3.18 \(\int \cos ^7(c+d x) (a+i a \tan (c+d x)) \, dx\)

Optimal. Leaf size=76 \[ -\frac{a \sin ^7(c+d x)}{7 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^7(c+d x)}{7 d} \]

[Out]

((-I/7)*a*Cos[c + d*x]^7)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[
c + d*x]^7)/(7*d)

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Rubi [A]  time = 0.0381817, antiderivative size = 76, 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} \[ -\frac{a \sin ^7(c+d x)}{7 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x]),x]

[Out]

((-I/7)*a*Cos[c + d*x]^7)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[
c + d*x]^7)/(7*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 ^7(c+d x) (a+i a \tan (c+d x)) \, dx &=-\frac{i a \cos ^7(c+d x)}{7 d}+a \int \cos ^7(c+d x) \, dx\\ &=-\frac{i a \cos ^7(c+d x)}{7 d}-\frac{a \operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{d}\\ &=-\frac{i a \cos ^7(c+d x)}{7 d}+\frac{a \sin (c+d x)}{d}-\frac{a \sin ^3(c+d x)}{d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^7(c+d x)}{7 d}\\ \end{align*}

Mathematica [A]  time = 0.0422304, size = 76, normalized size = 1. \[ -\frac{a \sin ^7(c+d x)}{7 d}+\frac{3 a \sin ^5(c+d x)}{5 d}-\frac{a \sin ^3(c+d x)}{d}+\frac{a \sin (c+d x)}{d}-\frac{i a \cos ^7(c+d x)}{7 d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x]),x]

[Out]

((-I/7)*a*Cos[c + d*x]^7)/d + (a*Sin[c + d*x])/d - (a*Sin[c + d*x]^3)/d + (3*a*Sin[c + d*x]^5)/(5*d) - (a*Sin[
c + d*x]^7)/(7*d)

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Maple [A]  time = 0.08, size = 57, normalized size = 0.8 \begin{align*}{\frac{1}{d} \left ( -{\frac{i}{7}}a \left ( \cos \left ( dx+c \right ) \right ) ^{7}+{\frac{a\sin \left ( dx+c \right ) }{7} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+{\frac{6\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{5}}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{5}} \right ) } \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*(a+I*a*tan(d*x+c)),x)

[Out]

1/d*(-1/7*I*a*cos(d*x+c)^7+1/7*a*(16/5+cos(d*x+c)^6+6/5*cos(d*x+c)^4+8/5*cos(d*x+c)^2)*sin(d*x+c))

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Maxima [A]  time = 1.11542, size = 78, normalized size = 1.03 \begin{align*} -\frac{5 i \, a \cos \left (d x + c\right )^{7} +{\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a}{35 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/35*(5*I*a*cos(d*x + c)^7 + (5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*a)/
d

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Fricas [A]  time = 1.07054, size = 297, normalized size = 3.91 \begin{align*} \frac{{\left (-5 i \, a e^{\left (12 i \, d x + 12 i \, c\right )} - 42 i \, a e^{\left (10 i \, d x + 10 i \, c\right )} - 175 i \, a e^{\left (8 i \, d x + 8 i \, c\right )} - 700 i \, a e^{\left (6 i \, d x + 6 i \, c\right )} + 525 i \, a e^{\left (4 i \, d x + 4 i \, c\right )} + 70 i \, a e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, a\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{2240 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

1/2240*(-5*I*a*e^(12*I*d*x + 12*I*c) - 42*I*a*e^(10*I*d*x + 10*I*c) - 175*I*a*e^(8*I*d*x + 8*I*c) - 700*I*a*e^
(6*I*d*x + 6*I*c) + 525*I*a*e^(4*I*d*x + 4*I*c) + 70*I*a*e^(2*I*d*x + 2*I*c) + 7*I*a)*e^(-5*I*d*x - 5*I*c)/d

