3.458 \(\int \frac{x^m}{\sqrt{1+c^2 x^2} (a+b \sinh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=48 \[ \frac{m \text{Unintegrable}\left (\frac{x^{m-1}}{a+b \sinh ^{-1}(c x)},x\right )}{b c}-\frac{x^m}{b c \left (a+b \sinh ^{-1}(c x)\right )} \]

[Out]

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

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Rubi [A]  time = 0.152405, 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 = {} \[ \int \frac{x^m}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 0.342317, size = 0, normalized size = 0. \[ \int \frac{x^m}{\sqrt{1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.177, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{m}}{ \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}{\frac{1}{\sqrt{{c}^{2}{x}^{2}+1}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((c^2*x^2 + 1)^(3/2)*x^m + (c^3*x^3 + c*x)*x^m)/((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)) + in
tegrate(((c^3*m*x^3 + c*(m - 1)*x)*(c^2*x^2 + 1)*x^m + (2*c^4*m*x^4 + 3*c^2*m*x^2 + m)*sqrt(c^2*x^2 + 1)*x^m +
(c^5*m*x^5 + c^3*(2*m + 1)*x^3 + c*(m + 1)*x)*x^m)/((c^2*x^2 + 1)^(3/2)*a*b*c^3*x^3 + 2*(a*b*c^4*x^4 + a*b*c^
2*x^2)*(c^2*x^2 + 1) + ((c^2*x^2 + 1)^(3/2)*b^2*c^3*x^3 + 2*(b^2*c^4*x^4 + b^2*c^2*x^2)*(c^2*x^2 + 1) + (b^2*c
^5*x^5 + 2*b^2*c^3*x^3 + b^2*c*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1)) + (a*b*c^5*x^5 + 2*a*b*c^3*x
^3 + a*b*c*x)*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} x^{m}}{a^{2} c^{2} x^{2} +{\left (b^{2} c^{2} x^{2} + b^{2}\right )} \operatorname{arsinh}\left (c x\right )^{2} + a^{2} + 2 \,{\left (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(x^m/(c^2*x^2+1)^(1/2)/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

integral(sqrt(c^2*x^2 + 1)*x^m/(a^2*c^2*x^2 + (b^2*c^2*x^2 + b^2)*arcsinh(c*x)^2 + a^2 + 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{x^{m}}{\left (a + b \operatorname{asinh}{\left (c x \right )}\right )^{2} \sqrt{c^{2} x^{2} + 1}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{m}}{\sqrt{c^{2} x^{2} + 1}{\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(x^m/(c^2*x^2+1)^(1/2)/(a+b*arcsinh(c*x))^2,x, algorithm="giac")

[Out]

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