3.1.40 \(\int \frac {(d+e x)^m}{(a+b \cosh ^{-1}(c x))^2} \, dx\) [40]

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {(d+e x)^m}{\left (a+b \cosh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

Unintegrable((e*x+d)^m/(a+b*arccosh(c*x))^2,x)

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Rubi [A]
time = 0.02, 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 {(d+e x)^m}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(d + e*x)^m/(a + b*ArcCosh[c*x])^2,x]

[Out]

Defer[Int][(d + e*x)^m/(a + b*ArcCosh[c*x])^2, x]

Rubi steps

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

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Mathematica [A]
time = 0.59, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(d+e x)^m}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(d + e*x)^m/(a + b*ArcCosh[c*x])^2,x]

[Out]

Integrate[(d + e*x)^m/(a + b*ArcCosh[c*x])^2, x]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^m/(a+b*arccosh(c*x))^2,x)

[Out]

int((e*x+d)^m/(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((e*x+d)^m/(a+b*arccosh(c*x))^2,x, algorithm="maxima")

[Out]

-((c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1)*(x*e + d)^m + (c^3*x^3 - c*x)*(x*e + d)^m)/(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 + sqrt(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^3*(m + 1)*x^3*e + c^3*d*x^2 - c*(m - 1)*x*e + c*d)*(c*x + 1)*(
c*x - 1)*(x*e + d)^m + (2*c^4*(m + 1)*x^4*e + 2*c^4*d*x^3 - c^2*(3*m + 1)*x^2*e - c^2*d*x + m*e)*sqrt(c*x + 1)
*sqrt(c*x - 1)*(x*e + d)^m + (c^5*(m + 1)*x^5*e + c^5*d*x^4 - 2*c^3*(m + 1)*x^3*e - 2*c^3*d*x^2 + c*(m + 1)*x*
e + c*d)*(x*e + d)^m)/(a*b*c^5*x^5*e + a*b*c^5*d*x^4 - 2*a*b*c^3*x^3*e - 2*a*b*c^3*d*x^2 + a*b*c*x*e + a*b*c*d
+ (a*b*c^3*x^3*e + a*b*c^3*d*x^2)*(c*x + 1)*(c*x - 1) + 2*(a*b*c^4*x^4*e + a*b*c^4*d*x^3 - a*b*c^2*x^2*e - a*
b*c^2*d*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*x^5*e + b^2*c^5*d*x^4 - 2*b^2*c^3*x^3*e - 2*b^2*c^3*d*x^2 +
b^2*c*x*e + b^2*c*d + (b^2*c^3*x^3*e + b^2*c^3*d*x^2)*(c*x + 1)*(c*x - 1) + 2*(b^2*c^4*x^4*e + b^2*c^4*d*x^3 -
b^2*c^2*x^2*e - b^2*c^2*d*x)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + 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((e*x+d)^m/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral((x*e + d)^m/(b^2*arccosh(c*x)^2 + 2*a*b*arccosh(c*x) + a^2), x)

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

[Out]

Integral((d + e*x)**m/(a + b*acosh(c*x))**2, x)

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Giac [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((e*x+d)^m/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

integrate((e*x + d)^m/(b*arccosh(c*x) + a)^2, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {{\left (d+e\,x\right )}^m}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^m/(a + b*acosh(c*x))^2,x)

[Out]

int((d + e*x)^m/(a + b*acosh(c*x))^2, x)

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