3.4.57 \(\int \frac {x^3}{(1-c^2 x^2)^{5/2} (a+b \cosh ^{-1}(c x))^2} \, dx\) [357]

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {x^3}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.10, 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 {x^3}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 35.32, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^3}{\left (1-c^2 x^2\right )^{5/2} \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[x^3/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2),x]

[Out]

Integrate[x^3/((1 - c^2*x^2)^(5/2)*(a + b*ArcCosh[c*x])^2), x]

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Maple [A]
time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {x^{3}}{\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x)

[Out]

int(x^3/(-c^2*x^2+1)^(5/2)/(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(x^3/(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

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

[Out]

integral(-sqrt(-c^2*x^2 + 1)*x^3/(a^2*c^6*x^6 - 3*a^2*c^4*x^4 + 3*a^2*c^2*x^2 + (b^2*c^6*x^6 - 3*b^2*c^4*x^4 +
3*b^2*c^2*x^2 - b^2)*arccosh(c*x)^2 - a^2 + 2*(a*b*c^6*x^6 - 3*a*b*c^4*x^4 + 3*a*b*c^2*x^2 - a*b)*arccosh(c*x
)), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{3}}{\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{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(x**3/(-c**2*x**2+1)**(5/2)/(a+b*acosh(c*x))**2,x)

[Out]

Integral(x**3/((-(c*x - 1)*(c*x + 1))**(5/2)*(a + b*acosh(c*x))**2), 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(x^3/(-c^2*x^2+1)^(5/2)/(a+b*arccosh(c*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 = -0.03 \begin {gather*} \int \frac {x^3}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (1-c^2\,x^2\right )}^{5/2}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/((a + b*acosh(c*x))^2*(1 - c^2*x^2)^(5/2)),x)

[Out]

int(x^3/((a + b*acosh(c*x))^2*(1 - c^2*x^2)^(5/2)), x)

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