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

Optimal result

Integrand size = 16, antiderivative size = 16 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\frac {(d+e x)^{1+m} \left (a+b \text {sech}^{-1}(c x)\right )}{e (1+m)}+\frac {b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {Int}\left (\frac {(d+e x)^{1+m}}{x \sqrt {1-c^2 x^2}},x\right )}{e (1+m)} \]

[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)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \]

[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*} \text {integral}& = \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 [N/A]

Not integrable

Time = 21.68 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.40 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00

\[\int \left (e x +d \right )^{m} \left (a +b \,\operatorname {arcsech}\left (c x \right )\right )d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} {\left (e x + d\right )}^{m} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 13.56 (sec) , antiderivative size = 15, normalized size of antiderivative = 0.94 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int \left (a + b \operatorname {asech}{\left (c x \right )}\right ) \left (d + e x\right )^{m}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.84 (sec) , antiderivative size = 222, normalized size of antiderivative = 13.88 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} {\left (e x + d\right )}^{m} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.12 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int { {\left (b \operatorname {arsech}\left (c x\right ) + a\right )} {\left (e x + d\right )}^{m} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 4.34 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.38 \[ \int (d+e x)^m \left (a+b \text {sech}^{-1}(c x)\right ) \, dx=\int \left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right )\,{\left (d+e\,x\right )}^m \,d x \]

[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)