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

Optimal. Leaf size=31 \[ \text{Unintegrable}\left (\left (d-c^2 d x^2\right )^2 (f x)^m \left (a+b \cosh ^{-1}(c x)\right )^n,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0900236, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., 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*}

Mathematica [A]  time = 1.11284, size = 0, normalized size = 0. \[ \int (f x)^m \left (d-c^2 d x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right )^n \, dx \]

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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Maple [A]  time = 0.305, size = 0, normalized size = 0. \begin{align*} \int \left ( fx \right ) ^{m} \left ( -{c}^{2}d{x}^{2}+d \right ) ^{2} \left ( a+b{\rm arccosh} \left (cx\right ) \right ) ^{n}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \mathit{sage}_{0} x \end{align*}

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]

sage0*x