\(\int \frac {1}{(1+c^2 x^2)^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx\) [452]

Optimal result

Integrand size = 24, antiderivative size = 24 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=-\frac {1}{b c \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}-\frac {4 c \text {Int}\left (\frac {x}{\left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))},x\right )}{b} \]

[Out]

-1/b/c/(c^2*x^2+1)^2/(a+b*arcsinh(c*x))-4*c*Unintegrable(x/(c^2*x^2+1)^3/(a+b*arcsinh(c*x)),x)/b

Rubi [N/A]

Not integrable

Time = 0.09 (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 = {} \[ \int \frac {1}{\left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

Int[1/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2),x]

[Out]

-(1/(b*c*(1 + c^2*x^2)^2*(a + b*ArcSinh[c*x]))) - (4*c*Defer[Int][x/((1 + c^2*x^2)^3*(a + b*ArcSinh[c*x])), x]
)/b

Rubi steps \begin{align*} \text {integral}& = -\frac {1}{b c \left (1+c^2 x^2\right )^2 (a+b \text {arcsinh}(c x))}-\frac {(4 c) \int \frac {x}{\left (1+c^2 x^2\right )^3 (a+b \text {arcsinh}(c x))} \, dx}{b} \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 4.41 (sec) , antiderivative size = 26, normalized size of antiderivative = 1.08 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx \]

[In]

Integrate[1/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2),x]

[Out]

Integrate[1/((1 + c^2*x^2)^(5/2)*(a + b*ArcSinh[c*x])^2), x]

Maple [N/A] (verified)

Not integrable

Time = 0.18 (sec) , antiderivative size = 22, normalized size of antiderivative = 0.92

\[\int \frac {1}{\left (c^{2} x^{2}+1\right )^{\frac {5}{2}} \left (a +b \,\operatorname {arcsinh}\left (c x \right )\right )^{2}}d x\]

[In]

int(1/(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x)

[Out]

int(1/(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x)

Fricas [N/A]

Not integrable

Time = 0.25 (sec) , antiderivative size = 134, normalized size of antiderivative = 5.58 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral(sqrt(c^2*x^2 + 1)/(a^2*c^6*x^6 + 3*a^2*c^4*x^4 + 3*a^2*c^2*x^2 + (b^2*c^6*x^6 + 3*b^2*c^4*x^4 + 3*b^2
*c^2*x^2 + b^2)*arcsinh(c*x)^2 + a^2 + 2*(a*b*c^6*x^6 + 3*a*b*c^4*x^4 + 3*a*b*c^2*x^2 + a*b)*arcsinh(c*x)), x)

Sympy [N/A]

Not integrable

Time = 3.83 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \left (c^{2} x^{2} + 1\right )^{\frac {5}{2}}}\, dx \]

[In]

integrate(1/(c**2*x**2+1)**(5/2)/(a+b*asinh(c*x))**2,x)

[Out]

Integral(1/((a + b*asinh(c*x))**2*(c**2*x**2 + 1)**(5/2)), x)

Maxima [N/A]

Not integrable

Time = 0.50 (sec) , antiderivative size = 554, normalized size of antiderivative = 23.08 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="maxima")

[Out]

-(c*x + sqrt(c^2*x^2 + 1))/((a*b*c^4*x^3 + a*b*c^2*x)*(c^2*x^2 + 1) + ((b^2*c^4*x^3 + b^2*c^2*x)*(c^2*x^2 + 1)
+ (b^2*c^5*x^4 + 2*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^5*x^4 + 2*a*
b*c^3*x^2 + a*b*c)*sqrt(c^2*x^2 + 1)) - integrate((4*c^4*x^4 + 3*c^2*x^2 + (4*c^2*x^2 + 1)*(c^2*x^2 + 1) + 4*(
2*c^3*x^3 + c*x)*sqrt(c^2*x^2 + 1) - 1)/((a*b*c^6*x^6 + 2*a*b*c^4*x^4 + a*b*c^2*x^2)*(c^2*x^2 + 1)^(3/2) + 2*(
a*b*c^7*x^7 + 3*a*b*c^5*x^5 + 3*a*b*c^3*x^3 + a*b*c*x)*(c^2*x^2 + 1) + ((b^2*c^6*x^6 + 2*b^2*c^4*x^4 + b^2*c^2
*x^2)*(c^2*x^2 + 1)^(3/2) + 2*(b^2*c^7*x^7 + 3*b^2*c^5*x^5 + 3*b^2*c^3*x^3 + b^2*c*x)*(c^2*x^2 + 1) + (b^2*c^8
*x^8 + 4*b^2*c^6*x^6 + 6*b^2*c^4*x^4 + 4*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^8*x^8 + 4*a*b*c^6*x^6 + 6*a*b*c^4*x^4 + 4*a*b*c^2*x^2 + a*b)*sqrt(c^2*x^2 + 1)), x)

Giac [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int { \frac {1}{{\left (c^{2} x^{2} + 1\right )}^{\frac {5}{2}} {\left (b \operatorname {arsinh}\left (c x\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(1/(c^2*x^2+1)^(5/2)/(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

integrate(1/((c^2*x^2 + 1)^(5/2)*(b*arcsinh(c*x) + a)^2), x)

Mupad [N/A]

Not integrable

Time = 2.58 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+c^2 x^2\right )^{5/2} (a+b \text {arcsinh}(c x))^2} \, dx=\int \frac {1}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (c^2\,x^2+1\right )}^{5/2}} \,d x \]

[In]

int(1/((a + b*asinh(c*x))^2*(c^2*x^2 + 1)^(5/2)),x)

[Out]

int(1/((a + b*asinh(c*x))^2*(c^2*x^2 + 1)^(5/2)), x)