3.1.76 \(\int \frac {(f+g x) (a+b \cosh ^{-1}(c x))^n}{\sqrt {1-c x} \sqrt {1+c x}} \, dx\) [76]

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

[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 = 0.41, antiderivative size = 200, normalized size of antiderivative = 1.00, 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 = {5982, 5980, 3398, 3388, 2212} \begin {gather*} \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} \text {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} \text {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}} \end {gather*}

Antiderivative was successfully verified.

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

Rule 2212

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(-F^(g*(e - c*(f/d))))*((c
+ d*x)^FracPart[m]/(d*((-f)*g*(Log[F]/d))^(IntPart[m] + 1)*((-f)*g*Log[F]*((c + d*x)/d))^FracPart[m]))*Gamma[m
+ 1, ((-f)*g*(Log[F]/d))*(c + d*x)], x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rule 3388

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 3398

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 5980

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 5982

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*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, 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} \text {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} \text {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 ) \text {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 ) \text {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 ) \text {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 = 0.17, size = 204, normalized size = 1.02 \begin {gather*} \frac {e^{-\frac {a}{b}} \sqrt {\frac {-1+c x}{1+c x}} (1+c x) \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 e^{a/b} f \left (a+b \cosh ^{-1}(c x)\right ) \left (-\frac {\left (a+b \cosh ^{-1}(c x)\right )^2}{b^2}\right )^n-b e^{\frac {2 a}{b}} g (1+n) \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^n \Gamma \left (1+n,\frac {a}{b}+\cosh ^{-1}(c x)\right )+b g (1+n) \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )^n \Gamma \left (1+n,-\frac {a+b \cosh ^{-1}(c x)}{b}\right )\right )}{2 b c^2 (1+n) \sqrt {1-c^2 x^2}} \end {gather*}

Antiderivative was successfully verified.

[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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Maple [F]
time = 180.00, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

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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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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]

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 \begin {gather*} \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 \end {gather*}

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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