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

Optimal. Leaf size=25 \[ \text {Int}\left (\frac {1}{(d+i c d x) \left (a+b \tan ^{-1}(c x)\right )},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.04, 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 = {} \[ \int \frac {1}{(d+i c d x) \left (a+b \tan ^{-1}(c x)\right )} \, dx \]

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*}

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Mathematica [A]  time = 3.40, size = 0, normalized size = 0.00 \[ \int \frac {1}{(d+i c d x) \left (a+b \tan ^{-1}(c x)\right )} \, dx \]

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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fricas [A]  time = 0.58, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {2}{-2 i \, a c d x - 2 \, a d + {\left (b c d x - i \, b d\right )} \log \left (-\frac {c x + i}{c x - i}\right )}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]

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]

sage0*x

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maple [A]  time = 1.47, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (i c d x +d \right ) \left (a +b \arctan \left (c x \right )\right )}\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (i \, c d x + d\right )} {\left (b \arctan \left (c x\right ) + a\right )}}\,{d x} \]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/((a + b*atan(c*x))*(d + c*d*x*1i)),x)

[Out]

int(1/((a + b*atan(c*x))*(d + c*d*x*1i)), x)

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

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]

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

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