Integrand size = 24, antiderivative size = 90 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {a^3 x}{8}-\frac {i a^6}{6 d (a-i a \tan (c+d x))^3}-\frac {i a^5}{8 d (a-i a \tan (c+d x))^2}-\frac {i a^4}{8 d (a-i a \tan (c+d x))} \]
1/8*a^3*x-1/6*I*a^6/d/(a-I*a*tan(d*x+c))^3-1/8*I*a^5/d/(a-I*a*tan(d*x+c))^ 2-1/8*I*a^4/d/(a-I*a*tan(d*x+c))
Time = 0.24 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.72 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {a^3 \left (-10+9 i \tan (c+d x)+3 \tan ^2(c+d x)+3 \arctan (\tan (c+d x)) (i+\tan (c+d x))^3\right )}{24 d (i+\tan (c+d x))^3} \]
(a^3*(-10 + (9*I)*Tan[c + d*x] + 3*Tan[c + d*x]^2 + 3*ArcTan[Tan[c + d*x]] *(I + Tan[c + d*x])^3))/(24*d*(I + Tan[c + d*x])^3)
Time = 0.28 (sec) , antiderivative size = 93, normalized size of antiderivative = 1.03, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {3042, 3968, 54, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (c+d x))^3}{\sec (c+d x)^6}dx\) |
\(\Big \downarrow \) 3968 |
\(\displaystyle -\frac {i a^7 \int \frac {1}{(a-i a \tan (c+d x))^4 (i \tan (c+d x) a+a)}d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 54 |
\(\displaystyle -\frac {i a^7 \int \left (\frac {1}{2 (a-i a \tan (c+d x))^4 a}+\frac {1}{4 (a-i a \tan (c+d x))^3 a^2}+\frac {1}{8 \left (\tan ^2(c+d x) a^2+a^2\right ) a^3}+\frac {1}{8 (a-i a \tan (c+d x))^2 a^3}\right )d(i a \tan (c+d x))}{d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {i a^7 \left (\frac {i \arctan (\tan (c+d x))}{8 a^4}+\frac {1}{8 a^3 (a-i a \tan (c+d x))}+\frac {1}{8 a^2 (a-i a \tan (c+d x))^2}+\frac {1}{6 a (a-i a \tan (c+d x))^3}\right )}{d}\) |
((-I)*a^7*(((I/8)*ArcTan[Tan[c + d*x]])/a^4 + 1/(6*a*(a - I*a*Tan[c + d*x] )^3) + 1/(8*a^2*(a - I*a*Tan[c + d*x])^2) + 1/(8*a^3*(a - I*a*Tan[c + d*x] ))))/d
3.1.43.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[E xpandIntegrand[(a + b*x)^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && ILtQ[m, 0] && IntegerQ[n] && !(IGtQ[n, 0] && LtQ[m + n + 2, 0])
Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[1/(a^(m - 2)*b*f) Subst[Int[(a - x)^(m/2 - 1)*(a + x )^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]
Time = 47.77 (sec) , antiderivative size = 62, normalized size of antiderivative = 0.69
method | result | size |
risch | \(\frac {a^{3} x}{8}-\frac {i a^{3} {\mathrm e}^{6 i \left (d x +c \right )}}{48 d}-\frac {3 i a^{3} {\mathrm e}^{4 i \left (d x +c \right )}}{32 d}-\frac {3 i a^{3} {\mathrm e}^{2 i \left (d x +c \right )}}{16 d}\) | \(62\) |
derivativedivides | \(\frac {-i a^{3} \left (-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )-3 a^{3} \left (-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {i a^{3} \left (\cos ^{6}\left (d x +c \right )\right )}{2}+a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(156\) |
