3.17 \(\int \cos ^5(c+d x) (a+i a \tan (c+d x)) \, dx\)

Optimal. Leaf size=62 \[ \frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^5(c+d x)}{5 d} \]

[Out]

((-I/5)*a*Cos[c + d*x]^5)/d + (a*Sin[c + d*x])/d - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)

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Rubi [A]  time = 0.0347422, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3486, 2633} \[ \frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^5*(a + I*a*Tan[c + d*x]),x]

[Out]

((-I/5)*a*Cos[c + d*x]^5)/d + (a*Sin[c + d*x])/d - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*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 2633

Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> -Dist[d^(-1), Subst[Int[Expand[(1 - x^2)^((n - 1)/2), x], x], x
, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]

Rubi steps

\begin{align*} \int \cos ^5(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac{i a \cos ^5(c+d x)}{5 d}+a \int \cos ^5(c+d x) \, dx\\ &=-\frac{i a \cos ^5(c+d x)}{5 d}-\frac{a \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=-\frac{i a \cos ^5(c+d x)}{5 d}+\frac{a \sin (c+d x)}{d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin ^5(c+d x)}{5 d}\\ \end{align*}

Mathematica [A]  time = 0.0150096, size = 62, normalized size = 1. \[ \frac{a \sin ^5(c+d x)}{5 d}-\frac{2 a \sin ^3(c+d x)}{3 d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^5(c+d x)}{5 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^5*(a + I*a*Tan[c + d*x]),x]

[Out]

((-I/5)*a*Cos[c + d*x]^5)/d + (a*Sin[c + d*x])/d - (2*a*Sin[c + d*x]^3)/(3*d) + (a*Sin[c + d*x]^5)/(5*d)

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Maple [A]  time = 0.078, size = 47, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -{\frac{i}{5}}a \left ( \cos \left ( dx+c \right ) \right ) ^{5}+{\frac{a\sin \left ( dx+c \right ) }{5} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^5*(a+I*a*tan(d*x+c)),x)

[Out]

1/d*(-1/5*I*a*cos(d*x+c)^5+1/5*a*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))

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Maxima [A]  time = 1.14187, size = 66, normalized size = 1.06 \begin{align*} -\frac{3 i \, a \cos \left (d x + c\right )^{5} -{\left (3 \, \sin \left (d x + c\right )^{5} - 10 \, \sin \left (d x + c\right )^{3} + 15 \, \sin \left (d x + c\right )\right )} a}{15 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+I*a*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/15*(3*I*a*cos(d*x + c)^5 - (3*sin(d*x + c)^5 - 10*sin(d*x + c)^3 + 15*sin(d*x + c))*a)/d

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Fricas [A]  time = 1.02029, size = 208, normalized size = 3.35 \begin{align*} \frac{{\left (-3 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 20 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} - 90 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 60 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 5 i \, a\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/240*(-3*I*a*e^(8*I*d*x + 8*I*c) - 20*I*a*e^(6*I*d*x + 6*I*c) - 90*I*a*e^(4*I*d*x + 4*I*c) + 60*I*a*e^(2*I*d*
x + 2*I*c) + 5*I*a)*e^(-3*I*d*x - 3*I*c)/d

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Sympy [A]  time = 1.00191, size = 185, normalized size = 2.98 \begin{align*} \begin{cases} \frac{\left (- 18432 i a d^{4} e^{9 i c} e^{5 i d x} - 122880 i a d^{4} e^{7 i c} e^{3 i d x} - 552960 i a d^{4} e^{5 i c} e^{i d x} + 368640 i a d^{4} e^{3 i c} e^{- i d x} + 30720 i a d^{4} e^{i c} e^{- 3 i d x}\right ) e^{- 4 i c}}{1474560 d^{5}} & \text{for}\: 1474560 d^{5} e^{4 i c} \neq 0 \\\frac{x \left (a e^{8 i c} + 4 a e^{6 i c} + 6 a e^{4 i c} + 4 a e^{2 i c} + a\right ) e^{- 3 i c}}{16} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**5*(a+I*a*tan(d*x+c)),x)

[Out]

Piecewise(((-18432*I*a*d**4*exp(9*I*c)*exp(5*I*d*x) - 122880*I*a*d**4*exp(7*I*c)*exp(3*I*d*x) - 552960*I*a*d**
4*exp(5*I*c)*exp(I*d*x) + 368640*I*a*d**4*exp(3*I*c)*exp(-I*d*x) + 30720*I*a*d**4*exp(I*c)*exp(-3*I*d*x))*exp(
-4*I*c)/(1474560*d**5), Ne(1474560*d**5*exp(4*I*c), 0)), (x*(a*exp(8*I*c) + 4*a*exp(6*I*c) + 6*a*exp(4*I*c) +
4*a*exp(2*I*c) + a)*exp(-3*I*c)/16, True))

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Giac [B]  time = 1.20179, size = 297, normalized size = 4.79 \begin{align*} -\frac{{\left (135 \, a e^{\left (3 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 90 \, a e^{\left (3 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 135 \, a e^{\left (3 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 90 \, a e^{\left (3 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 45 \, a e^{\left (3 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 45 \, a e^{\left (3 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 12 i \, a e^{\left (8 i \, d x + 6 i \, c\right )} + 80 i \, a e^{\left (6 i \, d x + 4 i \, c\right )} + 360 i \, a e^{\left (4 i \, d x + 2 i \, c\right )} - 240 i \, a e^{\left (2 i \, d x\right )} - 20 i \, a e^{\left (-2 i \, c\right )}\right )} e^{\left (-3 i \, d x - i \, c\right )}}{960 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^5*(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

-1/960*(135*a*e^(3*I*d*x + I*c)*log(I*e^(I*d*x + I*c) + 1) + 90*a*e^(3*I*d*x + I*c)*log(I*e^(I*d*x + I*c) - 1)
- 135*a*e^(3*I*d*x + I*c)*log(-I*e^(I*d*x + I*c) + 1) - 90*a*e^(3*I*d*x + I*c)*log(-I*e^(I*d*x + I*c) - 1) -
45*a*e^(3*I*d*x + I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 45*a*e^(3*I*d*x + I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 12*
I*a*e^(8*I*d*x + 6*I*c) + 80*I*a*e^(6*I*d*x + 4*I*c) + 360*I*a*e^(4*I*d*x + 2*I*c) - 240*I*a*e^(2*I*d*x) - 20*
I*a*e^(-2*I*c))*e^(-3*I*d*x - I*c)/d