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Sympy [A]  time = 1.24666, size = 255, normalized size = 3.36 \begin{align*} \begin{cases} \frac{\left (- 107374182400 i a d^{6} e^{16 i c} e^{7 i d x} - 901943132160 i a d^{6} e^{14 i c} e^{5 i d x} - 3758096384000 i a d^{6} e^{12 i c} e^{3 i d x} - 15032385536000 i a d^{6} e^{10 i c} e^{i d x} + 11274289152000 i a d^{6} e^{8 i c} e^{- i d x} + 1503238553600 i a d^{6} e^{6 i c} e^{- 3 i d x} + 150323855360 i a d^{6} e^{4 i c} e^{- 5 i d x}\right ) e^{- 9 i c}}{48103633715200 d^{7}} & \text{for}\: 48103633715200 d^{7} e^{9 i c} \neq 0 \\\frac{x \left (a e^{12 i c} + 6 a e^{10 i c} + 15 a e^{8 i c} + 20 a e^{6 i c} + 15 a e^{4 i c} + 6 a e^{2 i c} + a\right ) e^{- 5 i c}}{64} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**7*(a+I*a*tan(d*x+c)),x)

[Out]

Piecewise(((-107374182400*I*a*d**6*exp(16*I*c)*exp(7*I*d*x) - 901943132160*I*a*d**6*exp(14*I*c)*exp(5*I*d*x) -
3758096384000*I*a*d**6*exp(12*I*c)*exp(3*I*d*x) - 15032385536000*I*a*d**6*exp(10*I*c)*exp(I*d*x) + 1127428915
2000*I*a*d**6*exp(8*I*c)*exp(-I*d*x) + 1503238553600*I*a*d**6*exp(6*I*c)*exp(-3*I*d*x) + 150323855360*I*a*d**6
*exp(4*I*c)*exp(-5*I*d*x))*exp(-9*I*c)/(48103633715200*d**7), Ne(48103633715200*d**7*exp(9*I*c), 0)), (x*(a*ex
p(12*I*c) + 6*a*exp(10*I*c) + 15*a*exp(8*I*c) + 20*a*exp(6*I*c) + 15*a*exp(4*I*c) + 6*a*exp(2*I*c) + a)*exp(-5
*I*c)/64, True))

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Giac [B]  time = 1.21458, size = 329, normalized size = 4.33 \begin{align*} -\frac{{\left (1015 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 700 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 1015 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 700 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 315 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 315 \, a e^{\left (5 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 20 i \, a e^{\left (12 i \, d x + 8 i \, c\right )} + 168 i \, a e^{\left (10 i \, d x + 6 i \, c\right )} + 700 i \, a e^{\left (8 i \, d x + 4 i \, c\right )} + 2800 i \, a e^{\left (6 i \, d x + 2 i \, c\right )} - 280 i \, a e^{\left (2 i \, d x - 2 i \, c\right )} - 2100 i \, a e^{\left (4 i \, d x\right )} - 28 i \, a e^{\left (-4 i \, c\right )}\right )} e^{\left (-5 i \, d x - i \, c\right )}}{8960 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

-1/8960*(1015*a*e^(5*I*d*x + I*c)*log(I*e^(I*d*x + I*c) + 1) + 700*a*e^(5*I*d*x + I*c)*log(I*e^(I*d*x + I*c) -
1) - 1015*a*e^(5*I*d*x + I*c)*log(-I*e^(I*d*x + I*c) + 1) - 700*a*e^(5*I*d*x + I*c)*log(-I*e^(I*d*x + I*c) -
1) - 315*a*e^(5*I*d*x + I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 315*a*e^(5*I*d*x + I*c)*log(-I*e^(I*d*x) + e^(-I*c)
) + 20*I*a*e^(12*I*d*x + 8*I*c) + 168*I*a*e^(10*I*d*x + 6*I*c) + 700*I*a*e^(8*I*d*x + 4*I*c) + 2800*I*a*e^(6*I
*d*x + 2*I*c) - 280*I*a*e^(2*I*d*x - 2*I*c) - 2100*I*a*e^(4*I*d*x) - 28*I*a*e^(-4*I*c))*e^(-5*I*d*x - I*c)/d