default | \(\frac {-i a^{3} \left (-\frac {\left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{6}-\frac {\left (\cos ^{4}\left (d x +c \right )\right )}{12}\right )-3 a^{3} \left (-\frac {\left (\cos ^{5}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{6}+\frac {\left (\cos ^{3}\left (d x +c \right )+\frac {3 \cos \left (d x +c \right )}{2}\right ) \sin \left (d x +c \right )}{24}+\frac {d x}{16}+\frac {c}{16}\right )-\frac {i a^{3} \left (\cos ^{6}\left (d x +c \right )\right )}{2}+a^{3} \left (\frac {\left (\cos ^{5}\left (d x +c \right )+\frac {5 \left (\cos ^{3}\left (d x +c \right )\right )}{4}+\frac {15 \cos \left (d x +c \right )}{8}\right ) \sin \left (d x +c \right )}{6}+\frac {5 d x}{16}+\frac {5 c}{16}\right )}{d}\) | \(156\) |
1/8*a^3*x-1/48*I/d*a^3*exp(6*I*(d*x+c))-3/32*I/d*a^3*exp(4*I*(d*x+c))-3/16 *I/d*a^3*exp(2*I*(d*x+c))
Time = 0.23 (sec) , antiderivative size = 55, normalized size of antiderivative = 0.61 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {12 \, a^{3} d x - 2 i \, a^{3} e^{\left (6 i \, d x + 6 i \, c\right )} - 9 i \, a^{3} e^{\left (4 i \, d x + 4 i \, c\right )} - 18 i \, a^{3} e^{\left (2 i \, d x + 2 i \, c\right )}}{96 \, d} \]
1/96*(12*a^3*d*x - 2*I*a^3*e^(6*I*d*x + 6*I*c) - 9*I*a^3*e^(4*I*d*x + 4*I* c) - 18*I*a^3*e^(2*I*d*x + 2*I*c))/d
Time = 0.20 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.46 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {a^{3} x}{8} + \begin {cases} \frac {- 512 i a^{3} d^{2} e^{6 i c} e^{6 i d x} - 2304 i a^{3} d^{2} e^{4 i c} e^{4 i d x} - 4608 i a^{3} d^{2} e^{2 i c} e^{2 i d x}}{24576 d^{3}} & \text {for}\: d^{3} \neq 0 \\x \left (\frac {a^{3} e^{6 i c}}{8} + \frac {3 a^{3} e^{4 i c}}{8} + \frac {3 a^{3} e^{2 i c}}{8}\right ) & \text {otherwise} \end {cases} \]
a**3*x/8 + Piecewise(((-512*I*a**3*d**2*exp(6*I*c)*exp(6*I*d*x) - 2304*I*a **3*d**2*exp(4*I*c)*exp(4*I*d*x) - 4608*I*a**3*d**2*exp(2*I*c)*exp(2*I*d*x ))/(24576*d**3), Ne(d**3, 0)), (x*(a**3*exp(6*I*c)/8 + 3*a**3*exp(4*I*c)/8 + 3*a**3*exp(2*I*c)/8), True))
Time = 0.46 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.17 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {3 \, {\left (d x + c\right )} a^{3} + \frac {3 \, a^{3} \tan \left (d x + c\right )^{5} + 8 \, a^{3} \tan \left (d x + c\right )^{3} + 6 i \, a^{3} \tan \left (d x + c\right )^{2} + 21 \, a^{3} \tan \left (d x + c\right ) - 10 i \, a^{3}}{\tan \left (d x + c\right )^{6} + 3 \, \tan \left (d x + c\right )^{4} + 3 \, \tan \left (d x + c\right )^{2} + 1}}{24 \, d} \]
1/24*(3*(d*x + c)*a^3 + (3*a^3*tan(d*x + c)^5 + 8*a^3*tan(d*x + c)^3 + 6*I *a^3*tan(d*x + c)^2 + 21*a^3*tan(d*x + c) - 10*I*a^3)/(tan(d*x + c)^6 + 3* tan(d*x + c)^4 + 3*tan(d*x + c)^2 + 1))/d
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 457 vs. \(2 (70) = 140\).
