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

Optimal. Leaf size=27 \[ \frac {\text {Int}\left (\frac {1}{(c+d x) \left (a+b \sin ^{-1}(c+d x)\right )^2},x\right )}{e} \]

[Out]

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

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {1}{(c e+d e x) \left (a+b \sin ^{-1}(c+d x)\right )^2} \, dx \]

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*}

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Mathematica [A]  time = 2.80, size = 0, normalized size = 0.00 \[ \int \frac {1}{(c e+d e x) \left (a+b \sin ^{-1}(c+d x)\right )^2} \, dx \]

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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fricas [A]  time = 0.41, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {1}{a^{2} d e x + a^{2} c e + {\left (b^{2} d e x + b^{2} c e\right )} \arcsin \left (d x + c\right )^{2} + 2 \, {\left (a b d e x + a b c e\right )} \arcsin \left (d x + c\right )}, x\right ) \]

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

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)

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maple [A]  time = 0.16, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (d e x +c e \right ) \left (a +b \arcsin \left (d x +c \right )\right )^{2}}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {1}{\left (c\,e+d\,e\,x\right )\,{\left (a+b\,\mathrm {asin}\left (c+d\,x\right )\right )}^2} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

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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