3.5.22 \(\int \frac {\sqrt {d-c^2 d x^2} (a+b \cosh ^{-1}(c x))^n}{x} \, dx\) [422]

Optimal. Leaf size=212 \[ -\frac {d e^{-\frac {a}{b}} \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 \sqrt {d-c^2 d x^2}}+\frac {d e^{a/b} \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{2 \sqrt {d-c^2 d x^2}}+d \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x \sqrt {d-c^2 d x^2}},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.38, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 0.18, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n} \sqrt {-c^{2} d \,x^{2}+d}}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
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((a+b*arccosh(c*x))^n*(-c^2*d*x^2+d)^(1/2)/x,x, algorithm="maxima")

[Out]

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

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Fricas [A]
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((a+b*arccosh(c*x))^n*(-c^2*d*x^2+d)^(1/2)/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {- d \left (c x - 1\right ) \left (c x + 1\right )} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,sageVARx):;OUTP
UT:sym2poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,\sqrt {d-c^2\,d\,x^2}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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