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

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

[Out]

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

Verification is Not applicable to the result.

[In]

Int[1/((d + e*x)^2*(a + b*ArcCosh[c*x])^2),x]

[Out]

Defer[Int][1/((d + e*x)^2*(a + b*ArcCosh[c*x])^2), x]

Rubi steps

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

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Mathematica [A]  time = 103.18, size = 0, normalized size = 0.00 \[ \int \frac {1}{(d+e x)^2 \left (a+b \cosh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((d + e*x)^2*(a + b*ArcCosh[c*x])^2),x]

[Out]

Integrate[1/((d + e*x)^2*(a + b*ArcCosh[c*x])^2), x]

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fricas [A]  time = 0.52, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} e^{2} x^{2} + 2 \, a^{2} d e x + a^{2} d^{2} + {\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \operatorname {arcosh}\left (c x\right )^{2} + 2 \, {\left (a b e^{2} x^{2} + 2 \, a b d e x + a b d^{2}\right )} \operatorname {arcosh}\left (c x\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(a+b*arccosh(c*x))^2,x, algorithm="fricas")

[Out]

integral(1/(a^2*e^2*x^2 + 2*a^2*d*e*x + a^2*d^2 + (b^2*e^2*x^2 + 2*b^2*d*e*x + b^2*d^2)*arccosh(c*x)^2 + 2*(a*
b*e^2*x^2 + 2*a*b*d*e*x + a*b*d^2)*arccosh(c*x)), x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (e x + d\right )}^{2} {\left (b \operatorname {arcosh}\left (c x\right ) + a\right )}^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(a+b*arccosh(c*x))^2,x, algorithm="giac")

[Out]

integrate(1/((e*x + d)^2*(b*arccosh(c*x) + a)^2), x)

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maple [A]  time = 0.45, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (e x +d \right )^{2} \left (a +b \,\mathrm {arccosh}\left (c x \right )\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(e*x+d)^2/(a+b*arccosh(c*x))^2,x)

[Out]

int(1/(e*x+d)^2/(a+b*arccosh(c*x))^2,x)

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

[Out]

-(c^3*x^3 + (c^2*x^2 - 1)*sqrt(c*x + 1)*sqrt(c*x - 1) - c*x)/(a*b*c^3*e^2*x^4 + 2*a*b*c^3*d*e*x^3 - 2*a*b*c*d*
e*x - a*b*c*d^2 + (c^3*d^2 - c*e^2)*a*b*x^2 + (a*b*c^2*e^2*x^3 + 2*a*b*c^2*d*e*x^2 + a*b*c^2*d^2*x)*sqrt(c*x +
1)*sqrt(c*x - 1) + (b^2*c^3*e^2*x^4 + 2*b^2*c^3*d*e*x^3 - 2*b^2*c*d*e*x - b^2*c*d^2 + (c^3*d^2 - c*e^2)*b^2*x
^2 + (b^2*c^2*e^2*x^3 + 2*b^2*c^2*d*e*x^2 + b^2*c^2*d^2*x)*sqrt(c*x + 1)*sqrt(c*x - 1))*log(c*x + sqrt(c*x + 1
)*sqrt(c*x - 1))) - integrate((c^5*e*x^5 - c^5*d*x^4 - 2*c^3*e*x^3 + 2*c^3*d*x^2 + (c^3*e*x^3 - c^3*d*x^2 - 3*
c*e*x - c*d)*(c*x + 1)*(c*x - 1) + c*e*x + (2*c^4*e*x^4 - 2*c^4*d*x^3 - 5*c^2*e*x^2 + c^2*d*x + 2*e)*sqrt(c*x
+ 1)*sqrt(c*x - 1) - c*d)/(a*b*c^5*e^3*x^7 + 3*a*b*c^5*d*e^2*x^6 + (3*c^5*d^2*e - 2*c^3*e^3)*a*b*x^5 + 3*a*b*c
*d^2*e*x + (c^5*d^3 - 6*c^3*d*e^2)*a*b*x^4 + a*b*c*d^3 - (6*c^3*d^2*e - c*e^3)*a*b*x^3 - (2*c^3*d^3 - 3*c*d*e^
2)*a*b*x^2 + (a*b*c^3*e^3*x^5 + 3*a*b*c^3*d*e^2*x^4 + 3*a*b*c^3*d^2*e*x^3 + a*b*c^3*d^3*x^2)*(c*x + 1)*(c*x -
1) + 2*(a*b*c^4*e^3*x^6 + 3*a*b*c^4*d*e^2*x^5 - 3*a*b*c^2*d^2*e*x^2 - a*b*c^2*d^3*x + (3*c^4*d^2*e - c^2*e^3)*
a*b*x^4 + (c^4*d^3 - 3*c^2*d*e^2)*a*b*x^3)*sqrt(c*x + 1)*sqrt(c*x - 1) + (b^2*c^5*e^3*x^7 + 3*b^2*c^5*d*e^2*x^
6 + (3*c^5*d^2*e - 2*c^3*e^3)*b^2*x^5 + 3*b^2*c*d^2*e*x + (c^5*d^3 - 6*c^3*d*e^2)*b^2*x^4 + b^2*c*d^3 - (6*c^3
*d^2*e - c*e^3)*b^2*x^3 - (2*c^3*d^3 - 3*c*d*e^2)*b^2*x^2 + (b^2*c^3*e^3*x^5 + 3*b^2*c^3*d*e^2*x^4 + 3*b^2*c^3
*d^2*e*x^3 + b^2*c^3*d^3*x^2)*(c*x + 1)*(c*x - 1) + 2*(b^2*c^4*e^3*x^6 + 3*b^2*c^4*d*e^2*x^5 - 3*b^2*c^2*d^2*e
*x^2 - b^2*c^2*d^3*x + (3*c^4*d^2*e - c^2*e^3)*b^2*x^4 + (c^4*d^3 - 3*c^2*d*e^2)*b^2*x^3)*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 \[ \int \frac {1}{{\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*acosh(c*x))^2*(d + e*x)^2),x)

[Out]

int(1/((a + b*acosh(c*x))^2*(d + e*x)^2), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (a + b \operatorname {acosh}{\left (c x \right )}\right )^{2} \left (d + e x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)**2/(a+b*acosh(c*x))**2,x)

[Out]

Integral(1/((a + b*acosh(c*x))**2*(d + e*x)**2), x)

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