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

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

[Out]

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

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Rubi [A]  time = 0.54, 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 = {} \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.07, size = 0, normalized size = 0.00 \[ \int \frac {x^m \left (1-c^2 x^2\right )^{3/2}}{a+b \cosh ^{-1}(c x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (c^{2} x^{2} - 1\right )} \sqrt {-c^{2} x^{2} + 1} x^{m}}{b \operatorname {arcosh}\left (c x\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x)),x, algorithm="fricas")

[Out]

integral(-(c^2*x^2 - 1)*sqrt(-c^2*x^2 + 1)*x^m/(b*arccosh(c*x) + a), 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(x^m*(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*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 = 1.17, size = 0, normalized size = 0.00 \[ \int \frac {x^{m} \left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{a +b \,\mathrm {arccosh}\left (c x \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {x^m\,{\left (1-c^2\,x^2\right )}^{3/2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(-c**2*x**2+1)**(3/2)/(a+b*acosh(c*x)),x)

[Out]

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

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