\(\int \frac {1}{x^2 \text {arccosh}(a x)^4} \, dx\) [71]

Optimal result

Integrand size = 10, antiderivative size = 10 \[ \int \frac {1}{x^2 \text {arccosh}(a x)^4} \, dx=\text {Int}\left (\frac {1}{x^2 \text {arccosh}(a x)^4},x\right ) \]

[Out]

Unintegrable(1/x^2/arccosh(a*x)^4,x)

Rubi [N/A]

Not integrable

Time = 0.01 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x^2 \text {arccosh}(a x)^4} \, dx=\int \frac {1}{x^2 \text {arccosh}(a x)^4} \, dx \]

[In]

Int[1/(x^2*ArcCosh[a*x]^4),x]

[Out]

Defer[Int][1/(x^2*ArcCosh[a*x]^4), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {1}{x^2 \text {arccosh}(a x)^4} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 6.01 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arccosh}(a x)^4} \, dx=\int \frac {1}{x^2 \text {arccosh}(a x)^4} \, dx \]

[In]

Integrate[1/(x^2*ArcCosh[a*x]^4),x]

[Out]

Integrate[1/(x^2*ArcCosh[a*x]^4), x]

Maple [N/A] (verified)

Not integrable

Time = 0.11 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

\[\int \frac {1}{x^{2} \operatorname {arccosh}\left (a x \right )^{4}}d x\]

[In]

int(1/x^2/arccosh(a*x)^4,x)

[Out]

int(1/x^2/arccosh(a*x)^4,x)

Fricas [N/A]

Not integrable

Time = 0.24 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arccosh}(a x)^4} \, dx=\int { \frac {1}{x^{2} \operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]

[In]

integrate(1/x^2/arccosh(a*x)^4,x, algorithm="fricas")

[Out]

integral(1/(x^2*arccosh(a*x)^4), x)

Sympy [N/A]

Not integrable

Time = 5.36 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arccosh}(a x)^4} \, dx=\int \frac {1}{x^{2} \operatorname {acosh}^{4}{\left (a x \right )}}\, dx \]

[In]

integrate(1/x**2/acosh(a*x)**4,x)

[Out]

Integral(1/(x**2*acosh(a*x)**4), x)

Maxima [N/A]

Not integrable

Time = 2.58 (sec) , antiderivative size = 1996, normalized size of antiderivative = 199.60 \[ \int \frac {1}{x^2 \text {arccosh}(a x)^4} \, dx=\int { \frac {1}{x^{2} \operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]

[In]

integrate(1/x^2/arccosh(a*x)^4,x, algorithm="maxima")

[Out]

