/b)^n)/(-c*x+1)^(1/2)-1/2*exp(a/b)*g*(a+b*arcc

osh(c*x))^n*GAMMA(1+n,(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)/c^2/(((a+b*arccosh(c*x))/b)^n)/(-c*x+1)^(1/2)

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Rubi [A] time = 0.63, antiderivative size = 242, normalized size of antiderivative = 1.21, number of steps used = 7, number of rules used = 5, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {5837, 5832, 3317, 3307, 2181} \[ \frac {g e^{-\frac {a}{b}} \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,-\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x} \sqrt {c x+1}}-\frac {g e^{a/b} \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \text {Gamma}\left (n+1,\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x} \sqrt {c x+1}}+\frac {f \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{n+1}}{b c (n+1) \sqrt {1-c x} \sqrt {c x+1}} \]

Warning: Unable to verify antiderivative.

[In]

Int[((f + g*x)*(a + b*ArcCosh[c*x])^n)/(Sqrt[1 - c*x]*Sqrt[1 + c*x]),x]

[Out]

(f*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^(1 + n))/(b*c*(1 + n)*Sqrt[1 - c*x]*Sqrt[1 + c*x]) + (g*Sqrt[1 - c^2

*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(2*c^2*E^(a/b)*Sqrt[1 - c*x]*Sqrt[1 + c*

x]*(-((a + b*ArcCosh[c*x])/b))^n) - (E^(a/b)*g*Sqrt[1 - c^2*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*Ar

cCosh[c*x])/b])/(2*c^2*Sqrt[1 - c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n)

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +

d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*

Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] && !IntegerQ[m]

Rule 3307

Int[((c_.) + (d_.)*(x_))^(m_.)*sin[(e_.) + Pi*(k_.) + (f_.)*(x_)], x_Symbol] :> Dist[I/2, Int[(c + d*x)^m/(E^(

I*k*Pi)*E^(I*(e + f*x))), x], x] - Dist[I/2, Int[(c + d*x)^m*E^(I*k*Pi)*E^(I*(e + f*x)), x], x] /; FreeQ[{c, d

, e, f, m}, x] && IntegerQ[2*k]

Rule 3317

Int[((c_.) + (d_.)*(x_))^(m_.)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Int[ExpandIntegrand[

(c + d*x)^m, (a + b*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && IGtQ[n, 0] && (EqQ[n, 1] ||

IGtQ[m, 0] || NeQ[a^2 - b^2, 0])

Rule 5832

Int[(((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_) + (g_.)*(x_))^(m_.))/(Sqrt[(d1_) + (e1_.)*(x_)]*Sqrt[(d2_

) + (e2_.)*(x_)]), x_Symbol] :> Dist[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])

Rule 5837

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((d1_) + (e1_.)*(x_))^(p_)*((d2_) + (e2_.)*(x_))^(p_)*((f_) + (g

_.)*(x_))^(m_.), x_Symbol] :> Dist[((-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(1

- c^2*x^2)^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, d1, e1, d2, e2, f, g, n}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2, 0] && IntegerQ[m] && IntegerQ[p - 1/2

] && !(GtQ[d1, 0] && LtQ[d2, 0])

Rubi steps

\begin {align*} \int \frac {(f+g x) \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt {1-c x} \sqrt {1+c x}} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(f+g x) \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {1-c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int (a+b x)^n (c f+g \cosh (x)) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt {1-c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {1-c^2 x^2} \operatorname {Subst}\left (\int \left (c f (a+b x)^n+g (a+b x)^n \cosh (x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt {1-c x} \sqrt {1+c x}}\\ &=\frac {f \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt {1-c x} \sqrt {1+c x}}+\frac {\left (g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{c^2 \sqrt {1-c x} \sqrt {1+c x}}\\ &=\frac {f \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt {1-c x} \sqrt {1+c x}}+\frac {\left (g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {1-c x} \sqrt {1+c x}}+\frac {\left (g \sqrt {1-c^2 x^2}\right ) \operatorname {Subst}\left (\int e^x (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{2 c^2 \sqrt {1-c x} \sqrt {1+c x}}\\ &=\frac {f \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b c (1+n) \sqrt {1-c x} \sqrt {1+c x}}+\frac {e^{-\frac {a}{b}} g \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x} \sqrt {1+c x}}-\frac {e^{a/b} g \sqrt {1-c^2 x^2} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 c^2 \sqrt {1-c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A] time = 0.26, size = 204, normalized size = 1.02 \[ \frac {e^{-\frac {a}{b}} \sqrt {\frac {c x-1}{c x+1}} (c x+1) \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^{-n} \left (2 c f e^{a/b} \left (a+b \cosh ^{-1}(c x)\right ) \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n+b g (n+1) \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \Gamma \left (n+1,-\frac {a+b \cosh ^{-1}(c x)}{b}\right )-b g (n+1) e^{\frac {2 a}{b}} \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^n \Gamma \left (n+1,\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )}{2 b c^2 (n+1) \sqrt {1-c^2 x^2}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[((f + g*x)*(a + b*ArcCosh[c*x])^n)/(Sqrt[1 - c*x]*Sqrt[1 + c*x]),x]

