3.35 \(\int \tan ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx\)

Optimal. Leaf size=116 \[ -\frac{i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\frac{4 a^4 \tan (c+d x)}{d}+\frac{8 i a^4 \log (\cos (c+d x))}{d}-8 a^4 x-\frac{i (a+i a \tan (c+d x))^5}{5 a d}-\frac{i a (a+i a \tan (c+d x))^3}{3 d} \]

[Out]

-8*a^4*x + ((8*I)*a^4*Log[Cos[c + d*x]])/d + (4*a^4*Tan[c + d*x])/d - ((I/3)*a*(a + I*a*Tan[c + d*x])^3)/d - (
(I/5)*(a + I*a*Tan[c + d*x])^5)/(a*d) - (I*(a^2 + I*a^2*Tan[c + d*x])^2)/d

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Rubi [A]  time = 0.091607, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {3543, 3478, 3477, 3475} \[ -\frac{i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}+\frac{4 a^4 \tan (c+d x)}{d}+\frac{8 i a^4 \log (\cos (c+d x))}{d}-8 a^4 x-\frac{i (a+i a \tan (c+d x))^5}{5 a d}-\frac{i a (a+i a \tan (c+d x))^3}{3 d} \]

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^4,x]

[Out]

-8*a^4*x + ((8*I)*a^4*Log[Cos[c + d*x]])/d + (4*a^4*Tan[c + d*x])/d - ((I/3)*a*(a + I*a*Tan[c + d*x])^3)/d - (
(I/5)*(a + I*a*Tan[c + d*x])^5)/(a*d) - (I*(a^2 + I*a^2*Tan[c + d*x])^2)/d

Rule 3543

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

Rule 3478

Int[((a_) + (b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(a + b*Tan[c + d*x])^(n - 1))/(d*(n - 1)
), x] + Dist[2*a, Int[(a + b*Tan[c + d*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && EqQ[a^2 + b^2, 0] && G
tQ[n, 1]

Rule 3477

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

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 \tan ^2(c+d x) (a+i a \tan (c+d x))^4 \, dx &=-\frac{i (a+i a \tan (c+d x))^5}{5 a d}-\int (a+i a \tan (c+d x))^4 \, dx\\ &=-\frac{i a (a+i a \tan (c+d x))^3}{3 d}-\frac{i (a+i a \tan (c+d x))^5}{5 a d}-(2 a) \int (a+i a \tan (c+d x))^3 \, dx\\ &=-\frac{i a (a+i a \tan (c+d x))^3}{3 d}-\frac{i (a+i a \tan (c+d x))^5}{5 a d}-\frac{i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\left (4 a^2\right ) \int (a+i a \tan (c+d x))^2 \, dx\\ &=-8 a^4 x+\frac{4 a^4 \tan (c+d x)}{d}-\frac{i a (a+i a \tan (c+d x))^3}{3 d}-\frac{i (a+i a \tan (c+d x))^5}{5 a d}-\frac{i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}-\left (8 i a^4\right ) \int \tan (c+d x) \, dx\\ &=-8 a^4 x+\frac{8 i a^4 \log (\cos (c+d x))}{d}+\frac{4 a^4 \tan (c+d x)}{d}-\frac{i a (a+i a \tan (c+d x))^3}{3 d}-\frac{i (a+i a \tan (c+d x))^5}{5 a d}-\frac{i \left (a^2+i a^2 \tan (c+d x)\right )^2}{d}\\ \end{align*}

Mathematica [B]  time = 2.76358, size = 294, normalized size = 2.53 \[ -\frac{a^4 \sec (c) \sec ^5(c+d x) \left (345 \sin (2 c+d x)-275 \sin (2 c+3 d x)+120 \sin (4 c+3 d x)-79 \sin (4 c+5 d x)+150 d x \cos (2 c+3 d x)-90 i \cos (2 c+3 d x)+150 d x \cos (4 c+3 d x)-90 i \cos (4 c+3 d x)+30 d x \cos (4 c+5 d x)+30 d x \cos (6 c+5 d x)-75 i \cos (2 c+3 d x) \log \left (\cos ^2(c+d x)\right )+30 \cos (d x) \left (-5 i \log \left (\cos ^2(c+d x)\right )+10 d x-7 i\right )+30 \cos (2 c+d x) \left (-5 i \log \left (\cos ^2(c+d x)\right )+10 d x-7 i\right )-75 i \cos (4 c+3 d x) \log \left (\cos ^2(c+d x)\right )-15 i \cos (4 c+5 d x) \log \left (\cos ^2(c+d x)\right )-15 i \cos (6 c+5 d x) \log \left (\cos ^2(c+d x)\right )-445 \sin (d x)\right )}{120 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^2*(a + I*a*Tan[c + d*x])^4,x]