-1/6*(2*a^13*x^13 - 10*a^11*x^11 + 20*a^9*x^9 - 20*a^7*x^7 + 10*a^5*x^5 + 2*(a^8*x^8 - a^6*x^6)*(a*x + 1)^(5/2
)*(a*x - 1)^(5/2) - 2*a^3*x^3 + 2*(5*a^9*x^9 - 9*a^7*x^7 + 4*a^5*x^5)*(a*x + 1)^2*(a*x - 1)^2 + 4*(5*a^10*x^10
- 13*a^8*x^8 + 11*a^6*x^6 - 3*a^4*x^4)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 4*(5*a^11*x^11 - 17*a^9*x^9 + 21*a^7
*x^7 - 11*a^5*x^5 + 2*a^3*x^3)*(a*x + 1)*(a*x - 1) + (a^13*x^13 - 5*a^11*x^11 + 10*a^9*x^9 - 10*a^7*x^7 + 5*a^
5*x^5 + (a^8*x^8 - 13*a^6*x^6 + 27*a^4*x^4 - 15*a^2*x^2)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) - a^3*x^3 + (5*a^9*x^
9 - 57*a^7*x^7 + 124*a^5*x^5 - 90*a^3*x^3 + 18*a*x)*(a*x + 1)^2*(a*x - 1)^2 + (10*a^10*x^10 - 98*a^8*x^8 + 220
*a^6*x^6 - 189*a^4*x^4 + 63*a^2*x^2 - 6)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 2*(5*a^11*x^11 - 41*a^9*x^9 + 93*a^
7*x^7 - 89*a^5*x^5 + 38*a^3*x^3 - 6*a*x)*(a*x + 1)*(a*x - 1) + (5*a^12*x^12 - 33*a^10*x^10 + 73*a^8*x^8 - 74*a
^6*x^6 + 36*a^4*x^4 - 7*a^2*x^2)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2 + 2*(5*
a^12*x^12 - 21*a^10*x^10 + 34*a^8*x^8 - 26*a^6*x^6 + 9*a^4*x^4 - a^2*x^2)*sqrt(a*x + 1)*sqrt(a*x - 1) - (a^13*
x^13 - 5*a^11*x^11 + 10*a^9*x^9 - 10*a^7*x^7 + 5*a^5*x^5 + (a^8*x^8 - 4*a^6*x^6 + 3*a^4*x^4)*(a*x + 1)^(5/2)*(
a*x - 1)^(5/2) - a^3*x^3 + (5*a^9*x^9 - 21*a^7*x^7 + 24*a^5*x^5 - 8*a^3*x^3)*(a*x + 1)^2*(a*x - 1)^2 + (10*a^1
0*x^10 - 44*a^8*x^8 + 64*a^6*x^6 - 37*a^4*x^4 + 7*a^2*x^2)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + 2*(5*a^11*x^11 -
23*a^9*x^9 + 39*a^7*x^7 - 30*a^5*x^5 + 10*a^3*x^3 - a*x)*(a*x + 1)*(a*x - 1) + (5*a^12*x^12 - 24*a^10*x^10 + 4
5*a^8*x^8 - 41*a^6*x^6 + 18*a^4*x^4 - 3*a^2*x^2)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a*x
- 1)))/((a^13*x^14 - 5*a^11*x^12 + (a*x + 1)^(5/2)*(a*x - 1)^(5/2)*a^8*x^9 + 10*a^9*x^10 - 10*a^7*x^8 + 5*a^5
*x^6 - a^3*x^4 + 5*(a^9*x^10 - a^7*x^8)*(a*x + 1)^2*(a*x - 1)^2 + 10*(a^10*x^11 - 2*a^8*x^9 + a^6*x^7)*(a*x +
1)^(3/2)*(a*x - 1)^(3/2) + 10*(a^11*x^12 - 3*a^9*x^10 + 3*a^7*x^8 - a^5*x^6)*(a*x + 1)*(a*x - 1) + 5*(a^12*x^1
3 - 4*a^10*x^11 + 6*a^8*x^9 - 4*a^6*x^7 + a^4*x^5)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*x + sqrt(a*x + 1)*sqrt(a
*x - 1))^3) - integrate(1/6*(a^15*x^15 - 6*a^13*x^13 + 15*a^11*x^11 - 20*a^9*x^9 + 15*a^7*x^7 - 6*a^5*x^5 + (a
^9*x^9 - 39*a^7*x^7 + 135*a^5*x^5 - 105*a^3*x^3)*(a*x + 1)^3*(a*x - 1)^3 + (6*a^10*x^10 - 201*a^8*x^8 + 677*a^
6*x^6 - 663*a^4*x^4 + 174*a^2*x^2)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + a^3*x^3 + (15*a^11*x^11 - 420*a^9*x^9 + 1
373*a^7*x^7 - 1565*a^5*x^5 + 705*a^3*x^3 - 108*a*x)*(a*x + 1)^2*(a*x - 1)^2 + (20*a^12*x^12 - 450*a^10*x^10 +
1422*a^8*x^8 - 1787*a^6*x^6 + 1059*a^4*x^4 - 288*a^2*x^2 + 24)*(a*x + 1)^(3/2)*(a*x - 1)^(3/2) + (15*a^13*x^13
- 255*a^11*x^11 + 773*a^9*x^9 - 1026*a^7*x^7 + 714*a^5*x^5 - 257*a^3*x^3 + 36*a*x)*(a*x + 1)*(a*x - 1) + (6*a
^14*x^14 - 69*a^12*x^12 + 197*a^10*x^10 - 266*a^8*x^8 + 201*a^6*x^6 - 83*a^4*x^4 + 14*a^2*x^2)*sqrt(a*x + 1)*s
qrt(a*x - 1))/((a^15*x^17 - 6*a^13*x^15 + (a*x + 1)^3*(a*x - 1)^3*a^9*x^11 + 15*a^11*x^13 - 20*a^9*x^11 + 15*a
^7*x^9 - 6*a^5*x^7 + a^3*x^5 + 6*(a^10*x^12 - a^8*x^10)*(a*x + 1)^(5/2)*(a*x - 1)^(5/2) + 15*(a^11*x^13 - 2*a^
9*x^11 + a^7*x^9)*(a*x + 1)^2*(a*x - 1)^2 + 20*(a^12*x^14 - 3*a^10*x^12 + 3*a^8*x^10 - a^6*x^8)*(a*x + 1)^(3/2
)*(a*x - 1)^(3/2) + 15*(a^13*x^15 - 4*a^11*x^13 + 6*a^9*x^11 - 4*a^7*x^9 + a^5*x^7)*(a*x + 1)*(a*x - 1) + 6*(a
^14*x^16 - 5*a^12*x^14 + 10*a^10*x^12 - 10*a^8*x^10 + 5*a^6*x^8 - a^4*x^6)*sqrt(a*x + 1)*sqrt(a*x - 1))*log(a*
x + sqrt(a*x + 1)*sqrt(a*x - 1))), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arccosh}(a x)^4} \, dx=\int { \frac {1}{x^{2} \operatorname {arcosh}\left (a x\right )^{4}} \,d x } \]

[In]

integrate(1/x^2/arccosh(a*x)^4,x, algorithm="giac")

[Out]

integrate(1/(x^2*arccosh(a*x)^4), x)

Mupad [N/A]

Not integrable

Time = 2.66 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x^2 \text {arccosh}(a x)^4} \, dx=\int \frac {1}{x^2\,{\mathrm {acosh}\left (a\,x\right )}^4} \,d x \]

[In]

int(1/(x^2*acosh(a*x)^4),x)

[Out]

int(1/(x^2*acosh(a*x)^4), x)