\(\int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx\) [34]

Optimal result

Integrand size = 24, antiderivative size = 87 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {5 a^2 \sin (c+d x)}{7 d}-\frac {10 a^2 \sin ^3(c+d x)}{21 d}+\frac {a^2 \sin ^5(c+d x)}{7 d}-\frac {2 i \cos ^7(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d} \]

[Out]

5/7*a^2*sin(d*x+c)/d-10/21*a^2*sin(d*x+c)^3/d+1/7*a^2*sin(d*x+c)^5/d-2/7*I*cos(d*x+c)^7*(a^2+I*a^2*tan(d*x+c))
/d

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3577, 2713} \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {a^2 \sin ^5(c+d x)}{7 d}-\frac {10 a^2 \sin ^3(c+d x)}{21 d}+\frac {5 a^2 \sin (c+d x)}{7 d}-\frac {2 i \cos ^7(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d} \]

[In]

Int[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^2,x]

[Out]

(5*a^2*Sin[c + d*x])/(7*d) - (10*a^2*Sin[c + d*x]^3)/(21*d) + (a^2*Sin[c + d*x]^5)/(7*d) - (((2*I)/7)*Cos[c +
d*x]^7*(a^2 + I*a^2*Tan[c + d*x]))/d

Rule 2713

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]

Rule 3577

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[2*b*(d
*Sec[e + f*x])^m*((a + b*Tan[e + f*x])^(n - 1)/(f*m)), x] - Dist[b^2*((m + 2*n - 2)/(d^2*m)), Int[(d*Sec[e + f
*x])^(m + 2)*(a + b*Tan[e + f*x])^(n - 2), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n,
1] && ((IGtQ[n/2, 0] && ILtQ[m - 1/2, 0]) || (EqQ[n, 2] && LtQ[m, 0]) || (LeQ[m, -1] && GtQ[m + n, 0]) || (ILt
Q[m, 0] && LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && IntegerQ[2*m]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 i \cos ^7(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d}+\frac {1}{7} \left (5 a^2\right ) \int \cos ^5(c+d x) \, dx \\ & = -\frac {2 i \cos ^7(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d}-\frac {\left (5 a^2\right ) \text {Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{7 d} \\ & = \frac {5 a^2 \sin (c+d x)}{7 d}-\frac {10 a^2 \sin ^3(c+d x)}{21 d}+\frac {a^2 \sin ^5(c+d x)}{7 d}-\frac {2 i \cos ^7(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 i a^2 \cos ^7(c+d x)}{7 d}+\frac {a^2 \sin (c+d x)}{d}-\frac {4 a^2 \sin ^3(c+d x)}{3 d}+\frac {a^2 \sin ^5(c+d x)}{d}-\frac {2 a^2 \sin ^7(c+d x)}{7 d} \]

[In]

Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^2,x]

[Out]

(((-2*I)/7)*a^2*Cos[c + d*x]^7)/d + (a^2*Sin[c + d*x])/d - (4*a^2*Sin[c + d*x]^3)/(3*d) + (a^2*Sin[c + d*x]^5)
/d - (2*a^2*Sin[c + d*x]^7)/(7*d)

Maple [A] (verified)

Time = 49.18 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.17

method	result	size
risch	\(-\frac {i a^{2} {\mathrm e}^{7 i \left (d x +c \right )}}{224 d}-\frac {i a^{2} {\mathrm e}^{5 i \left (d x +c \right )}}{32 d}-\frac {5 i a^{2} \cos \left (d x +c \right )}{32 d}+\frac {15 a^{2} \sin \left (d x +c \right )}{32 d}-\frac {3 i a^{2} \cos \left (3 d x +3 c \right )}{32 d}+\frac {11 a^{2} \sin \left (3 d x +3 c \right )}{96 d}\)	\(102\)
derivativedivides	\(\frac {-a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {2 i a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+\frac {a^{2} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\)	\(111\)
default	\(\frac {-a^{2} \left (-\frac {\sin \left (d x +c \right ) \left (\cos ^{6}\left (d x +c \right )\right )}{7}+\frac {\left (\frac {8}{3}+\cos ^{4}\left (d x +c \right )+\frac {4 \left (\cos ^{2}\left (d x +c \right )\right )}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {2 i a^{2} \left (\cos ^{7}\left (d x +c \right )\right )}{7}+\frac {a^{2} \left (\frac {16}{5}+\cos ^{6}\left (d x +c \right )+\frac {6 \left (\cos ^{4}\left (d x +c \right )\right )}{5}+\frac {8 \left (\cos ^{2}\left (d x +c \right )\right )}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\)	\(111\)

[In]

int(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)

[Out]

-1/224*I/d*a^2*exp(7*I*(d*x+c))-1/32*I/d*a^2*exp(5*I*(d*x+c))-5/32*I/d*a^2*cos(d*x+c)+15/32*a^2*sin(d*x+c)/d-3
/32*I/d*a^2*cos(3*d*x+3*c)+11/96/d*a^2*sin(3*d*x+3*c)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.03 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {{\left (-3 i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} - 21 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 70 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 210 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 105 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, a^{2}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{672 \, d} \]

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")

[Out]

1/672*(-3*I*a^2*e^(10*I*d*x + 10*I*c) - 21*I*a^2*e^(8*I*d*x + 8*I*c) - 70*I*a^2*e^(6*I*d*x + 6*I*c) - 210*I*a^
2*e^(4*I*d*x + 4*I*c) + 105*I*a^2*e^(2*I*d*x + 2*I*c) + 7*I*a^2)*e^(-3*I*d*x - 3*I*c)/d

Sympy [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 238 vs. \(2 (76) = 152\).