[Out]

-(a^4*Sec[c]*Sec[c + d*x]^5*((-90*I)*Cos[2*c + 3*d*x] + 150*d*x*Cos[2*c + 3*d*x] - (90*I)*Cos[4*c + 3*d*x] + 1
50*d*x*Cos[4*c + 3*d*x] + 30*d*x*Cos[4*c + 5*d*x] + 30*d*x*Cos[6*c + 5*d*x] + 30*Cos[d*x]*(-7*I + 10*d*x - (5*
I)*Log[Cos[c + d*x]^2]) + 30*Cos[2*c + d*x]*(-7*I + 10*d*x - (5*I)*Log[Cos[c + d*x]^2]) - (75*I)*Cos[2*c + 3*d
*x]*Log[Cos[c + d*x]^2] - (75*I)*Cos[4*c + 3*d*x]*Log[Cos[c + d*x]^2] - (15*I)*Cos[4*c + 5*d*x]*Log[Cos[c + d*
x]^2] - (15*I)*Cos[6*c + 5*d*x]*Log[Cos[c + d*x]^2] - 445*Sin[d*x] + 345*Sin[2*c + d*x] - 275*Sin[2*c + 3*d*x]
+ 120*Sin[4*c + 3*d*x] - 79*Sin[4*c + 5*d*x]))/(120*d)

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Maple [A]  time = 0.005, size = 117, normalized size = 1. \begin{align*} 8\,{\frac{{a}^{4}\tan \left ( dx+c \right ) }{d}}+{\frac{{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d}}-{\frac{i{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{d}}-{\frac{7\,{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{3\,d}}+{\frac{4\,i{a}^{4} \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d}}-{\frac{4\,i{a}^{4}\ln \left ( 1+ \left ( \tan \left ( dx+c \right ) \right ) ^{2} \right ) }{d}}-8\,{\frac{{a}^{4}\arctan \left ( \tan \left ( dx+c \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^4,x)

[Out]

8*a^4*tan(d*x+c)/d+1/5/d*a^4*tan(d*x+c)^5-I/d*a^4*tan(d*x+c)^4-7/3/d*a^4*tan(d*x+c)^3+4*I/d*a^4*tan(d*x+c)^2-4
*I/d*a^4*ln(1+tan(d*x+c)^2)-8/d*a^4*arctan(tan(d*x+c))

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Maxima [A]  time = 1.65672, size = 128, normalized size = 1.1 \begin{align*} \frac{3 \, a^{4} \tan \left (d x + c\right )^{5} - 15 i \, a^{4} \tan \left (d x + c\right )^{4} - 35 \, a^{4} \tan \left (d x + c\right )^{3} + 60 i \, a^{4} \tan \left (d x + c\right )^{2} - 120 \,{\left (d x + c\right )} a^{4} - 60 i \, a^{4} \log \left (\tan \left (d x + c\right )^{2} + 1\right ) + 120 \, a^{4} \tan \left (d x + c\right )}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")

[Out]

1/15*(3*a^4*tan(d*x + c)^5 - 15*I*a^4*tan(d*x + c)^4 - 35*a^4*tan(d*x + c)^3 + 60*I*a^4*tan(d*x + c)^2 - 120*(
d*x + c)*a^4 - 60*I*a^4*log(tan(d*x + c)^2 + 1) + 120*a^4*tan(d*x + c))/d

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Fricas [B]  time = 2.24581, size = 664, normalized size = 5.72 \begin{align*} \frac{840 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 2220 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 2620 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 1460 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 316 i \, a^{4} +{\left (120 i \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} + 600 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 1200 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 1200 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 600 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 120 i \, a^{4}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right )}{15 \,{\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")

[Out]

