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

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 = 0.03, 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 = {} \[ \int \frac {(d+e x)^m}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

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 = 0.84, size = 0, normalized size = 0.00 \[ \int \frac {(d+e x)^m}{\left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

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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fricas [A]  time = 0.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (e x + d\right )}^{m}}{b^{2} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, a b \operatorname {arcosh}\left (c x\right ) + a^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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maple [A]  time = 1.50, 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 \[ -\frac {{\left (c^{2} x^{2} - 1\right )} \sqrt {c x + 1} \sqrt {c x - 1} {\left (e x + d\right )}^{m} + {\left (c^{3} x^{3} - c x\right )} {\left (e x + d\right )}^{m}}{a b c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} a b c^{2} x - a b c + {\left (b^{2} c^{3} x^{2} + \sqrt {c x + 1} \sqrt {c x - 1} b^{2} c^{2} x - b^{2} c\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )} + \int \frac {{\left (c^{3} e {\left (m + 1\right )} x^{3} + c^{3} d x^{2} - c e {\left (m - 1\right )} x + c d\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} {\left (e x + d\right )}^{m} + {\left (2 \, c^{4} e {\left (m + 1\right )} x^{4} + 2 \, c^{4} d x^{3} - c^{2} e {\left (3 \, m + 1\right )} x^{2} - c^{2} d x + e m\right )} \sqrt {c x + 1} \sqrt {c x - 1} {\left (e x + d\right )}^{m} + {\left (c^{5} e {\left (m + 1\right )} x^{5} + c^{5} d x^{4} - 2 \, c^{3} e {\left (m + 1\right )} x^{3} - 2 \, c^{3} d x^{2} + c e {\left (m + 1\right )} x + c d\right )} {\left (e x + d\right )}^{m}}{a b c^{5} e x^{5} + a b c^{5} d x^{4} - 2 \, a b c^{3} e x^{3} - 2 \, a b c^{3} d x^{2} + a b c e x + a b c d + {\left (a b c^{3} e x^{3} + a b c^{3} d x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 2 \, {\left (a b c^{4} e x^{4} + a b c^{4} d x^{3} - a b c^{2} e x^{2} - a b c^{2} d x\right )} \sqrt {c x + 1} \sqrt {c x - 1} + {\left (b^{2} c^{5} e x^{5} + b^{2} c^{5} d x^{4} - 2 \, b^{2} c^{3} e x^{3} - 2 \, b^{2} c^{3} d x^{2} + b^{2} c e x + b^{2} c d + {\left (b^{2} c^{3} e x^{3} + b^{2} c^{3} d x^{2}\right )} {\left (c x + 1\right )} {\left (c x - 1\right )} + 2 \, {\left (b^{2} c^{4} e x^{4} + b^{2} c^{4} d x^{3} - b^{2} c^{2} e x^{2} - b^{2} c^{2} d x\right )} \sqrt {c x + 1} \sqrt {c x - 1}\right )} \log \left (c x + \sqrt {c x + 1} \sqrt {c x - 1}\right )}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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)*(e*x + d)^m + (c^3*x^3 - c*x)*(e*x + 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*e*(m + 1)*x^3 + c^3*d*x^2 - c*e*(m - 1)*x + c*d)*(c*x + 1)*(
c*x - 1)*(e*x + d)^m + (2*c^4*e*(m + 1)*x^4 + 2*c^4*d*x^3 - c^2*e*(3*m + 1)*x^2 - c^2*d*x + e*m)*sqrt(c*x + 1)
*sqrt(c*x - 1)*(e*x + d)^m + (c^5*e*(m + 1)*x^5 + c^5*d*x^4 - 2*c^3*e*(m + 1)*x^3 - 2*c^3*d*x^2 + c*e*(m + 1)*
x + c*d)*(e*x + d)^m)/(a*b*c^5*e*x^5 + a*b*c^5*d*x^4 - 2*a*b*c^3*e*x^3 - 2*a*b*c^3*d*x^2 + a*b*c*e*x + a*b*c*d
+ (a*b*c^3*e*x^3 + a*b*c^3*d*x^2)*(c*x + 1)*(c*x - 1) + 2*(a*b*c^4*e*x^4 + a*b*c^4*d*x^3 - a*b*c^2*e*x^2 - a*
b*c^2*d*x)*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*e*x^5 + b^2*c^5*d*x^4 - 2*b^2*c^3*e*x^3 - 2*b^2*c^3*d*x^2 +
b^2*c*e*x + b^2*c*d + (b^2*c^3*e*x^3 + b^2*c^3*d*x^2)*(c*x + 1)*(c*x - 1) + 2*(b^2*c^4*e*x^4 + b^2*c^4*d*x^3 -
b^2*c^2*e*x^2 - 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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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int \frac {{\left (d+e\,x\right )}^m}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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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