Time = 0.31 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.74 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=\begin {cases} \frac {\left (- 75497472 i a^{2} d^{5} e^{11 i c} e^{7 i d x} - 528482304 i a^{2} d^{5} e^{9 i c} e^{5 i d x} - 1761607680 i a^{2} d^{5} e^{7 i c} e^{3 i d x} - 5284823040 i a^{2} d^{5} e^{5 i c} e^{i d x} + 2642411520 i a^{2} d^{5} e^{3 i c} e^{- i d x} + 176160768 i a^{2} d^{5} e^{i c} e^{- 3 i d x}\right ) e^{- 4 i c}}{16911433728 d^{6}} & \text {for}\: d^{6} e^{4 i c} \neq 0 \\\frac {x \left (a^{2} e^{10 i c} + 5 a^{2} e^{8 i c} + 10 a^{2} e^{6 i c} + 10 a^{2} e^{4 i c} + 5 a^{2} e^{2 i c} + a^{2}\right ) e^{- 3 i c}}{32} & \text {otherwise} \end {cases} \]

[In]

integrate(cos(d*x+c)**7*(a+I*a*tan(d*x+c))**2,x)

[Out]

Piecewise(((-75497472*I*a**2*d**5*exp(11*I*c)*exp(7*I*d*x) - 528482304*I*a**2*d**5*exp(9*I*c)*exp(5*I*d*x) - 1
761607680*I*a**2*d**5*exp(7*I*c)*exp(3*I*d*x) - 5284823040*I*a**2*d**5*exp(5*I*c)*exp(I*d*x) + 2642411520*I*a*
*2*d**5*exp(3*I*c)*exp(-I*d*x) + 176160768*I*a**2*d**5*exp(I*c)*exp(-3*I*d*x))*exp(-4*I*c)/(16911433728*d**6),
Ne(d**6*exp(4*I*c), 0)), (x*(a**2*exp(10*I*c) + 5*a**2*exp(8*I*c) + 10*a**2*exp(6*I*c) + 10*a**2*exp(4*I*c) +
5*a**2*exp(2*I*c) + a**2)*exp(-3*I*c)/32, True))

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.13 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {30 i \, a^{2} \cos \left (d x + c\right )^{7} + {\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} a^{2} + 3 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{2}}{105 \, d} \]

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")

[Out]

-1/105*(30*I*a^2*cos(d*x + c)^7 + (15*sin(d*x + c)^7 - 42*sin(d*x + c)^5 + 35*sin(d*x + c)^3)*a^2 + 3*(5*sin(d
*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*a^2)/d

Giac [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (75) = 150\).

Time = 0.67 (sec) , antiderivative size = 641, normalized size of antiderivative = 7.37 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2583 \, a^{2} e^{\left (7 i \, d x + 3 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 5166 \, a^{2} e^{\left (5 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 2583 \, a^{2} e^{\left (3 i \, d x - i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) + 2121 \, a^{2} e^{\left (7 i \, d x + 3 i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 4242 \, a^{2} e^{\left (5 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) + 2121 \, a^{2} e^{\left (3 i \, d x - i \, c\right )} \log \left (i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 2583 \, a^{2} e^{\left (7 i \, d x + 3 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 5166 \, a^{2} e^{\left (5 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 2583 \, a^{2} e^{\left (3 i \, d x - i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} + 1\right ) - 2121 \, a^{2} e^{\left (7 i \, d x + 3 i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 4242 \, a^{2} e^{\left (5 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 2121 \, a^{2} e^{\left (3 i \, d x - i \, c\right )} \log \left (-i \, e^{\left (i \, d x + i \, c\right )} - 1\right ) - 462 \, a^{2} e^{\left (7 i \, d x + 3 i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) - 924 \, a^{2} e^{\left (5 i \, d x + i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) - 462 \, a^{2} e^{\left (3 i \, d x - i \, c\right )} \log \left (i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 462 \, a^{2} e^{\left (7 i \, d x + 3 i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 924 \, a^{2} e^{\left (5 i \, d x + i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 462 \, a^{2} e^{\left (3 i \, d x - i \, c\right )} \log \left (-i \, e^{\left (i \, d x\right )} + e^{\left (-i \, c\right )}\right ) + 48 i \, a^{2} e^{\left (14 i \, d x + 10 i \, c\right )} + 432 i \, a^{2} e^{\left (12 i \, d x + 8 i \, c\right )} + 1840 i \, a^{2} e^{\left (10 i \, d x + 6 i \, c\right )} + 5936 i \, a^{2} e^{\left (8 i \, d x + 4 i \, c\right )} + 6160 i \, a^{2} e^{\left (6 i \, d x + 2 i \, c\right )} - 1904 i \, a^{2} e^{\left (2 i \, d x - 2 i \, c\right )} - 112 i \, a^{2} e^{\left (4 i \, d x\right )} - 112 i \, a^{2} e^{\left (-4 i \, c\right )}}{10752 \, {\left (d e^{\left (7 i \, d x + 3 i \, c\right )} + 2 \, d e^{\left (5 i \, d x + i \, c\right )} + d e^{\left (3 i \, d x - i \, c\right )}\right )}} \]

