3.1.88 \(\int \frac {\text {ArcTan}(c+(i+c) \tanh (a+b x))}{x} \, dx\) [88]

Optimal. Leaf size=22 \[ \text {Int}\left (\frac {\text {ArcTan}(c+(i+c) \tanh (a+b x))}{x},x\right ) \]

[Out]

CannotIntegrate(arctan(c+(I+c)*tanh(b*x+a))/x,x)

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Rubi [A]
time = 0.09, 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 = {} \begin {gather*} \int \frac {\text {ArcTan}(c+(i+c) \tanh (a+b x))}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 2.69, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\text {ArcTan}(c+(i+c) \tanh (a+b x))}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.08, size = 0, normalized size = 0.00 \[\int \frac {\arctan \left (c +\left (i+c \right ) \tanh \left (b x +a \right )\right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

I*b*x - 1/4*(4*pi - 4*I*a - 2*arctan(c) - I*log(c^2 + 1))*log(x) + 1/2*integrate(arctan(c*e^(2*b*x + 2*a))/x,
x) - 1/4*I*integrate(log(c^2*e^(4*b*x + 4*a) + 1)/x, x)

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.05 \begin {gather*} \int \frac {\mathrm {atan}\left (c+\mathrm {tanh}\left (a+b\,x\right )\,\left (c+1{}\mathrm {i}\right )\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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