3.4.39 \(\int \frac {(1-c^2 x^2)^{5/2}}{x^2 (a+b \cosh ^{-1}(c x))^2} \, dx\) [339]

Optimal. Leaf size=162 \[ -\frac {\sqrt {-1+c x} \sqrt {1+c x} \left (1-c^2 x^2\right )^{5/2}}{b c x^2 \left (a+b \cosh ^{-1}(c x)\right )}+\frac {2 \sqrt {1-c x} \text {Int}\left (\frac {\left (-1+c^2 x^2\right )^2}{x^3 \left (a+b \cosh ^{-1}(c x)\right )},x\right )}{b c \sqrt {-1+c x}}+\frac {4 c \sqrt {1-c x} \text {Int}\left (\frac {\left (-1+c^2 x^2\right )^2}{x \left (a+b \cosh ^{-1}(c x)\right )},x\right )}{b \sqrt {-1+c x}} \]

[Out]

-(-c^2*x^2+1)^(5/2)*(c*x-1)^(1/2)*(c*x+1)^(1/2)/b/c/x^2/(a+b*arccosh(c*x))+2*(-c*x+1)^(1/2)*Unintegrable((c^2*
x^2-1)^2/x^3/(a+b*arccosh(c*x)),x)/b/c/(c*x-1)^(1/2)+4*c*(-c*x+1)^(1/2)*Unintegrable((c^2*x^2-1)^2/x/(a+b*arcc
osh(c*x)),x)/b/(c*x-1)^(1/2)

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Rubi [A]
time = 0.43, 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 (1-c^2 x^2\right )^{5/2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcCosh[c*x])^2),x]

[Out]

-((Sqrt[-1 + c*x]*Sqrt[1 + c*x]*(1 - c^2*x^2)^(5/2))/(b*c*x^2*(a + b*ArcCosh[c*x]))) + (2*Sqrt[1 - c*x]*Defer[
Int][(-1 + c^2*x^2)^2/(x^3*(a + b*ArcCosh[c*x])), x])/(b*c*Sqrt[-1 + c*x]) + (4*c*Sqrt[1 - c*x]*Defer[Int][(-1
+ c^2*x^2)^2/(x*(a + b*ArcCosh[c*x])), x])/(b*Sqrt[-1 + c*x])

Rubi steps

\begin {align*} \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {(-1+c x)^{5/2} (1+c x)^{5/2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {(1-c x)^3 (1+c x)^{5/2} \sqrt {1-c^2 x^2}}{b c x^2 \sqrt {-1+c x} \left (a+b \cosh ^{-1}(c x)\right )}+\frac {\left (2 \sqrt {1-c^2 x^2}\right ) \int \frac {\left (-1+c^2 x^2\right )^2}{x^3 \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b c \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (4 c \sqrt {1-c^2 x^2}\right ) \int \frac {\left (-1+c^2 x^2\right )^2}{x \left (a+b \cosh ^{-1}(c x)\right )} \, dx}{b \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}

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Mathematica [A]
time = 11.89, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x^2 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcCosh[c*x])^2),x]

[Out]

Integrate[(1 - c^2*x^2)^(5/2)/(x^2*(a + b*ArcCosh[c*x])^2), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}{x^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*x^2+1)^(5/2)/x^2/(a+b*arccosh(c*x))^2,x)

[Out]

int((-c^2*x^2+1)^(5/2)/x^2/(a+b*arccosh(c*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*x^2+1)^(5/2)/x^2/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

-((c^6*x^6 - 3*c^4*x^4 + 3*c^2*x^2 - 1)*(c*x + 1)*sqrt(c*x - 1) + (c^7*x^7 - 3*c^5*x^5 + 3*c^3*x^3 - c*x)*sqrt
(c*x + 1))*sqrt(-c*x + 1)/(a*b*c^3*x^4 + sqrt(c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x^3 - a*b*c*x^2 + (b^2*c^3*x^4 +
sqrt(c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x^3 - b^2*c*x^2)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) + integrate(((4*c
^7*x^7 - 5*c^5*x^5 - 2*c^3*x^3 + 3*c*x)*(c*x + 1)^(3/2)*(c*x - 1) + 2*(4*c^8*x^8 - 8*c^6*x^6 + 3*c^4*x^4 + 2*c
^2*x^2 - 1)*(c*x + 1)*sqrt(c*x - 1) + (4*c^9*x^9 - 11*c^7*x^7 + 9*c^5*x^5 - c^3*x^3 - c*x)*sqrt(c*x + 1))*sqrt
(-c*x + 1)/(a*b*c^5*x^7 + (c*x + 1)*(c*x - 1)*a*b*c^3*x^5 - 2*a*b*c^3*x^5 + a*b*c*x^3 + 2*(a*b*c^4*x^6 - a*b*c
^2*x^4)*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^7 + (c*x + 1)*(c*x - 1)*b^2*c^3*x^5 - 2*b^2*c^3*x^5 + b^2*c*x
^3 + 2*(b^2*c^4*x^6 - b^2*c^2*x^4)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), 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*x^2+1)^(5/2)/x^2/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral((c^4*x^4 - 2*c^2*x^2 + 1)*sqrt(-c^2*x^2 + 1)/(b^2*x^2*arccosh(c*x)^2 + 2*a*b*x^2*arccosh(c*x) + a^2*x
^2), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{x^{2} \left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*x**2+1)**(5/2)/x**2/(a+b*acosh(c*x))**2,x)

[Out]

Integral((-(c*x - 1)*(c*x + 1))**(5/2)/(x**2*(a + b*acosh(c*x))**2), x)

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Giac [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*x^2+1)^(5/2)/x^2/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

integrate((-c^2*x^2 + 1)^(5/2)/((b*arccosh(c*x) + a)^2*x^2), x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (1-c^2\,x^2\right )}^{5/2}}{x^2\,{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - c^2*x^2)^(5/2)/(x^2*(a + b*acosh(c*x))^2),x)

[Out]

int((1 - c^2*x^2)^(5/2)/(x^2*(a + b*acosh(c*x))^2), x)

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