3.452 \(\int (f x)^m (d-c^2 d x^2)^2 (a+b \cosh ^{-1}(c x))^n \, dx\)

Optimal. Leaf size=32 \[ \text {Int}\left (\left (d-c^2 d x^2\right )^2 (f x)^m \left (a+b \cosh ^{-1}(c x)\right )^n,x\right ) \]

[Out]

Unintegrable((f*x)^m*(-c^2*d*x^2+d)^2*(a+b*arccosh(c*x))^n,x)

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Rubi [A]  time = 0.09, 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 = {} \[ \int (f x)^m \left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )^n \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][(f*x)^m*(d - c^2*d*x^2)^2*(a + b*ArcCosh[c*x])^n, x]

Rubi steps

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

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Mathematica [A]  time = 1.22, size = 0, normalized size = 0.00 \[ \int (f x)^m \left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )^n \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (c^{4} d^{2} x^{4} - 2 \, c^{2} d^{2} x^{2} + d^{2}\right )} \left (f x\right )^{m} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{n}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((c^4*d^2*x^4 - 2*c^2*d^2*x^2 + d^2)*(f*x)^m*(b*arccosh(c*x) + a)^n, 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*d*x^2+d)^2*(a+b*arccosh(c*x))^n,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 [F(-2)]  time = 180.00, size = 0, normalized size = 0.00 \[ \int \left (f x \right )^{m} \left (-c^{2} d \,x^{2}+d \right )^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{n}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((c^2*d*x^2 - d)^2*(f*x)^m*(b*arccosh(c*x) + a)^n, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

Timed out

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