\(\int \frac {1}{(1+a^2+2 a b x+b^2 x^2)^{3/2} \text {arcsinh}(a+b x)^2} \, dx\) [282]

Optimal result

Integrand size = 30, antiderivative size = 30 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2} \, dx=-\frac {1}{b \left (1+(a+b x)^2\right ) \text {arcsinh}(a+b x)}-2 \text {Int}\left (\frac {a+b x}{\left (1+(a+b x)^2\right )^2 \text {arcsinh}(a+b x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.09 (sec) , antiderivative size = 30, 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 = {} \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2} \, dx=\int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.21 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2} \, dx=\int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.32 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.93

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 82, normalized size of antiderivative = 2.73 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2} \, dx=\int { \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Sympy [N/A]

Not integrable

Time = 1.77 (sec) , antiderivative size = 32, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2} \, dx=\int \frac {1}{\left (a^{2} + 2 a b x + b^{2} x^{2} + 1\right )^{\frac {3}{2}} \operatorname {asinh}^{2}{\left (a + b x \right )}}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 2.54 (sec) , antiderivative size = 610, normalized size of antiderivative = 20.33 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2} \, dx=\int { \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]

[In]

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

[Out]

-(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))/(((b^2*x^2 + 2*a*b*x + a^2 + 1)*(b^2*x + a*b) + (b^3*x^2 + 2*a*
b^2*x + a^2*b + b)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*log(b*x + a + sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))) - inte
grate((2*b^4*x^4 + 8*a*b^3*x^3 + 2*a^4 + (12*a^2*b^2 + b^2)*x^2 + (2*b^2*x^2 + 4*a*b*x + 2*a^2 + 1)*(b^2*x^2 +
2*a*b*x + a^2 + 1) + a^2 + 2*(4*a^3*b + a*b)*x + 2*(2*b^3*x^3 + 6*a*b^2*x^2 + 2*a^3 + (6*a^2*b + b)*x + a)*sq
rt(b^2*x^2 + 2*a*b*x + a^2 + 1) - 1)/(((b^4*x^4 + 4*a*b^3*x^3 + a^4 + (6*a^2*b^2 + b^2)*x^2 + a^2 + 2*(2*a^3*b
+ a*b)*x)*(b^2*x^2 + 2*a*b*x + a^2 + 1)^(3/2) + 2*(b^5*x^5 + 5*a*b^4*x^4 + a^5 + 2*(5*a^2*b^3 + b^3)*x^3 + 2*
a^3 + 2*(5*a^3*b^2 + 3*a*b^2)*x^2 + (5*a^4*b + 6*a^2*b + b)*x + a)*(b^2*x^2 + 2*a*b*x + a^2 + 1) + (b^6*x^6 +
6*a*b^5*x^5 + a^6 + 3*(5*a^2*b^4 + b^4)*x^4 + 3*a^4 + 4*(5*a^3*b^3 + 3*a*b^3)*x^3 + 3*(5*a^4*b^2 + 6*a^2*b^2 +
b^2)*x^2 + 3*a^2 + 6*(a^5*b + 2*a^3*b + a*b)*x + 1)*sqrt(b^2*x^2 + 2*a*b*x + a^2 + 1))*log(b*x + a + sqrt(b^2
*x^2 + 2*a*b*x + a^2 + 1))), x)

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2} \, dx=\int { \frac {1}{{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac {3}{2}} \operatorname {arsinh}\left (b x + a\right )^{2}} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 2.63 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.00 \[ \int \frac {1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \text {arcsinh}(a+b x)^2} \, dx=\int \frac {1}{{\mathrm {asinh}\left (a+b\,x\right )}^2\,{\left (a^2+2\,a\,b\,x+b^2\,x^2+1\right )}^{3/2}} \,d x \]

[In]

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

[Out]

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