3.282 \(\int \frac{1}{(1+a^2+2 a b x+b^2 x^2)^{3/2} \sinh ^{-1}(a+b x)^2} \, dx\)

Optimal. Leaf size=54 \[ -2 \text{Unintegrable}\left (\frac{a+b x}{\left ((a+b x)^2+1\right )^2 \sinh ^{-1}(a+b x)},x\right )-\frac{1}{b \left ((a+b x)^2+1\right ) \sinh ^{-1}(a+b x)} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.116512, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \sinh ^{-1}(a+b x)^2} \, dx \]

Verification is Not applicable to the result.

[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*} \int \frac{1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \sinh ^{-1}(a+b x)^2} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{\left (1+x^2\right )^{3/2} \sinh ^{-1}(x)^2} \, dx,x,a+b x\right )}{b}\\ &=-\frac{1}{b \left (1+(a+b x)^2\right ) \sinh ^{-1}(a+b x)}-\frac{2 \operatorname{Subst}\left (\int \frac{x}{\left (1+x^2\right )^2 \sinh ^{-1}(x)} \, dx,x,a+b x\right )}{b}\\ \end{align*}

Mathematica [A]  time = 2.69618, size = 0, normalized size = 0. \[ \int \frac{1}{\left (1+a^2+2 a b x+b^2 x^2\right )^{3/2} \sinh ^{-1}(a+b x)^2} \, dx \]

Verification is Not applicable to the result.

[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 [A]  time = 0.097, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ({\it Arcsinh} \left ( bx+a \right ) \right ) ^{2}} \left ({b}^{2}{x}^{2}+2\,xab+{a}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} -\frac{b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left ({\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}{\left (b^{2} x + a b\right )} +{\left (b^{3} x^{2} + 2 \, a b^{2} x + a^{2} b + b\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} - \int \frac{2 \, b^{4} x^{4} + 8 \, a b^{3} x^{3} + 2 \, a^{4} +{\left (12 \, a^{2} b^{2} + b^{2}\right )} x^{2} +{\left (2 \, b^{2} x^{2} + 4 \, a b x + 2 \, a^{2} + 1\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )} + a^{2} + 2 \,{\left (4 \, a^{3} b + a b\right )} x + 2 \,{\left (2 \, b^{3} x^{3} + 6 \, a b^{2} x^{2} + 2 \, a^{3} +{\left (6 \, a^{2} b + b\right )} x + a\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1} - 1}{{\left ({\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + a^{4} +{\left (6 \, a^{2} b^{2} + b^{2}\right )} x^{2} + a^{2} + 2 \,{\left (2 \, a^{3} b + a b\right )} x\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )}^{\frac{3}{2}} + 2 \,{\left (b^{5} x^{5} + 5 \, a b^{4} x^{4} + a^{5} + 2 \,{\left (5 \, a^{2} b^{3} + b^{3}\right )} x^{3} + 2 \, a^{3} + 2 \,{\left (5 \, a^{3} b^{2} + 3 \, a b^{2}\right )} x^{2} +{\left (5 \, a^{4} b + 6 \, a^{2} b + b\right )} x + a\right )}{\left (b^{2} x^{2} + 2 \, a b x + a^{2} + 1\right )} +{\left (b^{6} x^{6} + 6 \, a b^{5} x^{5} + a^{6} + 3 \,{\left (5 \, a^{2} b^{4} + b^{4}\right )} x^{4} + 3 \, a^{4} + 4 \,{\left (5 \, a^{3} b^{3} + 3 \, a b^{3}\right )} x^{3} + 3 \,{\left (5 \, a^{4} b^{2} + 6 \, a^{2} b^{2} + b^{2}\right )} x^{2} + 3 \, a^{2} + 6 \,{\left (a^{5} b + 2 \, a^{3} b + a b\right )} x + 1\right )} \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )} \log \left (b x + a + \sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}\right )}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{b^{2} x^{2} + 2 \, a b x + a^{2} + 1}}{{\left (b^{4} x^{4} + 4 \, a b^{3} x^{3} + 2 \,{\left (3 \, a^{2} + 1\right )} b^{2} x^{2} + a^{4} + 4 \,{\left (a^{3} + a\right )} b x + 2 \, a^{2} + 1\right )} \operatorname{arsinh}\left (b x + a\right )^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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 \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \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} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)