3.4.28 \(\int \frac {\tanh ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx\) [328]

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

[Out]

CannotIntegrate(-arctanh(-1-I*d+d*tan(b*x+a))/x,x)

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Rubi [A]
time = 0.07, 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 {\tanh ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[ArcTanh[1 + I*d - d*Tan[a + b*x]]/x,x]

[Out]

Defer[Int][ArcTanh[1 + I*d - d*Tan[a + b*x]]/x, x]

Rubi steps

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

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Mathematica [A]
time = 1.20, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tanh ^{-1}(1+i d-d \tan (a+b x))}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[ArcTanh[1 + I*d - d*Tan[a + b*x]]/x,x]

[Out]

Integrate[ArcTanh[1 + I*d - d*Tan[a + b*x]]/x, x]

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Maple [A]
time = 0.12, size = 0, normalized size = 0.00 \[\int -\frac {\arctanh \left (-1-i d +d \tan \left (b x +a \right )\right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-arctanh(-1-I*d+d*tan(b*x+a))/x,x)

[Out]

int(-arctanh(-1-I*d+d*tan(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(-arctanh(-1-I*d+d*tan(b*x+a))/x,x, algorithm="maxima")

[Out]

-I*b*x + 1/4*(-I*pi - 4*I*a - 2*log(-d))*log(x) + 1/2*I*integrate(arctan2(d*cos(2*b*x + 2*a) + sin(2*b*x + 2*a
), -d*sin(2*b*x + 2*a) + cos(2*b*x + 2*a) + 1)/x, x) + 1/4*integrate(log((d^2 + 1)*cos(2*b*x + 2*a)^2 + (d^2 +
1)*sin(2*b*x + 2*a)^2 - 2*d*sin(2*b*x + 2*a) + 2*cos(2*b*x + 2*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(-arctanh(-1-I*d+d*tan(b*x+a))/x,x, algorithm="fricas")

[Out]

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

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\operatorname {atanh}{\left (- d \tan {\left (a + b x \right )} + i d + 1 \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(-atanh(-1-I*d+d*tan(b*x+a))/x,x)

[Out]

Integral(atanh(-d*tan(a + b*x) + I*d + 1)/x, x)

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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(-arctanh(-1-I*d+d*tan(b*x+a))/x,x, algorithm="giac")

[Out]

integrate(-arctanh(d*tan(b*x + a) - I*d - 1)/x, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {\mathrm {atanh}\left (1-d\,\mathrm {tan}\left (a+b\,x\right )+d\,1{}\mathrm {i}\right )}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(atanh(d*1i - d*tan(a + b*x) + 1)/x,x)

[Out]

int(atanh(d*1i - d*tan(a + b*x) + 1)/x, x)

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