3.9 \(\int \cos ^6(c+d x) (a+i a \tan (c+d x)) \, dx\)

Optimal. Leaf size=89 \[ -\frac{i a \cos ^6(c+d x)}{6 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16} \]

[Out]

(5*a*x)/16 - ((I/6)*a*Cos[c + d*x]^6)/d + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a*Cos[c + d*x]^3*Sin[c +
d*x])/(24*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)

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Rubi [A]  time = 0.0522472, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136, Rules used = {3486, 2635, 8} \[ -\frac{i a \cos ^6(c+d x)}{6 d}+\frac{a \sin (c+d x) \cos ^5(c+d x)}{6 d}+\frac{5 a \sin (c+d x) \cos ^3(c+d x)}{24 d}+\frac{5 a \sin (c+d x) \cos (c+d x)}{16 d}+\frac{5 a x}{16} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x]),x]

[Out]

(5*a*x)/16 - ((I/6)*a*Cos[c + d*x]^6)/d + (5*a*Cos[c + d*x]*Sin[c + d*x])/(16*d) + (5*a*Cos[c + d*x]^3*Sin[c +
d*x])/(24*d) + (a*Cos[c + d*x]^5*Sin[c + d*x])/(6*d)

Rule 3486

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(b*(d*Sec[
e + f*x])^m)/(f*m), x] + Dist[a, Int[(d*Sec[e + f*x])^m, x], x] /; FreeQ[{a, b, d, e, f, m}, x] && (IntegerQ[2
*m] || NeQ[a^2 + b^2, 0])

Rule 2635

Int[((b_.)*sin[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> -Simp[(b*Cos[c + d*x]*(b*Sin[c + d*x])^(n - 1))/(d*n),
x] + Dist[(b^2*(n - 1))/n, Int[(b*Sin[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n, 1] && Integer
Q[2*n]

Rule 8

Int[a_, x_Symbol] :> Simp[a*x, x] /; FreeQ[a, x]

Rubi steps

\begin{align*} \int \cos ^6(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac{i a \cos ^6(c+d x)}{6 d}+a \int \cos ^6(c+d x) \, dx\\ &=-\frac{i a \cos ^6(c+d x)}{6 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac{1}{6} (5 a) \int \cos ^4(c+d x) \, dx\\ &=-\frac{i a \cos ^6(c+d x)}{6 d}+\frac{5 a \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac{1}{8} (5 a) \int \cos ^2(c+d x) \, dx\\ &=-\frac{i a \cos ^6(c+d x)}{6 d}+\frac{5 a \cos (c+d x) \sin (c+d x)}{16 d}+\frac{5 a \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{6 d}+\frac{1}{16} (5 a) \int 1 \, dx\\ &=\frac{5 a x}{16}-\frac{i a \cos ^6(c+d x)}{6 d}+\frac{5 a \cos (c+d x) \sin (c+d x)}{16 d}+\frac{5 a \cos ^3(c+d x) \sin (c+d x)}{24 d}+\frac{a \cos ^5(c+d x) \sin (c+d x)}{6 d}\\ \end{align*}

Mathematica [A]  time = 0.05835, size = 56, normalized size = 0.63 \[ \frac{a \left (45 \sin (2 (c+d x))+9 \sin (4 (c+d x))+\sin (6 (c+d x))-32 i \cos ^6(c+d x)+60 c+60 d x\right )}{192 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^6*(a + I*a*Tan[c + d*x]),x]

[Out]

(a*(60*c + 60*d*x - (32*I)*Cos[c + d*x]^6 + 45*Sin[2*(c + d*x)] + 9*Sin[4*(c + d*x)] + Sin[6*(c + d*x)]))/(192
*d)

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Maple [A]  time = 0.084, size = 63, normalized size = 0.7 \begin{align*}{\frac{1}{d} \left ( -{\frac{i}{6}}a \left ( \cos \left ( dx+c \right ) \right ) ^{6}+a \left ({\frac{\sin \left ( dx+c \right ) }{6} \left ( \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{5\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{4}}+{\frac{15\,\cos \left ( dx+c \right ) }{8}} \right ) }+{\frac{5\,dx}{16}}+{\frac{5\,c}{16}} \right ) \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^6*(a+I*a*tan(d*x+c)),x)

[Out]

1/d*(-1/6*I*a*cos(d*x+c)^6+a*(1/6*(cos(d*x+c)^5+5/4*cos(d*x+c)^3+15/8*cos(d*x+c))*sin(d*x+c)+5/16*d*x+5/16*c))

