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

Optimal. Leaf size=34 \[ \frac {i a \tan (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d}-i a x \]

[Out]

-I*a*x-a*ln(cos(d*x+c))/d+I*a*tan(d*x+c)/d

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Rubi [A]  time = 0.02, antiderivative size = 34, 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 = {3525, 3475} \[ \frac {i a \tan (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d}-i a x \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3475

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

Rule 3525

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

Rubi steps

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

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Mathematica [A]  time = 0.02, size = 43, normalized size = 1.26 \[ -\frac {i a \tan ^{-1}(\tan (c+d x))}{d}+\frac {i a \tan (c+d x)}{d}-\frac {a \log (\cos (c+d x))}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 47, normalized size = 1.38 \[ -\frac {{\left (a e^{\left (2 i \, d x + 2 i \, c\right )} + a\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2 \, a}{d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 0.46, size = 58, normalized size = 1.71 \[ -\frac {a e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + a \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 2 \, a}{d e^{\left (2 i \, d x + 2 i \, c\right )} + d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.02, size = 46, normalized size = 1.35 \[ \frac {i a \tan \left (d x +c \right )}{d}+\frac {a \ln \left (1+\tan ^{2}\left (d x +c \right )\right )}{2 d}-\frac {i a \arctan \left (\tan \left (d x +c \right )\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.81, size = 37, normalized size = 1.09 \[ -\frac {2 i \, {\left (d x + c\right )} a - a \log \left (\tan \left (d x + c\right )^{2} + 1\right ) - 2 i \, a \tan \left (d x + c\right )}{2 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(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*I*a*tan(d*x + c))/d

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mupad [B]  time = 3.78, size = 25, normalized size = 0.74 \[ \frac {a\,\left (\ln \left (\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )+\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(a*(log(tan(c + d*x) + 1i) + tan(c + d*x)*1i))/d

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sympy [A]  time = 0.23, size = 44, normalized size = 1.29 \[ \frac {2 a}{- d e^{2 i c} e^{2 i d x} - d} - \frac {a \log {\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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