\(\int \frac {1}{x \text {arcsinh}(a x)^4} \, dx\) [72]

Optimal result

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

[Out]

Unintegrable(1/x/arcsinh(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 = {} \[ \int \frac {1}{x \text {arcsinh}(a x)^4} \, dx=\int \frac {1}{x \text {arcsinh}(a x)^4} \, dx \]

[In]

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

[Out]

Defer[Int][1/(x*ArcSinh[a*x]^4), x]

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

Mathematica [N/A]

Not integrable

Time = 1.99 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x \text {arcsinh}(a x)^4} \, dx=\int \frac {1}{x \text {arcsinh}(a x)^4} \, dx \]

[In]

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

[Out]

Integrate[1/(x*ArcSinh[a*x]^4), x]

Maple [N/A] (verified)

Not integrable

Time = 0.03 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00

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

[In]

int(1/x/arcsinh(a*x)^4,x)

[Out]

int(1/x/arcsinh(a*x)^4,x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x \text {arcsinh}(a x)^4} \, dx=\int { \frac {1}{x \operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]

[In]

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

[Out]

integral(1/(x*arcsinh(a*x)^4), x)

Sympy [N/A]

Not integrable

Time = 0.61 (sec) , antiderivative size = 10, normalized size of antiderivative = 1.00 \[ \int \frac {1}{x \text {arcsinh}(a x)^4} \, dx=\int \frac {1}{x \operatorname {asinh}^{4}{\left (a x \right )}}\, dx \]

[In]

integrate(1/x/asinh(a*x)**4,x)

[Out]

Integral(1/(x*asinh(a*x)**4), x)

Maxima [N/A]

Not integrable

Time = 1.29 (sec) , antiderivative size = 1611, normalized size of antiderivative = 161.10 \[ \int \frac {1}{x \text {arcsinh}(a x)^4} \, dx=\int { \frac {1}{x \operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]

[In]

