3.8 \(\int \cot ^2(c+d x) (a+i a \tan (c+d x)) \, dx\)

Optimal. Leaf size=32 \[ -\frac{a \cot (c+d x)}{d}+\frac{i a \log (\sin (c+d x))}{d}-a x \]

[Out]

-(a*x) - (a*Cot[c + d*x])/d + (I*a*Log[Sin[c + d*x]])/d

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Rubi [A]  time = 0.0441037, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3529, 3531, 3475} \[ -\frac{a \cot (c+d x)}{d}+\frac{i a \log (\sin (c+d x))}{d}-a x \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x]),x]

[Out]

-(a*x) - (a*Cot[c + d*x])/d + (I*a*Log[Sin[c + d*x]])/d

Rule 3529

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[((
b*c - a*d)*(a + b*Tan[e + f*x])^(m + 1))/(f*(m + 1)*(a^2 + b^2)), x] + Dist[1/(a^2 + b^2), Int[(a + b*Tan[e +
f*x])^(m + 1)*Simp[a*c + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c
- a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1]

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 ^2(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac{a \cot (c+d x)}{d}+\int \cot (c+d x) (i a-a \tan (c+d x)) \, dx\\ &=-a x-\frac{a \cot (c+d x)}{d}+(i a) \int \cot (c+d x) \, dx\\ &=-a x-\frac{a \cot (c+d x)}{d}+\frac{i a \log (\sin (c+d x))}{d}\\ \end{align*}

Mathematica [C]  time = 0.113574, size = 54, normalized size = 1.69 \[ -\frac{a \cot (c+d x) \, _2F_1\left (-\frac{1}{2},1;\frac{1}{2};-\tan ^2(c+d x)\right )}{d}+\frac{i a (\log (\tan (c+d x))+\log (\cos (c+d x)))}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^2*(a + I*a*Tan[c + d*x]),x]

[Out]

-((a*Cot[c + d*x]*Hypergeometric2F1[-1/2, 1, 1/2, -Tan[c + d*x]^2])/d) + (I*a*(Log[Cos[c + d*x]] + Log[Tan[c +
d*x]]))/d

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Maple [A]  time = 0.031, size = 39, normalized size = 1.2 \begin{align*}{\frac{ia\ln \left ( \sin \left ( dx+c \right ) \right ) }{d}}-ax-{\frac{\cot \left ( dx+c \right ) a}{d}}-{\frac{ac}{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^2*(a+I*a*tan(d*x+c)),x)

[Out]

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

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Maxima [A]  time = 2.30948, size = 66, normalized size = 2.06 \begin{align*} -\frac{2 \,{\left (d x + c\right )} a + i \, a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 i \, a \log \left (\tan \left (d x + c\right )\right ) + \frac{2 \, a}{\tan \left (d x + c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+I*a*tan(d*x+c)),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 2.41312, size = 135, normalized size = 4.22 \begin{align*} \frac{{\left (i \, a e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} - 1\right ) - 2 i \, a}{d e^{\left (2 i \, d x + 2 i \, c\right )} - d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 1.52719, size = 53, normalized size = 1.66 \begin{align*} \frac{i a \log{\left (e^{2 i d x} - e^{- 2 i c} \right )}}{d} - \frac{2 i a e^{- 2 i c}}{d \left (e^{2 i d x} - e^{- 2 i c}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**2*(a+I*a*tan(d*x+c)),x)

[Out]

I*a*log(exp(2*I*d*x) - exp(-2*I*c))/d - 2*I*a*exp(-2*I*c)/(d*(exp(2*I*d*x) - exp(-2*I*c)))

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Giac [B]  time = 1.25131, size = 103, normalized size = 3.22 \begin{align*} -\frac{4 i \, a \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + i\right ) - 2 i \, a \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) \right |}\right ) - a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - \frac{-2 i \, a \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - a}{\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )}}{2 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

-1/2*(4*I*a*log(tan(1/2*d*x + 1/2*c) + I) - 2*I*a*log(abs(tan(1/2*d*x + 1/2*c))) - a*tan(1/2*d*x + 1/2*c) - (-
2*I*a*tan(1/2*d*x + 1/2*c) - a)/tan(1/2*d*x + 1/2*c))/d