\(\int (d x)^m (a+b \text {csch}^{-1}(c x))^3 \, dx\) [39]

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx=\text {Int}\left ((d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3,x\right ) \]

[Out]

Unintegrable((d*x)^m*(a+b*arccsch(c*x))^3,x)

Rubi [N/A]

Not integrable

Time = 0.02 (sec) , antiderivative size = 16, 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 = {} \[ \int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx=\int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx \]

[In]

Int[(d*x)^m*(a + b*ArcCsch[c*x])^3,x]

[Out]

Defer[Int][(d*x)^m*(a + b*ArcCsch[c*x])^3, x]

Rubi steps \begin{align*} \text {integral}& = \int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 5.76 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx=\int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx \]

[In]

Integrate[(d*x)^m*(a + b*ArcCsch[c*x])^3,x]

[Out]

Integrate[(d*x)^m*(a + b*ArcCsch[c*x])^3, x]

Maple [N/A] (verified)

Not integrable

Time = 0.03 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

\[\int \left (d x \right )^{m} \left (a +b \,\operatorname {arccsch}\left (c x \right )\right )^{3}d x\]

[In]

int((d*x)^m*(a+b*arccsch(c*x))^3,x)

[Out]

int((d*x)^m*(a+b*arccsch(c*x))^3,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 44, normalized size of antiderivative = 2.75 \[ \int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{3} \left (d x\right )^{m} \,d x } \]

[In]

integrate((d*x)^m*(a+b*arccsch(c*x))^3,x, algorithm="fricas")

[Out]

integral((b^3*arccsch(c*x)^3 + 3*a*b^2*arccsch(c*x)^2 + 3*a^2*b*arccsch(c*x) + a^3)*(d*x)^m, x)

Sympy [N/A]

Not integrable

Time = 22.10 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx=\int \left (d x\right )^{m} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{3}\, dx \]

[In]

integrate((d*x)**m*(a+b*acsch(c*x))**3,x)

[Out]

Integral((d*x)**m*(a + b*acsch(c*x))**3, x)

Maxima [N/A]

Not integrable

Time = 7.73 (sec) , antiderivative size = 1351, normalized size of antiderivative = 84.44 \[ \int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{3} \left (d x\right )^{m} \,d x } \]

[In]

integrate((d*x)^m*(a+b*arccsch(c*x))^3,x, algorithm="maxima")

[Out]

