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

Optimal. Leaf size=415 \[ d^2 \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x \sqrt {d-c^2 d x^2}},x\right )+\frac {d^2 3^{-n-1} e^{-\frac {3 a}{b}} \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 \sqrt {d-c^2 d x^2}}-\frac {5 d^2 e^{-\frac {a}{b}} \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,-\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{8 \sqrt {d-c^2 d x^2}}+\frac {5 d^2 e^{a/b} \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{8 \sqrt {d-c^2 d x^2}}-\frac {d^2 3^{-n-1} e^{\frac {3 a}{b}} \sqrt {c x-1} \sqrt {c x+1} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (n+1,\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 \sqrt {d-c^2 d x^2}} \]

[Out]

1/8*3^(-1-n)*d^2*(a+b*arccosh(c*x))^n*GAMMA(1+n,-3*(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/exp(3*a/b
)/(((-a-b*arccosh(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)-5/8*d^2*(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*arccosh(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)+5/8*d^2*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)-1/8*3^(-1-n)*d^2*exp(3*a/b)*(a+b*arccosh(c*x))^n*GAMMA(1+n,3*(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^2*Unintegrable((a+b*arccosh(c*x))^n/x/(-c^2*
d*x^2+d)^(1/2),x)

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Rubi [A]  time = 1.84, 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 = {} \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

-(3^(-1 - n)*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-3*(a + b*ArcCosh[c*x]))/b])/(8*E^((3*
a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])/b))^n) + (5*d*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*
x])^n*Gamma[1 + n, -((a + b*ArcCosh[c*x])/b)])/(8*E^(a/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(-((a + b*ArcCosh[c*x])
/b))^n) - (5*d*E^(a/b)*Sqrt[d - c^2*d*x^2]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (a + b*ArcCosh[c*x])/b])/(8*Sqr
t[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])/b)^n) + (3^(-1 - n)*d*E^((3*a)/b)*Sqrt[d - c^2*d*x^2]*(a + b*A
rcCosh[c*x])^n*Gamma[1 + n, (3*(a + b*ArcCosh[c*x]))/b])/(8*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*((a + b*ArcCosh[c*x])
/b)^n) - (d*Sqrt[d - c^2*d*x^2]*Defer[Int][(a + b*ArcCosh[c*x])^n/(x*Sqrt[-1 + c*x]*Sqrt[1 + c*x]), x])/(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 )^n}{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 )^n}{x} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x \sqrt {-1+c x} \sqrt {1+c x}}-\frac {2 c^2 x \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^4 x^3 \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 {\left (d \sqrt {d-c^2 d x^2}\right ) \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 (2 c^2 d \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 {\left (c^4 d \sqrt {d-c^2 d x^2}\right ) \int \frac {x^3 \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 {\left (d \sqrt {d-c^2 d x^2}\right ) \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 (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh ^3(x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (2 d \sqrt {d-c^2 d x^2}\right ) \operatorname {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 {\left (d \sqrt {d-c^2 d x^2}\right ) \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 (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^x (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {3}{4} (a+b x)^n \cosh (x)+\frac {1}{4} (a+b x)^n \cosh (3 x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {d 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 )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {d 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 )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh (3 x) \, dx,x,\cosh ^{-1}(c x)\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int (a+b x)^n \cosh (x) \, dx,x,\cosh ^{-1}(c x)\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \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 {d 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 )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {d 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 )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-3 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{3 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^{-x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 d \sqrt {d-c^2 d x^2}\right ) \operatorname {Subst}\left (\int e^x (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \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 {3^{-1-n} d e^{-\frac {3 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 {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {5 d 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 )}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {5 d 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 )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3^{-1-n} d e^{\frac {3 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 {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d \sqrt {d-c^2 d x^2}\right ) \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 = 0.30, size = 0, normalized size = 0.00 \[ \int \frac {\left (d-c^2 d x^2\right )^{3/2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.60, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{x}, x\right ) \]

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))^n/x,x, algorithm="fricas")

[Out]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.49, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {3}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}}{x}\, dx \]

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))^n/x,x)

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-c^{2} d x^{2} + d\right )}^{\frac {3}{2}} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}}{x}\,{d x} \]

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))^n/x,x, algorithm="maxima")

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^n\,{\left (d-c^2\,d\,x^2\right )}^{3/2}}{x} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((a + b*acosh(c*x))^n*(d - c^2*d*x^2)^(3/2))/x,x)

[Out]

int(((a + b*acosh(c*x))^n*(d - c^2*d*x^2)^(3/2))/x, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- d \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{n}}{x}\, dx \]

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))**n/x,x)

[Out]

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

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