3.5.33 \(\int \frac {(d-c^2 d x^2)^{5/2} (a+b \cosh ^{-1}(c x))^n}{x^2} \, dx\) [433]

Optimal. Leaf size=486 \[ -\frac {15 c d^3 \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{8 b (1+n) \sqrt {d-c^2 d x^2}}-\frac {2^{-2 (3+n)} c d^3 e^{-\frac {4 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt {d-c^2 d x^2}}+\frac {2^{-2-n} c d^3 e^{-\frac {2 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (-\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,-\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt {d-c^2 d x^2}}-\frac {2^{-2-n} c d^3 e^{\frac {2 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt {d-c^2 d x^2}}+\frac {2^{-2 (3+n)} c d^3 e^{\frac {4 a}{b}} \sqrt {-1+c x} \sqrt {1+c x} \left (a+b \cosh ^{-1}(c x)\right )^n \left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )^{-n} \Gamma \left (1+n,\frac {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt {d-c^2 d x^2}}+d^3 \text {Int}\left (\frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt {d-c^2 d x^2}},x\right ) \]

[Out]

-15/8*c*d^3*(a+b*arccosh(c*x))^(1+n)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/(1+n)/(-c^2*d*x^2+d)^(1/2)-c*d^3*(a+b*arcco
sh(c*x))^n*GAMMA(1+n,-4*(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/(2^(6+2*n))/exp(4*a/b)/(((-a-b*arcco
sh(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)+2^(-2-n)*c*d^3*(a+b*arccosh(c*x))^n*GAMMA(1+n,-2*(a+b*arccosh(c*x))/b)*(c*
x-1)^(1/2)*(c*x+1)^(1/2)/exp(2*a/b)/(((-a-b*arccosh(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)-2^(-2-n)*c*d^3*exp(2*a/b)
*(a+b*arccosh(c*x))^n*GAMMA(1+n,2*(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)+c*d^3*exp(4*a/b)*(a+b*arccosh(c*x))^n*GAMMA(1+n,4*(a+b*arccosh(c*x))/b)*(c*x-1)^(1/2)*(c
*x+1)^(1/2)/(2^(6+2*n))/(((a+b*arccosh(c*x))/b)^n)/(-c^2*d*x^2+d)^(1/2)+d^3*Unintegrable((a+b*arccosh(c*x))^n/
x^2/(-c^2*d*x^2+d)^(1/2),x)

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Rubi [A]
time = 0.90, 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 {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n)/x^2,x]

[Out]

(-15*c*d^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^(1 + n))/(8*b*(1 + n)*Sqrt[d - c^2*d*x^2]) - (c*d
^3*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(3 + n
))*E^((4*a)/b)*Sqrt[d - c^2*d*x^2]*(-((a + b*ArcCosh[c*x])/b))^n) + (2^(-2 - n)*c*d^3*Sqrt[-1 + c*x]*Sqrt[1 +
c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (-2*(a + b*ArcCosh[c*x]))/b])/(E^((2*a)/b)*Sqrt[d - c^2*d*x^2]*(-((a
+ b*ArcCosh[c*x])/b))^n) - (2^(-2 - n)*c*d^3*E^((2*a)/b)*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*G
amma[1 + n, (2*(a + b*ArcCosh[c*x]))/b])/(Sqrt[d - c^2*d*x^2]*((a + b*ArcCosh[c*x])/b)^n) + (c*d^3*E^((4*a)/b)
*Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(a + b*ArcCosh[c*x])^n*Gamma[1 + n, (4*(a + b*ArcCosh[c*x]))/b])/(2^(2*(3 + n))*
Sqrt[d - c^2*d*x^2]*((a + b*ArcCosh[c*x])/b)^n) + d^3*Defer[Int][(a + b*ArcCosh[c*x])^n/(x^2*Sqrt[d - c^2*d*x^
2]), x]

Rubi steps

\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x^2} \, dx &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \left (\frac {3 c^2 \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 c^4 x^2 \left (a+b \cosh ^{-1}(c x)\right )^n}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {c^6 x^4 \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^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 c^2 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\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 (3 c^4 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^2 \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^6 d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {x^4 \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 {3 c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b (1+n) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \cosh ^4(x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \cosh ^2(x) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {3 c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{b (1+n) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {3}{8} (a+b x)^n+\frac {1}{2} (a+b x)^n \cosh (2 x)+\frac {1}{8} (a+b x)^n \cosh (4 x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int \left (\frac {1}{2} (a+b x)^n+\frac {1}{2} (a+b x)^n \cosh (2 x)\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {15 c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{8 b (1+n) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \cosh (4 x) \, dx,x,\cosh ^{-1}(c x)\right )}{8 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \cosh (2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int (a+b x)^n \cosh (2 x) \, dx,x,\cosh ^{-1}(c x)\right )}{2 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {15 c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{8 b (1+n) \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \sqrt {-1+c x} \sqrt {1+c x}} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{-4 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{4 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{16 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{-2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 c d^2 \sqrt {d-c^2 d x^2}\right ) \text {Subst}\left (\int e^{2 x} (a+b x)^n \, dx,x,\cosh ^{-1}(c x)\right )}{4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {15 c d^2 \sqrt {d-c^2 d x^2} \left (a+b \cosh ^{-1}(c x)\right )^{1+n}}{8 b (1+n) \sqrt {-1+c x} \sqrt {1+c x}}+\frac {4^{-3-n} c d^2 e^{-\frac {4 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 {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {2^{-2-n} c d^2 e^{-\frac {2 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 {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}+\frac {2^{-2-n} c d^2 e^{\frac {2 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 {2 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {4^{-3-n} c d^2 e^{\frac {4 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 {4 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{\sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (d^2 \sqrt {d-c^2 d x^2}\right ) \int \frac {\left (a+b \cosh ^{-1}(c x)\right )^n}{x^2 \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 = 0.43, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (d-c^2 d x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^n}{x^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n)/x^2,x]

[Out]

Integrate[((d - c^2*d*x^2)^(5/2)*(a + b*ArcCosh[c*x])^n)/x^2, x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-c^{2} d \,x^{2}+d \right )^{\frac {5}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}}{x^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n/x^2,x)

[Out]

int((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n/x^2,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((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n/x^2,x, algorithm="maxima")

[Out]

integrate((-c^2*d*x^2 + d)^(5/2)*(b*arccosh(c*x) + a)^n/x^2, 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((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n/x^2,x, algorithm="fricas")

[Out]

integral((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*sqrt(-c^2*d*x^2 + d)*(b*arccosh(c*x) + a)^n/x^2, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*d*x**2+d)**(5/2)*(a+b*acosh(c*x))**n/x**2,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6190 deep

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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((-c^2*d*x^2+d)^(5/2)*(a+b*arccosh(c*x))^n/x^2,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\,{\left (d-c^2\,d\,x^2\right )}^{5/2}}{x^2} \,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)^(5/2))/x^2,x)

[Out]

int(((a + b*acosh(c*x))^n*(d - c^2*d*x^2)^(5/2))/x^2, x)

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