b^3*d^m*x*x^m*log(sqrt(c^2*x^2 + 1) + 1)^3/(m + 1) + (d*x)^(m + 1)*a^3/(d*(m + 1)) - integrate((3*((b^3*d^m*(m
+ 1)*log(c) - a*b^2*d^m*(m + 1) - (a*b^2*c^2*d^m*(m + 1) - (d^m*(m + 1)*log(c) + d^m)*b^3*c^2)*x^2 + (b^3*c^2
*d^m*(m + 1)*x^2 + b^3*d^m*(m + 1))*log(x))*sqrt(c^2*x^2 + 1)*x^m + (b^3*d^m*(m + 1)*log(c) - a*b^2*d^m*(m + 1
) + (b^3*c^2*d^m*(m + 1)*log(c) - a*b^2*c^2*d^m*(m + 1))*x^2 + (b^3*c^2*d^m*(m + 1)*x^2 + b^3*d^m*(m + 1))*log
(x))*x^m)*log(sqrt(c^2*x^2 + 1) + 1)^2 + (b^3*d^m*(m + 1)*log(c)^3 - 3*a*b^2*d^m*(m + 1)*log(c)^2 + 3*a^2*b*d^
m*(m + 1)*log(c) + (b^3*c^2*d^m*(m + 1)*x^2 + b^3*d^m*(m + 1))*log(x)^3 + (b^3*c^2*d^m*(m + 1)*log(c)^3 - 3*a*
b^2*c^2*d^m*(m + 1)*log(c)^2 + 3*a^2*b*c^2*d^m*(m + 1)*log(c))*x^2 + 3*(b^3*d^m*(m + 1)*log(c) - a*b^2*d^m*(m
+ 1) + (b^3*c^2*d^m*(m + 1)*log(c) - a*b^2*c^2*d^m*(m + 1))*x^2)*log(x)^2 + 3*(b^3*d^m*(m + 1)*log(c)^2 - 2*a*
b^2*d^m*(m + 1)*log(c) + a^2*b*d^m*(m + 1) + (b^3*c^2*d^m*(m + 1)*log(c)^2 - 2*a*b^2*c^2*d^m*(m + 1)*log(c) +
a^2*b*c^2*d^m*(m + 1))*x^2)*log(x))*sqrt(c^2*x^2 + 1)*x^m + (b^3*d^m*(m + 1)*log(c)^3 - 3*a*b^2*d^m*(m + 1)*lo
g(c)^2 + 3*a^2*b*d^m*(m + 1)*log(c) + (b^3*c^2*d^m*(m + 1)*x^2 + b^3*d^m*(m + 1))*log(x)^3 + (b^3*c^2*d^m*(m +
1)*log(c)^3 - 3*a*b^2*c^2*d^m*(m + 1)*log(c)^2 + 3*a^2*b*c^2*d^m*(m + 1)*log(c))*x^2 + 3*(b^3*d^m*(m + 1)*log
(c) - a*b^2*d^m*(m + 1) + (b^3*c^2*d^m*(m + 1)*log(c) - a*b^2*c^2*d^m*(m + 1))*x^2)*log(x)^2 + 3*(b^3*d^m*(m +
1)*log(c)^2 - 2*a*b^2*d^m*(m + 1)*log(c) + a^2*b*d^m*(m + 1) + (b^3*c^2*d^m*(m + 1)*log(c)^2 - 2*a*b^2*c^2*d^
m*(m + 1)*log(c) + a^2*b*c^2*d^m*(m + 1))*x^2)*log(x))*x^m - 3*((b^3*d^m*(m + 1)*log(c)^2 - 2*a*b^2*d^m*(m + 1
)*log(c) + a^2*b*d^m*(m + 1) + (b^3*c^2*d^m*(m + 1)*log(c)^2 - 2*a*b^2*c^2*d^m*(m + 1)*log(c) + a^2*b*c^2*d^m*
(m + 1))*x^2 + (b^3*c^2*d^m*(m + 1)*x^2 + b^3*d^m*(m + 1))*log(x)^2 + 2*(b^3*d^m*(m + 1)*log(c) - a*b^2*d^m*(m
+ 1) + (b^3*c^2*d^m*(m + 1)*log(c) - a*b^2*c^2*d^m*(m + 1))*x^2)*log(x))*sqrt(c^2*x^2 + 1)*x^m + (b^3*d^m*(m
+ 1)*log(c)^2 - 2*a*b^2*d^m*(m + 1)*log(c) + a^2*b*d^m*(m + 1) + (b^3*c^2*d^m*(m + 1)*log(c)^2 - 2*a*b^2*c^2*d
^m*(m + 1)*log(c) + a^2*b*c^2*d^m*(m + 1))*x^2 + (b^3*c^2*d^m*(m + 1)*x^2 + b^3*d^m*(m + 1))*log(x)^2 + 2*(b^3
*d^m*(m + 1)*log(c) - a*b^2*d^m*(m + 1) + (b^3*c^2*d^m*(m + 1)*log(c) - a*b^2*c^2*d^m*(m + 1))*x^2)*log(x))*x^
m)*log(sqrt(c^2*x^2 + 1) + 1))/(c^2*(m + 1)*x^2 + (c^2*(m + 1)*x^2 + m + 1)*sqrt(c^2*x^2 + 1) + m + 1), x)

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx=\int { {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{3} \left (d x\right )^{m} \,d x } \]

[In]

integrate((d*x)^m*(a+b*arccsch(c*x))^3,x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)^3*(d*x)^m, x)

Mupad [N/A]

Not integrable

Time = 5.47 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx=\int {\left (d\,x\right )}^m\,{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]

[In]

int((d*x)^m*(a + b*asinh(1/(c*x)))^3,x)

[Out]

int((d*x)^m*(a + b*asinh(1/(c*x)))^3, x)