3.7 \(\int \cot (c+d x) (a+i a \tan (c+d x)) \, dx\)

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

[Out]

I*a*x + (a*Log[Sin[c + d*x]])/d

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Rubi [A]  time = 0.0218355, antiderivative size = 19, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {3531, 3475} \[ \frac{a \log (\sin (c+d x))}{d}+i a x \]

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 3531

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]

Rule 3475

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

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*}

Mathematica [A]  time = 0.0152012, size = 27, normalized size = 1.42 \[ \frac{a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d}+i a x \]

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 = 0.035, size = 27, normalized size = 1.4 \begin{align*} iax+{\frac{iac}{d}}+{\frac{a\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [B]  time = 2.21664, size = 50, normalized size = 2.63 \begin{align*} -\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{align*}

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 = 2.53015, size = 46, normalized size = 2.42 \begin{align*} \frac{a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right )}{d} \end{align*}

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 = 0.795676, size = 20, normalized size = 1.05 \begin{align*} \frac{a \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} \end{align*}

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 [B]  time = 1.2689, size = 47, normalized size = 2.47 \begin{align*} -\frac{2 \, a \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right ) - a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right )}{d} \end{align*}

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(abs(tan(1/2*d*x + 1/2*c))))/d