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

Optimal. Leaf size=33 \[ \text {Int}\left (\frac {\left (1-c^2 x^2\right )^{3/2} (f x)^m}{\left (a+b \cosh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.53, 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 = {} \[ \int \frac {(f x)^m \left (1-c^2 x^2\right )^{3/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 1.20, size = 0, normalized size = 0.00 \[ \int \frac {(f x)^m \left (1-c^2 x^2\right )^{3/2}}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (c^{2} x^{2} - 1\right )} \sqrt {-c^{2} x^{2} + 1} \left (f x\right )^{m}}{b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(-(c^2*x^2 - 1)*sqrt(-c^2*x^2 + 1)*(f*x)^m/(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2), 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((f*x)^m*(-c^2*x^2+1)^(3/2)/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,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 = 1.33, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x \right )^{m} \left (-c^{2} x^{2}+1\right )^{\frac {3}{2}}}{\left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left ({\left (c^{4} f^{m} x^{4} - 2 \, c^{2} f^{m} x^{2} + f^{m}\right )} {\left (c x + 1\right )} \sqrt {c x - 1} x^{m} + {\left (c^{5} f^{m} x^{5} - 2 \, c^{3} f^{m} x^{3} + c f^{m} x\right )} \sqrt {c x + 1} x^{m}\right )} \sqrt {-c x + 1}}{a b c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} a b c^{2} x - a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )} - \int \frac {{\left ({\left (c^{5} f^{m} {\left (m + 4\right )} x^{5} - c^{3} f^{m} {\left (2 \, m + 3\right )} x^{3} + c f^{m} {\left (m - 1\right )} x\right )} {\left (c x + 1\right )}^{\frac {3}{2}} {\left (c x - 1\right )} x^{m} + {\left (2 \, c^{6} f^{m} {\left (m + 4\right )} x^{6} - c^{4} f^{m} {\left (5 \, m + 12\right )} x^{4} + 4 \, c^{2} f^{m} {\left (m + 1\right )} x^{2} - f^{m} m\right )} {\left (c x + 1\right )} \sqrt {c x - 1} x^{m} + {\left (c^{7} f^{m} {\left (m + 4\right )} x^{7} - 3 \, c^{5} f^{m} {\left (m + 3\right )} x^{5} + 3 \, c^{3} f^{m} {\left (m + 2\right )} x^{3} - c f^{m} {\left (m + 1\right )} x\right )} \sqrt {c x + 1} x^{m}\right )} \sqrt {-c x + 1}}{a b c^{5} x^{5} + {\left (c x + 1\right )} {\left (c x - 1\right )} a b c^{3} x^{3} - 2 \, a b c^{3} x^{3} + a b c x + 2 \, {\left (a b c^{4} x^{4} - a b c^{2} x^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1} + {\left (b^{2} c^{5} x^{5} + {\left (c x + 1\right )} {\left (c x - 1\right )} b^{2} c^{3} x^{3} - 2 \, b^{2} c^{3} x^{3} + b^{2} c x + 2 \, {\left (b^{2} c^{4} x^{4} - b^{2} c^{2} x^{2}\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((c^4*f^m*x^4 - 2*c^2*f^m*x^2 + f^m)*(c*x + 1)*sqrt(c*x - 1)*x^m + (c^5*f^m*x^5 - 2*c^3*f^m*x^3 + c*f^m*x)*sqr
t(c*x + 1)*x^m)*sqrt(-c*x + 1)/(a*b*c^3*x^2 + sqrt(c*x + 1)*sqrt(c*x - 1)*a*b*c^2*x - a*b*c + (b^2*c^3*x^2 + s
qrt(c*x + 1)*sqrt(c*x - 1)*b^2*c^2*x - b^2*c)*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))) - integrate(((c^5*f^m*(m
+ 4)*x^5 - c^3*f^m*(2*m + 3)*x^3 + c*f^m*(m - 1)*x)*(c*x + 1)^(3/2)*(c*x - 1)*x^m + (2*c^6*f^m*(m + 4)*x^6 -
c^4*f^m*(5*m + 12)*x^4 + 4*c^2*f^m*(m + 1)*x^2 - f^m*m)*(c*x + 1)*sqrt(c*x - 1)*x^m + (c^7*f^m*(m + 4)*x^7 - 3
*c^5*f^m*(m + 3)*x^5 + 3*c^3*f^m*(m + 2)*x^3 - c*f^m*(m + 1)*x)*sqrt(c*x + 1)*x^m)*sqrt(-c*x + 1)/(a*b*c^5*x^5
+ (c*x + 1)*(c*x - 1)*a*b*c^3*x^3 - 2*a*b*c^3*x^3 + a*b*c*x + 2*(a*b*c^4*x^4 - a*b*c^2*x^2)*sqrt(c*x + 1)*sqr
t(c*x - 1) + (b^2*c^5*x^5 + (c*x + 1)*(c*x - 1)*b^2*c^3*x^3 - 2*b^2*c^3*x^3 + b^2*c*x + 2*(b^2*c^4*x^4 - b^2*c
^2*x^2)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1)*sqrt(c*x - 1))), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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