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^2,x, algorithm="giac")

[Out]

-1/10752*(2583*a^2*e^(7*I*d*x + 3*I*c)*log(I*e^(I*d*x + I*c) + 1) + 5166*a^2*e^(5*I*d*x + I*c)*log(I*e^(I*d*x
+ I*c) + 1) + 2583*a^2*e^(3*I*d*x - I*c)*log(I*e^(I*d*x + I*c) + 1) + 2121*a^2*e^(7*I*d*x + 3*I*c)*log(I*e^(I*
d*x + I*c) - 1) + 4242*a^2*e^(5*I*d*x + I*c)*log(I*e^(I*d*x + I*c) - 1) + 2121*a^2*e^(3*I*d*x - I*c)*log(I*e^(
I*d*x + I*c) - 1) - 2583*a^2*e^(7*I*d*x + 3*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 5166*a^2*e^(5*I*d*x + I*c)*log(
-I*e^(I*d*x + I*c) + 1) - 2583*a^2*e^(3*I*d*x - I*c)*log(-I*e^(I*d*x + I*c) + 1) - 2121*a^2*e^(7*I*d*x + 3*I*c
)*log(-I*e^(I*d*x + I*c) - 1) - 4242*a^2*e^(5*I*d*x + I*c)*log(-I*e^(I*d*x + I*c) - 1) - 2121*a^2*e^(3*I*d*x -
I*c)*log(-I*e^(I*d*x + I*c) - 1) - 462*a^2*e^(7*I*d*x + 3*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 924*a^2*e^(5*I*d
*x + I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 462*a^2*e^(3*I*d*x - I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 462*a^2*e^(7*I
*d*x + 3*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 924*a^2*e^(5*I*d*x + I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 462*a^2*
e^(3*I*d*x - I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 48*I*a^2*e^(14*I*d*x + 10*I*c) + 432*I*a^2*e^(12*I*d*x + 8*I*
c) + 1840*I*a^2*e^(10*I*d*x + 6*I*c) + 5936*I*a^2*e^(8*I*d*x + 4*I*c) + 6160*I*a^2*e^(6*I*d*x + 2*I*c) - 1904*
I*a^2*e^(2*I*d*x - 2*I*c) - 112*I*a^2*e^(4*I*d*x) - 112*I*a^2*e^(-4*I*c))/(d*e^(7*I*d*x + 3*I*c) + 2*d*e^(5*I*
d*x + I*c) + d*e^(3*I*d*x - I*c))

Mupad [B] (verification not implemented)

Time = 4.10 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.94 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {2\,a^2\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-2{}\mathrm {i}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {256\,a^2\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\mathrm {i}\right )}{7\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7}-\frac {8\,a^2\,\left (4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-9{}\mathrm {i}\right )}{3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2}-\frac {128\,a^2\,\left (6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-7{}\mathrm {i}\right )}{7\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6}+\frac {16\,a^2\,\left (8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-15{}\mathrm {i}\right )}{3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3}-\frac {32\,a^2\,\left (22\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-35{}\mathrm {i}\right )}{7\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4}+\frac {32\,a^2\,\left (31\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-42{}\mathrm {i}\right )}{7\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \]

[In]

int(cos(c + d*x)^7*(a + a*tan(c + d*x)*1i)^2,x)

[Out]

(2*a^2*(tan(c/2 + (d*x)/2) - 2i))/(d*(tan(c/2 + (d*x)/2)^2 + 1)) + (256*a^2*(tan(c/2 + (d*x)/2) - 1i))/(7*d*(t
an(c/2 + (d*x)/2)^2 + 1)^7) - (8*a^2*(4*tan(c/2 + (d*x)/2) - 9i))/(3*d*(tan(c/2 + (d*x)/2)^2 + 1)^2) - (128*a^
2*(6*tan(c/2 + (d*x)/2) - 7i))/(7*d*(tan(c/2 + (d*x)/2)^2 + 1)^6) + (16*a^2*(8*tan(c/2 + (d*x)/2) - 15i))/(3*d
*(tan(c/2 + (d*x)/2)^2 + 1)^3) - (32*a^2*(22*tan(c/2 + (d*x)/2) - 35i))/(7*d*(tan(c/2 + (d*x)/2)^2 + 1)^4) + (
32*a^2*(31*tan(c/2 + (d*x)/2) - 42i))/(7*d*(tan(c/2 + (d*x)/2)^2 + 1)^5)