3.51 \(\int (d x)^m (a+b \sec ^{-1}(c x))^3 \, dx\)

Optimal. Leaf size=18 \[ \text{Unintegrable}\left ((d x)^m \left (a+b \sec ^{-1}(c x)\right )^3,x\right ) \]

[Out]

Unintegrable[(d*x)^m*(a + b*ArcSec[c*x])^3, x]

________________________________________________________________________________________

Rubi [A]  time = 0.0220289, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int (d x)^m \left (a+b \sec ^{-1}(c x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d*x)^m*(a + b*ArcSec[c*x])^3,x]

[Out]

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

Rubi steps

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

Mathematica [A]  time = 4.39341, size = 0, normalized size = 0. \[ \int (d x)^m \left (a+b \sec ^{-1}(c x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

Maple [A]  time = 1.964, size = 0, normalized size = 0. \begin{align*} \int \left ( dx \right ) ^{m} \left ( a+b{\rm arcsec} \left (cx\right ) \right ) ^{3}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b^{3} \operatorname{arcsec}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname{arcsec}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname{arcsec}\left (c x\right ) + a^{3}\right )} \left (d x\right )^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arcsec}\left (c x\right ) + a\right )}^{3} \left (d x\right )^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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