3.1.7 \(\int \cot (c+d x) (a+i a \tan (c+d x)) \, dx\) [7]

Optimal. Leaf size=19 \[ i a x+\frac {a \log (\sin (c+d x))}{d} \]

[Out]

I*a*x+a*ln(sin(d*x+c))/d

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Rubi [A]
time = 0.02, antiderivative size = 19, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {3612, 3556} \begin {gather*} \frac {a \log (\sin (c+d x))}{d}+i a x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]*(a + I*a*Tan[c + d*x]),x]

[Out]

I*a*x + (a*Log[Sin[c + d*x]])/d

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3612

Int[((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(a*c +
b*d)*(x/(a^2 + b^2)), x] + Dist[(b*c - a*d)/(a^2 + b^2), Int[(b - a*Tan[e + f*x])/(a + b*Tan[e + f*x]), x], x]
/; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[a*c + b*d, 0]

Rubi steps

\begin {align*} \int \cot (c+d x) (a+i a \tan (c+d x)) \, dx &=i a x+a \int \cot (c+d x) \, dx\\ &=i a x+\frac {a \log (\sin (c+d x))}{d}\\ \end {align*}

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Mathematica [A]
time = 0.02, size = 27, normalized size = 1.42 \begin {gather*} i a x+\frac {a (\log (\cos (c+d x))+\log (\tan (c+d x)))}{d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]*(a + I*a*Tan[c + d*x]),x]

[Out]

I*a*x + (a*(Log[Cos[c + d*x]] + Log[Tan[c + d*x]]))/d

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Maple [A]
time = 0.15, size = 24, normalized size = 1.26

method	result	size
derivativedivides	\(\frac {i a \left (d x +c \right )+a \ln \left (\sin \left (d x +c \right )\right )}{d}\)	\(24\)
default	\(\frac {i a \left (d x +c \right )+a \ln \left (\sin \left (d x +c \right )\right )}{d}\)	\(24\)
risch	\(-\frac {2 i a c}{d}+\frac {a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(27\)
norman	\(i a x +\frac {a \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(36\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)*(a+I*a*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d*(I*a*(d*x+c)+a*ln(sin(d*x+c)))

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 37 vs. \(2 (17) = 34\).
time = 0.49, size = 37, normalized size = 1.95 \begin {gather*} -\frac {-2 i \, {\left (d x + c\right )} a + a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 \, a \log \left (\tan \left (d x + c\right )\right )}{2 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+I*a*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/2*(-2*I*(d*x + c)*a + a*log(tan(d*x + c)^2 + 1) - 2*a*log(tan(d*x + c)))/d

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Fricas [A]
time = 0.54, size = 17, normalized size = 0.89 \begin {gather*} \frac {a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

a*log(e^(2*I*d*x + 2*I*c) - 1)/d

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Sympy [A]
time = 0.09, size = 20, normalized size = 1.05 \begin {gather*} \frac {a \log {\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+I*a*tan(d*x+c)),x)

[Out]

a*log(exp(2*I*d*x) - exp(-2*I*c))/d

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Giac [A]
time = 0.47, size = 34, normalized size = 1.79 \begin {gather*} -\frac {2 \, a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + i\right ) - a \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)*(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

-(2*a*log(tan(1/2*d*x + 1/2*c) + I) - a*log(tan(1/2*d*x + 1/2*c)))/d

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Mupad [B]
time = 3.78, size = 19, normalized size = 1.00 \begin {gather*} \frac {a\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\,2{}\mathrm {i}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)*(a + a*tan(c + d*x)*1i),x)

[Out]

(a*atan(2*tan(c + d*x) + 1i)*2i)/d

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