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Maxima [A]  time = 1.67519, size = 111, normalized size = 1.25 \begin{align*} \frac{15 \,{\left (d x + c\right )} a + \frac{15 \, a \tan \left (d x + c\right )^{5} + 40 \, a \tan \left (d x + c\right )^{3} + 33 \, a \tan \left (d x + c\right ) - 8 i \, a}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1}}{48 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+I*a*tan(d*x+c)),x, algorithm="maxima")

[Out]

1/48*(15*(d*x + c)*a + (15*a*tan(d*x + c)^5 + 40*a*tan(d*x + c)^3 + 33*a*tan(d*x + c) - 8*I*a)/(tan(d*x + c)^6
+ 3*tan(d*x + c)^4 + 3*tan(d*x + c)^2 + 1))/d

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Fricas [A]  time = 1.06969, size = 252, normalized size = 2.83 \begin{align*} \frac{{\left (120 \, a d x e^{\left (4 i \, d x + 4 i \, c\right )} - 2 i \, a e^{\left (10 i \, d x + 10 i \, c\right )} - 15 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 60 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 30 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 3 i \, a\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/384*(120*a*d*x*e^(4*I*d*x + 4*I*c) - 2*I*a*e^(10*I*d*x + 10*I*c) - 15*I*a*e^(8*I*d*x + 8*I*c) - 60*I*a*e^(6*
I*d*x + 6*I*c) + 30*I*a*e^(2*I*d*x + 2*I*c) + 3*I*a)*e^(-4*I*d*x - 4*I*c)/d

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Sympy [A]  time = 0.782871, size = 212, normalized size = 2.38 \begin{align*} \frac{5 a x}{16} + \begin{cases} \frac{\left (- 33554432 i a d^{4} e^{12 i c} e^{6 i d x} - 251658240 i a d^{4} e^{10 i c} e^{4 i d x} - 1006632960 i a d^{4} e^{8 i c} e^{2 i d x} + 503316480 i a d^{4} e^{4 i c} e^{- 2 i d x} + 50331648 i a d^{4} e^{2 i c} e^{- 4 i d x}\right ) e^{- 6 i c}}{6442450944 d^{5}} & \text{for}\: 6442450944 d^{5} e^{6 i c} \neq 0 \\x \left (- \frac{5 a}{16} + \frac{\left (a e^{10 i c} + 5 a e^{8 i c} + 10 a e^{6 i c} + 10 a e^{4 i c} + 5 a e^{2 i c} + a\right ) e^{- 4 i c}}{32}\right ) & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**6*(a+I*a*tan(d*x+c)),x)

[Out]

5*a*x/16 + Piecewise(((-33554432*I*a*d**4*exp(12*I*c)*exp(6*I*d*x) - 251658240*I*a*d**4*exp(10*I*c)*exp(4*I*d*
x) - 1006632960*I*a*d**4*exp(8*I*c)*exp(2*I*d*x) + 503316480*I*a*d**4*exp(4*I*c)*exp(-2*I*d*x) + 50331648*I*a*
d**4*exp(2*I*c)*exp(-4*I*d*x))*exp(-6*I*c)/(6442450944*d**5), Ne(6442450944*d**5*exp(6*I*c), 0)), (x*(-5*a/16
+ (a*exp(10*I*c) + 5*a*exp(8*I*c) + 10*a*exp(6*I*c) + 10*a*exp(4*I*c) + 5*a*exp(2*I*c) + a)*exp(-4*I*c)/32), T
rue))

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Giac [A]  time = 1.13989, size = 171, normalized size = 1.92 \begin{align*} \frac{{\left (120 \, a d x e^{\left (4 i \, d x + 2 i \, c\right )} + 12 i \, a e^{\left (4 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 i \, a e^{\left (4 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) - 2 i \, a e^{\left (10 i \, d x + 8 i \, c\right )} - 15 i \, a e^{\left (8 i \, d x + 6 i \, c\right )} - 60 i \, a e^{\left (6 i \, d x + 4 i \, c\right )} + 30 i \, a e^{\left (2 i \, d x\right )} + 3 i \, a e^{\left (-2 i \, c\right )}\right )} e^{\left (-4 i \, d x - 2 i \, c\right )}}{384 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^6*(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

1/384*(120*a*d*x*e^(4*I*d*x + 2*I*c) + 12*I*a*e^(4*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 12*I*a*e^(4*I
*d*x + 2*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) - 2*I*a*e^(10*I*d*x + 8*I*c) - 15*I*a*e^(8*I*d*x + 6*I*c) - 60*I*a
*e^(6*I*d*x + 4*I*c) + 30*I*a*e^(2*I*d*x) + 3*I*a*e^(-2*I*c))*e^(-4*I*d*x - 2*I*c)/d