3.1.26 \(\int \frac {1}{(d+e x)^2 (a+b \text {ArcSin}(c x))^2} \, dx\) [26]

Optimal. Leaf size=21 \[ \text {Int}\left (\frac {1}{(d+e x)^2 (a+b \text {ArcSin}(c x))^2},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {1}{(d+e x)^2 (a+b \text {ArcSin}(c x))^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 8.41, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{(d+e x)^2 (a+b \text {ArcSin}(c x))^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 1.97, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (e x +d \right )^{2} \left (a +b \arcsin \left (c x \right )\right )^{2}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-((a*b*c*x^2*e^2 + 2*a*b*c*d*x*e + a*b*c*d^2 + (b^2*c*x^2*e^2 + 2*b^2*c*d*x*e + b^2*c*d^2)*arctan2(c*x, sqrt(c
*x + 1)*sqrt(-c*x + 1)))*integrate((c^2*x^2*e - c^2*d*x - 2*e)*sqrt(c*x + 1)*sqrt(-c*x + 1)/(a*b*c^3*x^5*e^3 +
3*a*b*c^3*d*x^4*e^2 - 3*a*b*c*d^2*x*e - a*b*c*d^3 + (3*a*b*c^3*d^2*e - a*b*c*e^3)*x^3 + (a*b*c^3*d^3 - 3*a*b*
c*d*e^2)*x^2 + (b^2*c^3*x^5*e^3 + 3*b^2*c^3*d*x^4*e^2 - 3*b^2*c*d^2*x*e - b^2*c*d^3 + (3*b^2*c^3*d^2*e - b^2*c
*e^3)*x^3 + (b^2*c^3*d^3 - 3*b^2*c*d*e^2)*x^2)*arctan2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1))), x) + sqrt(c*x + 1)
*sqrt(-c*x + 1))/(a*b*c*x^2*e^2 + 2*a*b*c*d*x*e + a*b*c*d^2 + (b^2*c*x^2*e^2 + 2*b^2*c*d*x*e + b^2*c*d^2)*arct
an2(c*x, sqrt(c*x + 1)*sqrt(-c*x + 1)))

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/(a^2*x^2*e^2 + 2*a^2*d*x*e + a^2*d^2 + (b^2*x^2*e^2 + 2*b^2*d*x*e + b^2*d^2)*arcsin(c*x)^2 + 2*(a*b
*x^2*e^2 + 2*a*b*d*x*e + a*b*d^2)*arcsin(c*x)), x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2} \left (d + e x\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2\,{\left (d+e\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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