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

Optimal. Leaf size=86 \[ \frac{b \sqrt{\frac{1}{c x+1}} \sqrt{c x+1} \text{Unintegrable}\left (\frac{(d+e x)^{m+1}}{x \sqrt{1-c^2 x^2}},x\right )}{e (m+1)}+\frac{(d+e x)^{m+1} \left (a+b \text{sech}^{-1}(c x)\right )}{e (m+1)} \]

[Out]

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

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Rubi [A]  time = 0.0481412, 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 = {} \[ \int (d+e x)^m \left (a+b \text{sech}^{-1}(c x)\right ) \, dx \]

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*}

Mathematica [A]  time = 1.83619, size = 0, normalized size = 0. \[ \int (d+e x)^m \left (a+b \text{sech}^{-1}(c x)\right ) \, dx \]

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 = 1.454, size = 0, normalized size = 0. \begin{align*} \int \left ( ex+d \right ) ^{m} \left ( a+b{\rm arcsech} \left (cx\right ) \right ) \, dx \end{align*}

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

[Out]

Exception raised: ValueError

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

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)*(e*x + d)^m, x)

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

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., size = 0, normalized size = 0. \begin{align*} \int{\left (b \operatorname{arsech}\left (c x\right ) + a\right )}{\left (e x + d\right )}^{m}\,{d x} \end{align*}

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)