Time = 0.81 (sec) , antiderivative size = 457, normalized size of antiderivative = 5.08 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {12 \, a^{3} d x e^{\left (8 i \, d x + 4 i \, c\right )} + 48 \, a^{3} d x e^{\left (6 i \, d x + 2 i \, c\right )} + 48 \, a^{3} d x e^{\left (2 i \, d x - 2 i \, c\right )} + 72 \, a^{3} d x e^{\left (4 i \, d x\right )} + 12 \, a^{3} d x e^{\left (-4 i \, c\right )} - 3 i \, a^{3} e^{\left (8 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 i \, a^{3} e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 12 i \, a^{3} e^{\left (2 i \, d x - 2 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 18 i \, a^{3} e^{\left (4 i \, d x\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 3 i \, a^{3} e^{\left (-4 i \, c\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) + 3 i \, a^{3} e^{\left (8 i \, d x + 4 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 12 i \, a^{3} e^{\left (6 i \, d x + 2 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 12 i \, a^{3} e^{\left (2 i \, d x - 2 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 18 i \, a^{3} e^{\left (4 i \, d x\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) + 3 i \, a^{3} e^{\left (-4 i \, c\right )} \log \left (e^{\left (2 i \, d x\right )} + e^{\left (-2 i \, c\right )}\right ) - 2 i \, a^{3} e^{\left (14 i \, d x + 10 i \, c\right )} - 17 i \, a^{3} e^{\left (12 i \, d x + 8 i \, c\right )} - 66 i \, a^{3} e^{\left (10 i \, d x + 6 i \, c\right )} - 134 i \, a^{3} e^{\left (8 i \, d x + 4 i \, c\right )} - 146 i \, a^{3} e^{\left (6 i \, d x + 2 i \, c\right )} - 18 i \, a^{3} e^{\left (2 i \, d x - 2 i \, c\right )} - 81 i \, a^{3} e^{\left (4 i \, d x\right )}}{96 \, {\left (d e^{\left (8 i \, d x + 4 i \, c\right )} + 4 \, d e^{\left (6 i \, d x + 2 i \, c\right )} + 4 \, d e^{\left (2 i \, d x - 2 i \, c\right )} + 6 \, d e^{\left (4 i \, d x\right )} + d e^{\left (-4 i \, c\right )}\right )}} \]
1/96*(12*a^3*d*x*e^(8*I*d*x + 4*I*c) + 48*a^3*d*x*e^(6*I*d*x + 2*I*c) + 48 *a^3*d*x*e^(2*I*d*x - 2*I*c) + 72*a^3*d*x*e^(4*I*d*x) + 12*a^3*d*x*e^(-4*I *c) - 3*I*a^3*e^(8*I*d*x + 4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 12*I*a^3* e^(6*I*d*x + 2*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 12*I*a^3*e^(2*I*d*x - 2 *I*c)*log(e^(2*I*d*x + 2*I*c) + 1) - 18*I*a^3*e^(4*I*d*x)*log(e^(2*I*d*x + 2*I*c) + 1) - 3*I*a^3*e^(-4*I*c)*log(e^(2*I*d*x + 2*I*c) + 1) + 3*I*a^3*e ^(8*I*d*x + 4*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 12*I*a^3*e^(6*I*d*x + 2 *I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) + 12*I*a^3*e^(2*I*d*x - 2*I*c)*log(e^( 2*I*d*x) + e^(-2*I*c)) + 18*I*a^3*e^(4*I*d*x)*log(e^(2*I*d*x) + e^(-2*I*c) ) + 3*I*a^3*e^(-4*I*c)*log(e^(2*I*d*x) + e^(-2*I*c)) - 2*I*a^3*e^(14*I*d*x + 10*I*c) - 17*I*a^3*e^(12*I*d*x + 8*I*c) - 66*I*a^3*e^(10*I*d*x + 6*I*c) - 134*I*a^3*e^(8*I*d*x + 4*I*c) - 146*I*a^3*e^(6*I*d*x + 2*I*c) - 18*I*a^ 3*e^(2*I*d*x - 2*I*c) - 81*I*a^3*e^(4*I*d*x))/(d*e^(8*I*d*x + 4*I*c) + 4*d *e^(6*I*d*x + 2*I*c) + 4*d*e^(2*I*d*x - 2*I*c) + 6*d*e^(4*I*d*x) + d*e^(-4 *I*c))
Time = 4.15 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.86 \[ \int \cos ^6(c+d x) (a+i a \tan (c+d x))^3 \, dx=\frac {a^3\,x}{8}-\frac {\frac {a^3\,{\mathrm {tan}\left (c+d\,x\right )}^2}{8}+\frac {a^3\,\mathrm {tan}\left (c+d\,x\right )\,3{}\mathrm {i}}{8}-\frac {5\,a^3}{12}}{d\,\left (-{\mathrm {tan}\left (c+d\,x\right )}^3-{\mathrm {tan}\left (c+d\,x\right )}^2\,3{}\mathrm {i}+3\,\mathrm {tan}\left (c+d\,x\right )+1{}\mathrm {i}\right )} \]