\(\int \frac {1}{(1+c^2 x^2)^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx\) [446]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {1}{b c \left (1+c^2 x^2\right ) (a+b \text {arcsinh}(c x))}-\frac {2 c \text {Int}\left (\frac {x}{\left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))},x\right )}{b} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 24, 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 = {} \[ \int \frac {1}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

Int[1/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2),x]

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.19 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

Integrate[1/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2),x]

[Out]

Integrate[1/((1 + c^2*x^2)^(3/2)*(a + b*ArcSinh[c*x])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.06 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

\[\int \frac {1}{\left (c^{2} x^{2}+1\right )^{\frac {3}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}d x\]

[In]

int(1/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x))^2,x)

[Out]

int(1/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x))^2,x)

Fricas [N/A]

Not integrable

Time = 0.27 (sec) , antiderivative size = 102, normalized size of antiderivative = 4.25 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral(sqrt(c^2*x^2 + 1)/(a^2*c^4*x^4 + 2*a^2*c^2*x^2 + (b^2*c^4*x^4 + 2*b^2*c^2*x^2 + b^2)*arcsinh(c*x)^2 +
a^2 + 2*(a*b*c^4*x^4 + 2*a*b*c^2*x^2 + a*b)*arcsinh(c*x)), x)

Sympy [N/A]

Not integrable

Time = 2.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \left (c^{2} x^{2} + 1\right )^{\frac {3}{2}}}\, dx \]

[In]

integrate(1/(c**2*x**2+1)**(3/2)/(a+b*asinh(c*x))**2,x)

[Out]

Integral(1/((a + b*asinh(c*x))**2*(c**2*x**2 + 1)**(3/2)), x)

Maxima [N/A]

Not integrable

Time = 0.45 (sec) , antiderivative size = 446, normalized size of antiderivative = 18.58 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

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

Giac [N/A]

Not integrable

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + 1\right )}^{\frac {3}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(c^2*x^2+1)^(3/2)/(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

integrate(1/((c^2*x^2 + 1)^(3/2)*(b*arcsinh(c*x) + a)^2), x)

Mupad [N/A]

Not integrable

Time = 2.54 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{3/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (c^2\,x^2+1\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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