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

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

[Out]

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

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Rubi [A]  time = 0.0638499, 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 = {} \[ \int \frac{1}{(c e+d e x) \left (a+b \sin ^{-1}(c+d x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

Defer[Subst][Defer[Int][1/(x*(a + b*ArcSin[x])), 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 )} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{1}{e x \left (a+b \sin ^{-1}(x)\right )} \, dx,x,c+d x\right )}{d}\\ &=\frac{\operatorname{Subst}\left (\int \frac{1}{x \left (a+b \sin ^{-1}(x)\right )} \, dx,x,c+d x\right )}{d e}\\ \end{align*}

Mathematica [A]  time = 0.7783, size = 0, normalized size = 0. \[ \int \frac{1}{(c e+d e x) \left (a+b \sin ^{-1}(c+d x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.095, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( dex+ce \right ) \left ( a+b\arcsin \left ( dx+c \right ) \right ) }}\, 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)),x)

[Out]

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

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Maxima [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 )}}\,{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)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{1}{a d e x + a c e +{\left (b d e x + 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)),x, algorithm="fricas")

[Out]

integral(1/(a*d*e*x + a*c*e + (b*d*e*x + 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 c + a d x + b c \operatorname{asin}{\left (c + d x \right )} + b d x \operatorname{asin}{\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)),x)

[Out]

Integral(1/(a*c + a*d*x + b*c*asin(c + d*x) + b*d*x*asin(c + d*x)), 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 )}}\,{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)),x, algorithm="giac")

[Out]

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