1/15*(840*I*a^4*e^(8*I*d*x + 8*I*c) + 2220*I*a^4*e^(6*I*d*x + 6*I*c) + 2620*I*a^4*e^(4*I*d*x + 4*I*c) + 1460*I
*a^4*e^(2*I*d*x + 2*I*c) + 316*I*a^4 + (120*I*a^4*e^(10*I*d*x + 10*I*c) + 600*I*a^4*e^(8*I*d*x + 8*I*c) + 1200
*I*a^4*e^(6*I*d*x + 6*I*c) + 1200*I*a^4*e^(4*I*d*x + 4*I*c) + 600*I*a^4*e^(2*I*d*x + 2*I*c) + 120*I*a^4)*log(e
^(2*I*d*x + 2*I*c) + 1))/(d*e^(10*I*d*x + 10*I*c) + 5*d*e^(8*I*d*x + 8*I*c) + 10*d*e^(6*I*d*x + 6*I*c) + 10*d*
e^(4*I*d*x + 4*I*c) + 5*d*e^(2*I*d*x + 2*I*c) + d)

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Sympy [B]  time = 7.76811, size = 226, normalized size = 1.95 \begin{align*} \frac{8 i a^{4} \log{\left (e^{2 i d x} + e^{- 2 i c} \right )}}{d} + \frac{\frac{56 i a^{4} e^{- 2 i c} e^{8 i d x}}{d} + \frac{148 i a^{4} e^{- 4 i c} e^{6 i d x}}{d} + \frac{524 i a^{4} e^{- 6 i c} e^{4 i d x}}{3 d} + \frac{292 i a^{4} e^{- 8 i c} e^{2 i d x}}{3 d} + \frac{316 i a^{4} e^{- 10 i c}}{15 d}}{e^{10 i d x} + 5 e^{- 2 i c} e^{8 i d x} + 10 e^{- 4 i c} e^{6 i d x} + 10 e^{- 6 i c} e^{4 i d x} + 5 e^{- 8 i c} e^{2 i d x} + e^{- 10 i c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**2*(a+I*a*tan(d*x+c))**4,x)

[Out]

8*I*a**4*log(exp(2*I*d*x) + exp(-2*I*c))/d + (56*I*a**4*exp(-2*I*c)*exp(8*I*d*x)/d + 148*I*a**4*exp(-4*I*c)*ex
p(6*I*d*x)/d + 524*I*a**4*exp(-6*I*c)*exp(4*I*d*x)/(3*d) + 292*I*a**4*exp(-8*I*c)*exp(2*I*d*x)/(3*d) + 316*I*a
**4*exp(-10*I*c)/(15*d))/(exp(10*I*d*x) + 5*exp(-2*I*c)*exp(8*I*d*x) + 10*exp(-4*I*c)*exp(6*I*d*x) + 10*exp(-6
*I*c)*exp(4*I*d*x) + 5*exp(-8*I*c)*exp(2*I*d*x) + exp(-10*I*c))

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Giac [B]  time = 1.69824, size = 370, normalized size = 3.19 \begin{align*} \frac{120 i \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 600 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 1200 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 1200 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 600 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 840 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} + 2220 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} + 2620 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} + 1460 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 120 i \, a^{4} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 316 i \, a^{4}}{15 \,{\left (d e^{\left (10 i \, d x + 10 i \, c\right )} + 5 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 10 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 5 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^2*(a+I*a*tan(d*x+c))^4,x, algorithm="giac")

[Out]

1/15*(120*I*a^4*e^(10*I*d*x + 10*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 600*I*a^4*e^(8*I*d*x + 8*I*c)*log(e^(2*I*
d*x + 2*I*c) + 1) + 1200*I*a^4*e^(6*I*d*x + 6*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 1200*I*a^4*e^(4*I*d*x + 4*I*
c)*log(e^(2*I*d*x + 2*I*c) + 1) + 600*I*a^4*e^(2*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 840*I*a^4*e^(8*
I*d*x + 8*I*c) + 2220*I*a^4*e^(6*I*d*x + 6*I*c) + 2620*I*a^4*e^(4*I*d*x + 4*I*c) + 1460*I*a^4*e^(2*I*d*x + 2*I
*c) + 120*I*a^4*log(e^(2*I*d*x + 2*I*c) + 1) + 316*I*a^4)/(d*e^(10*I*d*x + 10*I*c) + 5*d*e^(8*I*d*x + 8*I*c) +
10*d*e^(6*I*d*x + 6*I*c) + 10*d*e^(4*I*d*x + 4*I*c) + 5*d*e^(2*I*d*x + 2*I*c) + d)