3.69 \(\int (d x)^m (a+b \text {sech}^{-1}(c x))^3 \, dx\)

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

[Out]

Unintegrable((d*x)^m*(a+b*arcsech(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 = {} \[ \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d*x)^m*(a + b*ArcSech[c*x])^3,x]

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 5.61, size = 0, normalized size = 0.00 \[ \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^3 \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.70, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{3} \operatorname {arsech}\left (c x\right )^{3} + 3 \, a b^{2} \operatorname {arsech}\left (c x\right )^{2} + 3 \, a^{2} b \operatorname {arsech}\left (c x\right ) + a^{3}\right )} \left (d x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \operatorname {arsech}\left (c x\right ) + a\right )}^{3} \left (d x\right )^{m}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 2.89, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )^{3}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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*arcsech(c*x))^3,x, algorithm="maxima")

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int {\left (d\,x\right )}^m\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^3 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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