3.39 \(\int (d x)^m (a+b \text {csch}^{-1}(c x))^3 \, dx\)
Optimal. Leaf size=19 \[ \text {Int}\left ((d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3,x\right ) \]
[Out]
Unintegrable((d*x)^m*(a+b*arccsch(c*x))^3,x)
________________________________________________________________________________________
Rubi [A] time = 0.02, 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 =
{} \[ \int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx \]
Verification is Not applicable to the result.
[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*} \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\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 6.87, size = 0, normalized size = 0.00 \[ \int (d x)^m \left (a+b \text {csch}^{-1}(c x)\right )^3 \, dx \]
Verification is Not applicable to the result.
[In]
Integrate[(d*x)^m*(a + b*ArcCsch[c*x])^3,x]
[Out]
Integrate[(d*x)^m*(a + b*ArcCsch[c*x])^3, x]
________________________________________________________________________________________
fricas [A] time = 2.69, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \operatorname {arcsch}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname {arcsch}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname {arcsch}\left (c x\right ) + a^{3}\right )} \left (d x\right )^{m}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
giac [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}^{3} \left (d x\right )^{m}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
maple [A] time = 0.05, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \,\mathrm {arccsch}\left (c x \right )\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((d*x)^m*(a+b*arccsch(c*x))^3,x)
[Out]
int((d*x)^m*(a+b*arccsch(c*x))^3,x)
________________________________________________________________________________________
maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
mupad [A] time = 0.00, size = -1, normalized size = -0.05 \[ \int {\left (d\,x\right )}^m\,{\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[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)
________________________________________________________________________________________
sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} \left (a + b \operatorname {acsch}{\left (c x \right )}\right )^{3}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((d*x)**m*(a+b*acsch(c*x))**3,x)
[Out]
Integral((d*x)**m*(a + b*acsch(c*x))**3, x)
________________________________________________________________________________________