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

Optimal. Leaf size=26 \[ \frac{\text{Unintegrable}\left (\frac{1}{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2},x\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.0592427, 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 = {} \[ \int \frac{1}{(c e+d e x) \left (a+b \sin ^{-1}(c+d x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 2.56167, size = 0, normalized size = 0. \[ \int \frac{1}{(c e+d e x) \left (a+b \sin ^{-1}(c+d x)\right )^2} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.088, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dex+ce \right ) \left ( a+b\arcsin \left ( dx+c \right ) \right ) ^{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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