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

Optimal. Leaf size=61 \[ -\frac{2 c \text{Unintegrable}\left (\frac{x}{\left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )},x\right )}{b}-\frac{1}{b c \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.110704, 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 = {} \[ \int \frac{1}{\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

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

Mathematica [A]  time = 2.33559, size = 0, normalized size = 0. \[ \int \frac{1}{\left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[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]

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Maple [A]  time = 0.102, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}} \left ({c}^{2}{x}^{2}+1 \right ) ^{-{\frac{3}{2}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)

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

Verification of antiderivative is not currently implemented for this CAS.

[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)