\(\int \frac {1}{(c+a^2 c x^2)^{3/2} \text {arcsinh}(a x)^{3/2}} \, dx\) [505]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2}} \, dx=-\frac {2 \sqrt {1+a^2 x^2}}{a \left (c+a^2 c x^2\right )^{3/2} \sqrt {\text {arcsinh}(a x)}}-\frac {4 a \sqrt {1+a^2 x^2} \text {Int}\left (\frac {x}{\left (1+a^2 x^2\right )^2 \sqrt {\text {arcsinh}(a x)}},x\right )}{c \sqrt {c+a^2 c x^2}} \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 23, 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 = {} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 1.91 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2}} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.26 (sec) , antiderivative size = 19, normalized size of antiderivative = 0.83

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

[In]

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

[Out]

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

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

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

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [N/A]

Not integrable

Time = 25.93 (sec) , antiderivative size = 24, normalized size of antiderivative = 1.04 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 0.41 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2}} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (a x\right )^{\frac {3}{2}}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.69 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.91 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \text {arcsinh}(a x)^{3/2}} \, dx=\int \frac {1}{{\mathrm {asinh}\left (a\,x\right )}^{3/2}\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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