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

Optimal. Leaf size=19 \[ \text {Int}\left ((d x)^m \left (a+b \csc ^{-1}(c x)\right )^3,x\right ) \]

[Out]

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

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

Verification is not applicable to the result.

[In]

Int[(d*x)^m*(a + b*ArcCsc[c*x])^3,x]

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 4.88, size = 0, normalized size = 0.00 \begin {gather*} \int (d x)^m \left (a+b \csc ^{-1}(c x)\right )^3 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 1.76, size = 0, normalized size = 0.00 \[\int \left (d x \right )^{m} \left (a +b \,\mathrm {arccsc}\left (c x \right )\right )^{3}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x)^(m + 1)*a^3/(d*(m + 1)) + 1/4*(4*b^3*d^m*x*x^m*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^3 - 3*b^3*d^m*x*x
^m*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))*log(c^2*x^2)^2 - 4*(m + 1)*integrate(-3/4*((a*b^2*d^m*m + a*b^2*d^m
- (a*b^2*c^2*d^m*m + a*b^2*c^2*d^m)*x^2)*x^m*log(c^2*x^2)^2 + 4*((b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))
+ a*b^2)*d^m*m - ((b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a*b^2)*c^2*d^m*m + (b^3*arctan2(1, sqrt(c*x
+ 1)*sqrt(c*x - 1)) + a*b^2)*c^2*d^m)*x^2 + (b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a*b^2)*d^m)*x^m*log
(x)^2 + 8*((b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a*b^2)*d^m*m*log(c) - ((b^3*arctan2(1, sqrt(c*x + 1)
*sqrt(c*x - 1)) + a*b^2)*c^2*d^m*m*log(c) + (b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a*b^2)*c^2*d^m*log(
c))*x^2 + (b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a*b^2)*d^m*log(c))*x^m*log(x) + (4*b^3*d^m*x^m*arctan
2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^2 - b^3*d^m*x^m*log(c^2*x^2)^2)*sqrt(c*x + 1)*sqrt(c*x - 1) - 4*((a*b^2*arct
an2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^2 + a^2*b*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) - (b^3*arctan2(1, sqrt(c
*x + 1)*sqrt(c*x - 1)) + a*b^2)*log(c)^2)*d^m*m + (((b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a*b^2)*c^2*
log(c)^2 - (a*b^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1))^2 + a^2*b*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*c
^2)*d^m*m + ((b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a*b^2)*c^2*log(c)^2 - (a*b^2*arctan2(1, sqrt(c*x +
1)*sqrt(c*x - 1))^2 + a^2*b*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)))*c^2)*d^m)*x^2 + (a*b^2*arctan2(1, sqrt(c
*x + 1)*sqrt(c*x - 1))^2 + a^2*b*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) - (b^3*arctan2(1, sqrt(c*x + 1)*sqrt(
c*x - 1)) + a*b^2)*log(c)^2)*d^m)*x^m - 4*(((b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a*b^2)*d^m*m - ((b^
3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a*b^2)*c^2*d^m*m + (b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) +
a*b^2)*c^2*d^m)*x^2 + (b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a*b^2)*d^m)*x^m*log(x) + ((b^3*arctan2(1,
sqrt(c*x + 1)*sqrt(c*x - 1)) + a*b^2)*d^m*m*log(c) - ((b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + a*b^2)*c
^2*d^m*m*log(c) + (b^3*c^2*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + (b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x -
1)) + a*b^2)*c^2*log(c))*d^m)*x^2 + (b^3*arctan2(1, sqrt(c*x + 1)*sqrt(c*x - 1)) + (b^3*arctan2(1, sqrt(c*x +
1)*sqrt(c*x - 1)) + a*b^2)*log(c))*d^m)*x^m)*log(c^2*x^2))/((c^2*m + c^2)*x^2 - m - 1), x))/(m + 1)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (d x\right )^{m} \left (a + b \operatorname {acsc}{\left (c x \right )}\right )^{3}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int {\left (d\,x\right )}^m\,{\left (a+b\,\mathrm {asin}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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