\(\int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx\) [68]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx=\text {Int}\left (\frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}},x\right ) \]

[Out]

Unintegrable((a+b*arccsch(c*x))/x^2/(e*x+d)^(3/2),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 = {} \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx=\int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx \]

[In]

Int[(a + b*ArcCsch[c*x])/(x^2*(d + e*x)^(3/2)),x]

[Out]

Defer[Int][(a + b*ArcCsch[c*x])/(x^2*(d + e*x)^(3/2)), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 19.34 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx=\int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx \]

[In]

Integrate[(a + b*ArcCsch[c*x])/(x^2*(d + e*x)^(3/2)),x]

[Out]

Integrate[(a + b*ArcCsch[c*x])/(x^2*(d + e*x)^(3/2)), x]

Maple [N/A] (verified)

Not integrable

Time = 0.16 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.90

\[\int \frac {a +b \,\operatorname {arccsch}\left (c x \right )}{x^{2} \left (e x +d \right )^{\frac {3}{2}}}d x\]

[In]

int((a+b*arccsch(c*x))/x^2/(e*x+d)^(3/2),x)

[Out]

int((a+b*arccsch(c*x))/x^2/(e*x+d)^(3/2),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 42, normalized size of antiderivative = 2.00 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

[In]

integrate((a+b*arccsch(c*x))/x^2/(e*x+d)^(3/2),x, algorithm="fricas")

[Out]

integral(sqrt(e*x + d)*(b*arccsch(c*x) + a)/(e^2*x^4 + 2*d*e*x^3 + d^2*x^2), x)

Sympy [N/A]

Not integrable

Time = 59.65 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx=\int \frac {a + b \operatorname {acsch}{\left (c x \right )}}{x^{2} \left (d + e x\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate((a+b*acsch(c*x))/x**2/(e*x+d)**(3/2),x)

[Out]

Integral((a + b*acsch(c*x))/(x**2*(d + e*x)**(3/2)), x)

Maxima [N/A]

Not integrable

Time = 2.67 (sec) , antiderivative size = 252, normalized size of antiderivative = 12.00 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

[In]

integrate((a+b*arccsch(c*x))/x^2/(e*x+d)^(3/2),x, algorithm="maxima")

[Out]

1/2*b*((3*e^2*log((sqrt(e*x + d) - sqrt(d))/(sqrt(e*x + d) + sqrt(d)))/d^(5/2) + 2*(3*(e*x + d)*e^2 - 2*d*e^2)
/((e*x + d)^(3/2)*d^2 - sqrt(e*x + d)*d^3))*log(c)/e - 2*integrate(log(x)/(sqrt(e*x + d)*e*x^3 + sqrt(e*x + d)
*d*x^2), x) + 2*integrate(log(sqrt(c^2*x^2 + 1) + 1)/(sqrt(e*x + d)*e*x^3 + sqrt(e*x + d)*d*x^2), x)) - 1/2*a*
(2*(3*(e*x + d)*e - 2*d*e)/((e*x + d)^(3/2)*d^2 - sqrt(e*x + d)*d^3) + 3*e*log((sqrt(e*x + d) - sqrt(d))/(sqrt
(e*x + d) + sqrt(d)))/d^(5/2))

Giac [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx=\int { \frac {b \operatorname {arcsch}\left (c x\right ) + a}{{\left (e x + d\right )}^{\frac {3}{2}} x^{2}} \,d x } \]

[In]

integrate((a+b*arccsch(c*x))/x^2/(e*x+d)^(3/2),x, algorithm="giac")

[Out]

integrate((b*arccsch(c*x) + a)/((e*x + d)^(3/2)*x^2), x)

Mupad [N/A]

Not integrable

Time = 4.99 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.19 \[ \int \frac {a+b \text {csch}^{-1}(c x)}{x^2 (d+e x)^{3/2}} \, dx=\int \frac {a+b\,\mathrm {asinh}\left (\frac {1}{c\,x}\right )}{x^2\,{\left (d+e\,x\right )}^{3/2}} \,d x \]

[In]

int((a + b*asinh(1/(c*x)))/(x^2*(d + e*x)^(3/2)),x)

[Out]

int((a + b*asinh(1/(c*x)))/(x^2*(d + e*x)^(3/2)), x)