3.133 \(\int \frac{1}{(d+i c d x) (a+b \tan ^{-1}(c x))} \, dx\)

Optimal. Leaf size=24 \[ \text{Unintegrable}\left (\frac{1}{(d+i c d x) \left (a+b \tan ^{-1}(c x)\right )},x\right ) \]

[Out]

Unintegrable[1/((d + I*c*d*x)*(a + b*ArcTan[c*x])), x]

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Rubi [A]  time = 0.0373987, 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 = {} \[ \int \frac{1}{(d+i c d x) \left (a+b \tan ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[1/((d + I*c*d*x)*(a + b*ArcTan[c*x])),x]

[Out]

Defer[Int][1/((d + I*c*d*x)*(a + b*ArcTan[c*x])), x]

Rubi steps

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

Mathematica [A]  time = 3.12181, size = 0, normalized size = 0. \[ \int \frac{1}{(d+i c d x) \left (a+b \tan ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[1/((d + I*c*d*x)*(a + b*ArcTan[c*x])),x]

[Out]

Integrate[1/((d + I*c*d*x)*(a + b*ArcTan[c*x])), x]

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Maple [A]  time = 0.671, size = 0, normalized size = 0. \begin{align*} \int{\frac{1}{ \left ( d+icdx \right ) \left ( a+b\arctan \left ( cx \right ) \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(d+I*c*d*x)/(a+b*arctan(c*x)),x)

[Out]

int(1/(d+I*c*d*x)/(a+b*arctan(c*x)),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (i \, c d x + d\right )}{\left (b \arctan \left (c x\right ) + a\right )}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d+I*c*d*x)/(a+b*arctan(c*x)),x, algorithm="maxima")

[Out]

integrate(1/((I*c*d*x + d)*(b*arctan(c*x) + a)), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d+I*c*d*x)/(a+b*arctan(c*x)),x, algorithm="fricas")

[Out]

integral(-2/(-2*I*a*c*d*x - 2*a*d + (b*c*d*x - I*b*d)*log(-(c*x + I)/(c*x - I))), x)

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d+I*c*d*x)/(a+b*atan(c*x)),x)

[Out]

Exception raised: AttributeError

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(d+I*c*d*x)/(a+b*arctan(c*x)),x, algorithm="giac")

[Out]

integrate(1/((I*c*d*x + d)*(b*arctan(c*x) + a)), x)