3.28 \(\int \frac{1}{(d+e x) (a+b \sinh ^{-1}(c x))^2} \, dx\)

Optimal. Leaf size=20 \[ \text{Unintegrable}\left (\frac{1}{(d+e x) \left (a+b \sinh ^{-1}(c x)\right )^2},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0314303, 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 = {} \[ \int \frac{1}{(d+e x) \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Int][1/((d + e*x)*(a + b*ArcSinh[c*x])^2), x]

Rubi steps

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

Mathematica [A]  time = 3.07591, size = 0, normalized size = 0. \[ \int \frac{1}{(d+e x) \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.171, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( ex+d \right ) \left ( a+b{\it Arcsinh} \left ( cx \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(c^3*x^3 + c*x + (c^2*x^2 + 1)^(3/2))/(a*b*c^3*e*x^3 + a*b*c^3*d*x^2 + a*b*c*e*x + a*b*c*d + (b^2*c^3*e*x^3 +
b^2*c^3*d*x^2 + b^2*c*e*x + b^2*c*d + (b^2*c^2*e*x^2 + b^2*c^2*d*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2
+ 1)) + (a*b*c^2*e*x^2 + a*b*c^2*d*x)*sqrt(c^2*x^2 + 1)) + integrate((c^5*d*x^4 + 2*c^3*d*x^2 + (c^3*d*x^2 -
2*c*e*x - c*d)*(c^2*x^2 + 1) + c*d + (2*c^4*d*x^3 - 2*c^2*e*x^2 + c^2*d*x - e)*sqrt(c^2*x^2 + 1))/(a*b*c^5*e^2
*x^6 + 2*a*b*c^5*d*e*x^5 + 4*a*b*c^3*d*e*x^3 + (c^5*d^2 + 2*c^3*e^2)*a*b*x^4 + 2*a*b*c*d*e*x + a*b*c*d^2 + (2*
c^3*d^2 + c*e^2)*a*b*x^2 + (a*b*c^3*e^2*x^4 + 2*a*b*c^3*d*e*x^3 + a*b*c^3*d^2*x^2)*(c^2*x^2 + 1) + (b^2*c^5*e^
2*x^6 + 2*b^2*c^5*d*e*x^5 + 4*b^2*c^3*d*e*x^3 + (c^5*d^2 + 2*c^3*e^2)*b^2*x^4 + 2*b^2*c*d*e*x + b^2*c*d^2 + (2
*c^3*d^2 + c*e^2)*b^2*x^2 + (b^2*c^3*e^2*x^4 + 2*b^2*c^3*d*e*x^3 + b^2*c^3*d^2*x^2)*(c^2*x^2 + 1) + 2*(b^2*c^4
*e^2*x^5 + 2*b^2*c^4*d*e*x^4 + 2*b^2*c^2*d*e*x^2 + b^2*c^2*d^2*x + (c^4*d^2 + c^2*e^2)*b^2*x^3)*sqrt(c^2*x^2 +
1))*log(c*x + sqrt(c^2*x^2 + 1)) + 2*(a*b*c^4*e^2*x^5 + 2*a*b*c^4*d*e*x^4 + 2*a*b*c^2*d*e*x^2 + a*b*c^2*d^2*x
+ (c^4*d^2 + c^2*e^2)*a*b*x^3)*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{1}{a^{2} e x + a^{2} d +{\left (b^{2} e x + b^{2} d\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b e x + a b d\right )} \operatorname{arsinh}\left (c x\right )}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(a^2*e*x + a^2*d + (b^2*e*x + b^2*d)*arcsinh(c*x)^2 + 2*(a*b*e*x + a*b*d)*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 (d + e x\right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(1/((a + b*asinh(c*x))**2*(d + e*x)), x)

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

[Out]

integrate(1/((e*x + d)*(b*arcsinh(c*x) + a)^2), x)