3.70 \(\int (d x)^m (a+b \text {sech}^{-1}(c x))^2 \, dx\)

Optimal. Leaf size=19 \[ \text {Int}\left ((d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^2,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.03, 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 = {} \[ \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Int[(d*x)^m*(a + b*ArcSech[c*x])^2,x]

[Out]

Defer[Int][(d*x)^m*(a + b*ArcSech[c*x])^2, x]

Rubi steps

\begin {align*} \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx &=\int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx\\ \end {align*}

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Mathematica [A]  time = 3.61, size = 0, normalized size = 0.00 \[ \int (d x)^m \left (a+b \text {sech}^{-1}(c x)\right )^2 \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(d*x)^m*(a + b*ArcSech[c*x])^2,x]

[Out]

Integrate[(d*x)^m*(a + b*ArcSech[c*x])^2, x]

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fricas [A]  time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b^{2} \operatorname {arsech}\left (c x\right )^{2} + 2 \, a b \operatorname {arsech}\left (c x\right ) + a^{2}\right )} \left (d x\right )^{m}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*arcsech(c*x))^2,x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*arcsech(c*x))^2,x, algorithm="giac")

[Out]

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

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maple [A]  time = 2.68, size = 0, normalized size = 0.00 \[ \int \left (d x \right )^{m} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {b^{2} d^{m} x x^{m} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )^{2}}{m + 1} + \frac {\left (d x\right )^{m + 1} a^{2}}{d {\left (m + 1\right )}} - \int -\frac {{\left (b^{2} c^{2} d^{m} {\left (m + 1\right )} x^{2} - b^{2} d^{m} {\left (m + 1\right )}\right )} x^{m} \log \relax (x)^{2} - 2 \, {\left (b^{2} d^{m} {\left (m + 1\right )} \log \relax (c) - a b d^{m} {\left (m + 1\right )} - {\left (b^{2} c^{2} d^{m} {\left (m + 1\right )} \log \relax (c) - a b c^{2} d^{m} {\left (m + 1\right )}\right )} x^{2}\right )} x^{m} \log \relax (x) + {\left ({\left (b^{2} c^{2} d^{m} {\left (m + 1\right )} x^{2} - b^{2} d^{m} {\left (m + 1\right )}\right )} x^{m} \log \relax (x)^{2} - 2 \, {\left (b^{2} d^{m} {\left (m + 1\right )} \log \relax (c) - a b d^{m} {\left (m + 1\right )} - {\left (b^{2} c^{2} d^{m} {\left (m + 1\right )} \log \relax (c) - a b c^{2} d^{m} {\left (m + 1\right )}\right )} x^{2}\right )} x^{m} \log \relax (x) - {\left (b^{2} d^{m} {\left (m + 1\right )} \log \relax (c)^{2} - 2 \, a b d^{m} {\left (m + 1\right )} \log \relax (c) - {\left (b^{2} c^{2} d^{m} {\left (m + 1\right )} \log \relax (c)^{2} - 2 \, a b c^{2} d^{m} {\left (m + 1\right )} \log \relax (c)\right )} x^{2}\right )} x^{m}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - {\left (b^{2} d^{m} {\left (m + 1\right )} \log \relax (c)^{2} - 2 \, a b d^{m} {\left (m + 1\right )} \log \relax (c) - {\left (b^{2} c^{2} d^{m} {\left (m + 1\right )} \log \relax (c)^{2} - 2 \, a b c^{2} d^{m} {\left (m + 1\right )} \log \relax (c)\right )} x^{2}\right )} x^{m} - 2 \, {\left ({\left (b^{2} c^{2} d^{m} {\left (m + 1\right )} x^{2} - b^{2} d^{m} {\left (m + 1\right )}\right )} x^{m} \log \relax (x) + {\left ({\left (b^{2} c^{2} d^{m} {\left (m + 1\right )} x^{2} - b^{2} d^{m} {\left (m + 1\right )}\right )} x^{m} \log \relax (x) - {\left (b^{2} d^{m} {\left (m + 1\right )} \log \relax (c) - a b d^{m} {\left (m + 1\right )} + {\left (a b c^{2} d^{m} {\left (m + 1\right )} - {\left (d^{m} {\left (m + 1\right )} \log \relax (c) + d^{m}\right )} b^{2} c^{2}\right )} x^{2}\right )} x^{m}\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - {\left (b^{2} d^{m} {\left (m + 1\right )} \log \relax (c) - a b d^{m} {\left (m + 1\right )} - {\left (b^{2} c^{2} d^{m} {\left (m + 1\right )} \log \relax (c) - a b c^{2} d^{m} {\left (m + 1\right )}\right )} x^{2}\right )} x^{m}\right )} \log \left (\sqrt {c x + 1} \sqrt {-c x + 1} + 1\right )}{c^{2} {\left (m + 1\right )} x^{2} + {\left (c^{2} {\left (m + 1\right )} x^{2} - m - 1\right )} \sqrt {c x + 1} \sqrt {-c x + 1} - m - 1}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)^m*(a+b*arcsech(c*x))^2,x, algorithm="maxima")

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.05 \[ \int {\left (d\,x\right )}^m\,{\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )}^2 \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x)^m*(a + b*acosh(1/(c*x)))^2,x)

[Out]

int((d*x)^m*(a + b*acosh(1/(c*x)))^2, x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (d x\right )^{m} \left (a + b \operatorname {asech}{\left (c x \right )}\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x)**m*(a+b*asech(c*x))**2,x)

[Out]

Integral((d*x)**m*(a + b*asech(c*x))**2, x)

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