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

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

[Out]

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

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Rubi [A]  time = 0.0291871, 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 = {} \[ \int \frac{1}{(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 5.08055, size = 0, normalized size = 0. \[ \int \frac{1}{(d+e x)^2 \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.25, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( ex+d \right ) ^{2} \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)^2/(a+b*arcsinh(c*x))^2,x)

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(e*x+d)^2/(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^2*x^4 + 2*a*b*c^3*d*e*x^3 + 2*a*b*c*d*e*x + a*b*c*d^2 + (c^3
*d^2 + c*e^2)*a*b*x^2 + (b^2*c^3*e^2*x^4 + 2*b^2*c^3*d*e*x^3 + 2*b^2*c*d*e*x + b^2*c*d^2 + (c^3*d^2 + c*e^2)*b
^2*x^2 + (b^2*c^2*e^2*x^3 + 2*b^2*c^2*d*e*x^2 + b^2*c^2*d^2*x)*sqrt(c^2*x^2 + 1))*log(c*x + sqrt(c^2*x^2 + 1))
+ (a*b*c^2*e^2*x^3 + 2*a*b*c^2*d*e*x^2 + a*b*c^2*d^2*x)*sqrt(c^2*x^2 + 1)) - integrate((c^5*e*x^5 - c^5*d*x^4
+ 2*c^3*e*x^3 - 2*c^3*d*x^2 + c*e*x + (c^3*e*x^3 - c^3*d*x^2 + 3*c*e*x + c*d)*(c^2*x^2 + 1) - c*d + (2*c^4*e*
x^4 - 2*c^4*d*x^3 + 5*c^2*e*x^2 - c^2*d*x + 2*e)*sqrt(c^2*x^2 + 1))/(a*b*c^5*e^3*x^7 + 3*a*b*c^5*d*e^2*x^6 + (
3*c^5*d^2*e + 2*c^3*e^3)*a*b*x^5 + 3*a*b*c*d^2*e*x + (c^5*d^3 + 6*c^3*d*e^2)*a*b*x^4 + a*b*c*d^3 + (6*c^3*d^2*
e + c*e^3)*a*b*x^3 + (2*c^3*d^3 + 3*c*d*e^2)*a*b*x^2 + (a*b*c^3*e^3*x^5 + 3*a*b*c^3*d*e^2*x^4 + 3*a*b*c^3*d^2*
e*x^3 + a*b*c^3*d^3*x^2)*(c^2*x^2 + 1) + (b^2*c^5*e^3*x^7 + 3*b^2*c^5*d*e^2*x^6 + (3*c^5*d^2*e + 2*c^3*e^3)*b^
2*x^5 + 3*b^2*c*d^2*e*x + (c^5*d^3 + 6*c^3*d*e^2)*b^2*x^4 + b^2*c*d^3 + (6*c^3*d^2*e + c*e^3)*b^2*x^3 + (2*c^3
*d^3 + 3*c*d*e^2)*b^2*x^2 + (b^2*c^3*e^3*x^5 + 3*b^2*c^3*d*e^2*x^4 + 3*b^2*c^3*d^2*e*x^3 + b^2*c^3*d^3*x^2)*(c
^2*x^2 + 1) + 2*(b^2*c^4*e^3*x^6 + 3*b^2*c^4*d*e^2*x^5 + 3*b^2*c^2*d^2*e*x^2 + b^2*c^2*d^3*x + (3*c^4*d^2*e +
c^2*e^3)*b^2*x^4 + (c^4*d^3 + 3*c^2*d*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^3*x^6 + 3*a*b*c^4*d*e^2*x^5 + 3*a*b*c^2*d^2*e*x^2 + a*b*c^2*d^3*x + (3*c^4*d^2*e + c^2*e^3)*a*b*x^4 + (c^
4*d^3 + 3*c^2*d*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^{2} x^{2} + 2 \, a^{2} d e x + a^{2} d^{2} +{\left (b^{2} e^{2} x^{2} + 2 \, b^{2} d e x + b^{2} d^{2}\right )} \operatorname{arsinh}\left (c x\right )^{2} + 2 \,{\left (a b e^{2} x^{2} + 2 \, a b d e x + a b d^{2}\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)^2/(a+b*arcsinh(c*x))^2,x, algorithm="fricas")

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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