Integrand size = 24, antiderivative size = 87 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {5 a^2 \sin (c+d x)}{7 d}-\frac {10 a^2 \sin ^3(c+d x)}{21 d}+\frac {a^2 \sin ^5(c+d x)}{7 d}-\frac {2 i \cos ^7(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d} \] Output:
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
Time = 0.12 (sec) , antiderivative size = 85, normalized size of antiderivative = 0.98 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {2 i a^2 \cos ^7(c+d x)}{7 d}+\frac {a^2 \sin (c+d x)}{d}-\frac {4 a^2 \sin ^3(c+d x)}{3 d}+\frac {a^2 \sin ^5(c+d x)}{d}-\frac {2 a^2 \sin ^7(c+d x)}{7 d} \] Input:
Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^2,x]
Output:
(((-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)
Time = 0.32 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.90, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {3042, 3977, 3042, 3113, 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.
\(\displaystyle \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (c+d x))^2}{\sec (c+d x)^7}dx\) |
\(\Big \downarrow \) 3977 |
\(\displaystyle \frac {5}{7} a^2 \int \cos ^5(c+d x)dx-\frac {2 i \cos ^7(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {5}{7} a^2 \int \sin \left (c+d x+\frac {\pi }{2}\right )^5dx-\frac {2 i \cos ^7(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d}\) |
\(\Big \downarrow \) 3113 |
\(\displaystyle -\frac {5 a^2 \int \left (\sin ^4(c+d x)-2 \sin ^2(c+d x)+1\right )d(-\sin (c+d x))}{7 d}-\frac {2 i \cos ^7(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle -\frac {5 a^2 \left (-\frac {1}{5} \sin ^5(c+d x)+\frac {2}{3} \sin ^3(c+d x)-\sin (c+d x)\right )}{7 d}-\frac {2 i \cos ^7(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d}\) |
Input:
Int[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^2,x]
Output:
(-5*a^2*(-Sin[c + d*x] + (2*Sin[c + d*x]^3)/3 - