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

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

[Out]

CannotIntegrate(-arctan(-c-(1-I*c)*cot(b*x+a))/x,x)

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Rubi [A]  time = 0.12, 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 = {} \[ \int \frac {\tan ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.76, size = 0, normalized size = 0.00 \[ \int \frac {\tan ^{-1}(c+(1-i c) \cot (a+b x))}{x} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {i \, \log \left (-\frac {{\left (c + i\right )} e^{\left (2 i \, b x + 2 i \, a\right )}}{c e^{\left (2 i \, b x + 2 i \, a\right )} + i}\right )}{2 \, x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(1/2*I*log(-(c + I)*e^(2*I*b*x + 2*I*a)/(c*e^(2*I*b*x + 2*I*a) + I))/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.80, size = 0, normalized size = 0.00 \[ \int -\frac {\arctan \left (-c -\left (-i c +1\right ) \cot \left (b x +a \right )\right )}{x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c-1>0)', see `assume?` for mor
e details)Is c-1 zero or nonzero?

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [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(-atan(-c-(1-I*c)*cot(b*x+a))/x,x)

[Out]

Timed out

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