3.1.87 \(\int (d+e x)^m (a+b \text {sech}^{-1}(c x)) \, dx\) [87]

Optimal. Leaf size=87 \[ \frac {(d+e x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{e (1+m)}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {Int}\left (\frac {(d+e x)^{1+m}}{x \sqrt {1-c^2 x^2}},x\right )}{e (1+m)} \]

[Out]

(e*x+d)^(1+m)*(a+b*arcsech(c*x))/e/(1+m)+b*(1/(c*x+1))^(1/2)*(c*x+1)^(1/2)*Unintegrable((e*x+d)^(1+m)/x/(-c^2*
x^2+1)^(1/2),x)/e/(1+m)

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

Verification is not applicable to the result.

[In]

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

[Out]

((d + e*x)^(1 + m)*(a + b*ArcSech[c*x]))/(e*(1 + m)) + (b*Sqrt[(1 + c*x)^(-1)]*Sqrt[1 + c*x]*Defer[Int][(d + e
*x)^(1 + m)/(x*Sqrt[1 - c^2*x^2]), x])/(e*(1 + m))

Rubi steps

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

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Mathematica [A]
time = 9.06, size = 0, normalized size = 0.00 \begin {gather*} \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.29, size = 0, normalized size = 0.00 \[\int \left (e x +d \right )^{m} \left (a +b \,\mathrm {arcsech}\left (c x \right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

[Out]

b*(((x*e + d)*(x*e + d)^m*log(sqrt(c*x + 1)*sqrt(-c*x + 1) + 1) - (x*e + d)*(x*e + d)^m*log(x))*e^(-1)/(m + 1)
- integrate((c^2*(m + 1)*x^3*e*log(c) - ((m + 1)*log(c) - 1)*x*e + d)*(x*e + d)^m/(c^2*(m + 1)*x^3*e - (m + 1
)*x*e), x) + integrate((c^2*x^2*e + c^2*d*x)*(x*e + d)^m/(c^2*(m + 1)*x^2*e + (c^2*(m + 1)*x^2*e - (m + 1)*e)*
sqrt(c*x + 1)*sqrt(-c*x + 1) - (m + 1)*e), x)) + (x*e + d)^(m + 1)*a*e^(-1)/(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((e*x+d)^m*(a+b*arcsech(c*x)),x, algorithm="fricas")

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*asech(c*x))*(d + e*x)**m, 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((e*x+d)^m*(a+b*arcsech(c*x)),x, algorithm="giac")

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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