[Out]

(Sqrt[(-1 + c*x)/(1 + c*x)]*(1 + c*x)*(a + b*ArcCosh[c*x])^n*(2*c*E^(a/b)*f*(a + b*ArcCosh[c*x])*(-((a + b*Arc

Cosh[c*x])^2/b^2))^n - b*E^((2*a)/b)*g*(1 + n)*(-((a + b*ArcCosh[c*x])/b))^n*Gamma[1 + n, a/b + ArcCosh[c*x]]

+ b*g*(1 + n)*(a/b + ArcCosh[c*x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)]))/(2*b*c^2*E^(a/b)*(1 + n)*Sqrt[1

- c^2*x^2]*(-((a + b*ArcCosh[c*x])^2/b^2))^n)

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fricas [F] time = 0.75, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {c x + 1} \sqrt {-c x + 1} {\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{c^{2} x^{2} - 1}, x\right ) \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-sqrt(c*x + 1)*sqrt(-c*x + 1)*(g*x + f)*(b*arccosh(c*x) + a)^n/(c^2*x^2 - 1), x)

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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn

ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab

s(t_nostep)]Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assum

ing a near 0Unable to build a single algebraic extension for simplifying.Trying rational simplification only.

This might return a wrong answer if simplifying 0/0!Simplification assuming t_nostep near 0Simplification assu

ming t_nostep near 0Simplification assuming a near 0Warning: vanishing non integral power expansionWarning: va

nishing non integral power expansionUnable to build a single algebraic extension for simplifying.Trying ration

al simplification only. This might return a wrong answer if simplifying 0/0!Simplification assuming t_nostep n

ear 0Simplification assuming t_nostep near 0Simplification assuming a near 0Unable to build a single algebraic

extension for simplifying.Trying rational simplification only. This might return a wrong answer if simplifyin

g 0/0!Simplification assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assuming a

near 0Warning: vanishing non integral power expansionWarning: vanishing non integral power expansionUnable to

build a single algebraic extension for simplifying.Trying rational simplification only. This might return a wr

ong answer if simplifying 0/0!Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_

nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_no

step^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nost

ep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep

^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4

-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2

*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t

_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_n

ostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nos

tep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_noste

p^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^

2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)

-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1

)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/

2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)

Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Un

able to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unab

le to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable

to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable t

o check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to

check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to ch

eck sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to chec

k sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check

sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check si

gn: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign

: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign:

(8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8

*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*p

i/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/

(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(s

ign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sig

n(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(

t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_

nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_no

step^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nost

ep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep

^4-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4

-2*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2

*t_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t

_nostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_n

ostep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nos

tep^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_noste

p^2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^

2)-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)

-1)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1

)/2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/

2)>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)

>(-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(

-8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8

*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*p

i/(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/

(sign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(s

ign(t_nostep^4-2*t_nostep^2)-1)/2)Unable to check sign: (8*pi/(sign(t_nostep^4-2*t_nostep^2)-1)/2)>(-8*pi/(sig

n(t_nostep^4-2*t_nostep^2)-1)/2)Evaluation time: 12.83sym2poly/r2sym(const gen & e,const index_m & i,const vec

teur & l) Error: Bad Argument Value

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maple [F] time = 0.86, size = 0, normalized size = 0.00 \[ \int \frac {\left (g x +f \right ) \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}}{\sqrt {-c x +1}\, \sqrt {c x +1}}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (g x + f\right )} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{\sqrt {c x + 1} \sqrt {-c x + 1}}\,{d x} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((g*x + f)*(b*arccosh(c*x) + a)^n/(sqrt(c*x + 1)*sqrt(-c*x + 1)), x)

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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\left (f+g\,x\right )\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n}{\sqrt {1-c\,x}\,\sqrt {c\,x+1}} \,d x \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n} \left (f + g x\right )}{\sqrt {- c x + 1} \sqrt {c x + 1}}\, dx \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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