\(\int \frac {1}{\sqrt {d+e x^2} (a+b \text {arcsinh}(c x))^2} \, dx\) [661]

Optimal result

Integrand size = 22, antiderivative size = 22 \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\text {Int}\left (\frac {1}{\sqrt {d+e x^2} (a+b \text {arcsinh}(c x))^2},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 22, 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 = {} \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\sqrt {d+e x^2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

Int[1/(Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 6.73 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\sqrt {d+e x^2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

Integrate[1/(Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2),x]

[Out]

Integrate[1/(Sqrt[d + e*x^2]*(a + b*ArcSinh[c*x])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.66 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.91

\[\int \frac {1}{\left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2} \sqrt {e \,x^{2}+d}}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 67, normalized size of antiderivative = 3.05 \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{\sqrt {e x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.23 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \sqrt {d + e x^{2}}}\, dx \]

[In]

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

[Out]

Integral(1/((a + b*asinh(c*x))**2*sqrt(d + e*x**2)), x)

Maxima [N/A]

Not integrable

Time = 0.78 (sec) , antiderivative size = 591, normalized size of antiderivative = 26.86 \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{\sqrt {e x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

-(c^3*x^3 + c*x + (c^2*x^2 + 1)^(3/2))/(sqrt(c^2*x^2 + 1)*sqrt(e*x^2 + d)*a*b*c^2*x + (sqrt(c^2*x^2 + 1)*sqrt(
e*x^2 + d)*b^2*c^2*x + (b^2*c^3*x^2 + b^2*c)*sqrt(e*x^2 + d))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^3*x^2 + a*
b*c)*sqrt(e*x^2 + d)) + integrate((c^5*d*x^4 + 2*c^3*d*x^2 + (c^2*x^2 + 1)*((c^3*d - 2*c*e)*x^2 - c*d) + c*d +
sqrt(c^2*x^2 + 1)*(2*(c^4*d - c^2*e)*x^3 + (c^2*d - e)*x))/((a*b*c^3*e*x^4 + a*b*c^3*d*x^2)*(c^2*x^2 + 1)*sqr
t(e*x^2 + d) + 2*(a*b*c^4*e*x^5 + a*b*c^2*d*x + (c^4*d + c^2*e)*a*b*x^3)*sqrt(c^2*x^2 + 1)*sqrt(e*x^2 + d) + (
(b^2*c^3*e*x^4 + b^2*c^3*d*x^2)*(c^2*x^2 + 1)*sqrt(e*x^2 + d) + 2*(b^2*c^4*e*x^5 + b^2*c^2*d*x + (c^4*d + c^2*
e)*b^2*x^3)*sqrt(c^2*x^2 + 1)*sqrt(e*x^2 + d) + (b^2*c^5*e*x^6 + (c^5*d + 2*c^3*e)*b^2*x^4 + (2*c^3*d + c*e)*b
^2*x^2 + b^2*c*d)*sqrt(e*x^2 + d))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^5*e*x^6 + (c^5*d + 2*c^3*e)*a*b*x^4 +
(2*c^3*d + c*e)*a*b*x^2 + a*b*c*d)*sqrt(e*x^2 + d)), x)

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{\sqrt {e x^{2} + d} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.66 (sec) , antiderivative size = 22, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\sqrt {d+e x^2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {e\,x^2+d}} \,d x \]

[In]

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

[Out]

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