\(\int \frac {\sqrt {d+e x} (a+b \text {csch}^{-1}(c x))}{x^2} \, dx\) [55]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {\sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx=\text {Int}\left (\frac {\sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{x^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 21, 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 = {} \[ \int \frac {\sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx=\int \frac {\sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx \]

[In]

Int[(Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/x^2,x]

[Out]

Defer[Int][(Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/x^2, x]

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

Mathematica [N/A]

Not integrable

Time = 6.81 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {\sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx=\int \frac {\sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx \]

[In]

Integrate[(Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/x^2,x]

[Out]

Integrate[(Sqrt[d + e*x]*(a + b*ArcCsch[c*x]))/x^2, x]

Maple [N/A] (verified)

Not integrable

Time = 0.17 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

\[\int \frac {\left (a +b \,\operatorname {arccsch}\left (c x \right )\right ) \sqrt {e x +d}}{x^{2}}d x\]

[In]

int((a+b*arccsch(c*x))*(e*x+d)^(1/2)/x^2,x)

[Out]

int((a+b*arccsch(c*x))*(e*x+d)^(1/2)/x^2,x)

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]

[In]

integrate((a+b*arccsch(c*x))*(e*x+d)^(1/2)/x^2,x, algorithm="fricas")

[Out]

integral(sqrt(e*x + d)*(b*arccsch(c*x) + a)/x^2, x)

Sympy [N/A]

Not integrable

Time = 14.32 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {\sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx=\int \frac {\left (a + b \operatorname {acsch}{\left (c x \right )}\right ) \sqrt {d + e x}}{x^{2}}\, dx \]

[In]

integrate((a+b*acsch(c*x))*(e*x+d)**(1/2)/x**2,x)

[Out]

Integral((a + b*acsch(c*x))*sqrt(d + e*x)/x**2, x)

Maxima [N/A]

Not integrable

Time = 2.34 (sec) , antiderivative size = 149, normalized size of antiderivative = 7.10 \[ \int \frac {\sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]

[In]

integrate((a+b*arccsch(c*x))*(e*x+d)^(1/2)/x^2,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {\sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx=\int { \frac {\sqrt {e x + d} {\left (b \operatorname {arcsch}\left (c x\right ) + a\right )}}{x^{2}} \,d x } \]

[In]

integrate((a+b*arccsch(c*x))*(e*x+d)^(1/2)/x^2,x, algorithm="giac")

[Out]

integrate(sqrt(e*x + d)*(b*arccsch(c*x) + a)/x^2, x)

Mupad [N/A]

Not integrable

Time = 4.84 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {\sqrt {d+e x} \left (a+b \text {csch}^{-1}(c x)\right )}{x^2} \, dx=\int \frac {\left (a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )\right )\,\sqrt {d+e\,x}}{x^2} \,d x \]

[In]

int(((a + b*asinh(1/(c*x)))*(d + e*x)^(1/2))/x^2,x)

[Out]

int(((a + b*asinh(1/(c*x)))*(d + e*x)^(1/2))/x^2, x)