3.1.8 \(\int \cot ^2(c+d x) (a+i a \tan (c+d x)) \, dx\) [8]

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

[Out]

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

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Rubi [A]
time = 0.03, antiderivative size = 32, normalized size of antiderivative = 1.00, 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 = {3610, 3612, 3556} \begin {gather*} -\frac {a \cot (c+d x)}{d}+\frac {i a \log (\sin (c+d x))}{d}-a x \end {gather*}

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 3556

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

Rule 3610

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 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 ^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*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 0.13, size = 54, normalized size = 1.69 \begin {gather*} -\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 (\cos (c+d x))+\log (\tan (c+d x)))}{d} \end {gather*}

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 = 0.15, size = 35, normalized size = 1.09

method	result	size
derivativedivides	\(\frac {i a \ln \left (\sin \left (d x +c \right )\right )+a \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\)	\(35\)
default	\(\frac {i a \ln \left (\sin \left (d x +c \right )\right )+a \left (-\cot \left (d x +c \right )-d x -c \right )}{d}\)	\(35\)
risch	\(\frac {2 a c}{d}-\frac {2 i a}{d \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}+\frac {i a \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}-1\right )}{d}\)	\(48\)
norman	\(\frac {-\frac {a}{d}-a x \tan \left (d x +c \right )}{\tan \left (d x +c \right )}+\frac {i a \ln \left (\tan \left (d x +c \right )\right )}{d}-\frac {i a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}\)	\(60\)

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,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.49, size = 49, normalized size = 1.53 \begin {gather*} -\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 {gather*}

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 = 0.44, size = 51, normalized size = 1.59 \begin {gather*} \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 {gather*}

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 [A]
time = 0.15, size = 46, normalized size = 1.44 \begin {gather*} - \frac {2 i a}{d e^{2 i c} e^{2 i d x} - d} + \frac {i 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)**2*(a+I*a*tan(d*x+c)),x)

[Out]

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

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 75 vs. \(2 (30) = 60\).
time = 0.61, size = 75, normalized size = 2.34 \begin {gather*} -\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 (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\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 {gather*}

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(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

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Mupad [B]
time = 3.73, size = 27, normalized size = 0.84 \begin {gather*} -\frac {a\,\left (\mathrm {cot}\left (c+d\,x\right )+2\,\mathrm {atan}\left (2\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )\right )}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(c + d*x)^2*(a + a*tan(c + d*x)*1i),x)

[Out]

-(a*(cot(c + d*x) + 2*atan(2*tan(c + d*x) + 1i)))/d

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