integrate(1/x/arcsinh(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^3*x^3 + 2*(a^8*x^8 + a^6*x^6)*(a
^2*x^2 + 1)^(5/2) + 2*(5*a^9*x^9 + 9*a^7*x^7 + 4*a^5*x^5)*(a^2*x^2 + 1)^2 + (4*(a^6*x^6 + 3*a^4*x^4 + 2*a^2*x^
2)*(a^2*x^2 + 1)^(5/2) + (16*a^7*x^7 + 46*a^5*x^5 + 37*a^3*x^3 + 7*a*x)*(a^2*x^2 + 1)^2 + (24*a^8*x^8 + 66*a^6
*x^6 + 59*a^4*x^4 + 19*a^2*x^2 + 2)*(a^2*x^2 + 1)^(3/2) + (16*a^9*x^9 + 42*a^7*x^7 + 39*a^5*x^5 + 16*a^3*x^3 +
3*a*x)*(a^2*x^2 + 1) + (4*a^10*x^10 + 10*a^8*x^8 + 9*a^6*x^6 + 4*a^4*x^4 + a^2*x^2)*sqrt(a^2*x^2 + 1))*log(a*
x + sqrt(a^2*x^2 + 1))^2 + 4*(5*a^10*x^10 + 13*a^8*x^8 + 11*a^6*x^6 + 3*a^4*x^4)*(a^2*x^2 + 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^2*x^2 + 1) - (2*(a^6*x^6 + a^4*x^4)*(a^2*x^2 +
1)^(5/2) + (8*a^7*x^7 + 13*a^5*x^5 + 5*a^3*x^3)*(a^2*x^2 + 1)^2 + (12*a^8*x^8 + 27*a^6*x^6 + 19*a^4*x^4 + 4*a^
2*x^2)*(a^2*x^2 + 1)^(3/2) + (8*a^9*x^9 + 23*a^7*x^7 + 23*a^5*x^5 + 9*a^3*x^3 + a*x)*(a^2*x^2 + 1) + (2*a^10*x
^10 + 7*a^8*x^8 + 9*a^6*x^6 + 5*a^4*x^4 + a^2*x^2)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1)) + 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^2*x^2 + 1))/((a^13*x^13 + 5*a^11*
x^11 + (a^2*x^2 + 1)^(5/2)*a^8*x^8 + 10*a^9*x^9 + 10*a^7*x^7 + 5*a^5*x^5 + a^3*x^3 + 5*(a^9*x^9 + a^7*x^7)*(a^
2*x^2 + 1)^2 + 10*(a^10*x^10 + 2*a^8*x^8 + a^6*x^6)*(a^2*x^2 + 1)^(3/2) + 10*(a^11*x^11 + 3*a^9*x^9 + 3*a^7*x^
7 + a^5*x^5)*(a^2*x^2 + 1) + 5*(a^12*x^12 + 4*a^10*x^10 + 6*a^8*x^8 + 4*a^6*x^6 + a^4*x^4)*sqrt(a^2*x^2 + 1))*
log(a*x + sqrt(a^2*x^2 + 1))^3) - integrate(1/6*(8*(a^7*x^7 + 6*a^5*x^5 + 6*a^3*x^3)*(a^2*x^2 + 1)^3 + (40*a^8
*x^8 + 204*a^6*x^6 + 228*a^4*x^4 + 57*a^2*x^2)*(a^2*x^2 + 1)^(5/2) + 2*(40*a^9*x^9 + 168*a^7*x^7 + 200*a^5*x^5
+ 87*a^3*x^3 + 15*a*x)*(a^2*x^2 + 1)^2 + 2*(40*a^10*x^10 + 132*a^8*x^8 + 156*a^6*x^6 + 91*a^4*x^4 + 30*a^2*x^
2 + 3)*(a^2*x^2 + 1)^(3/2) + 2*(20*a^11*x^11 + 48*a^9*x^9 + 48*a^7*x^7 + 35*a^5*x^5 + 18*a^3*x^3 + 3*a*x)*(a^2
*x^2 + 1) + (8*a^12*x^12 + 12*a^10*x^10 + 4*a^8*x^8 + 5*a^6*x^6 + 6*a^4*x^4 + a^2*x^2)*sqrt(a^2*x^2 + 1))/((a^
15*x^16 + 6*a^13*x^14 + 15*a^11*x^12 + (a^2*x^2 + 1)^3*a^9*x^10 + 20*a^9*x^10 + 15*a^7*x^8 + 6*a^5*x^6 + a^3*x
^4 + 6*(a^10*x^11 + a^8*x^9)*(a^2*x^2 + 1)^(5/2) + 15*(a^11*x^12 + 2*a^9*x^10 + a^7*x^8)*(a^2*x^2 + 1)^2 + 20*
(a^12*x^13 + 3*a^10*x^11 + 3*a^8*x^9 + a^6*x^7)*(a^2*x^2 + 1)^(3/2) + 15*(a^13*x^14 + 4*a^11*x^12 + 6*a^9*x^10
+ 4*a^7*x^8 + a^5*x^6)*(a^2*x^2 + 1) + 6*(a^14*x^15 + 5*a^12*x^13 + 10*a^10*x^11 + 10*a^8*x^9 + 5*a^6*x^7 + a
^4*x^5)*sqrt(a^2*x^2 + 1))*log(a*x + sqrt(a^2*x^2 + 1))), x)

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x \text {arcsinh}(a x)^4} \, dx=\int { \frac {1}{x \operatorname {arsinh}\left (a x\right )^{4}} \,d x } \]

[In]

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

[Out]

integrate(1/(x*arcsinh(a*x)^4), x)

Mupad [N/A]

Not integrable

Time = 2.60 (sec) , antiderivative size = 12, normalized size of antiderivative = 1.20 \[ \int \frac {1}{x \text {arcsinh}(a x)^4} \, dx=\int \frac {1}{x\,{\mathrm {asinh}\left (a\,x\right )}^4} \,d x \]

[In]

int(1/(x*asinh(a*x)^4),x)

[Out]

int(1/(x*asinh(a*x)^4), x)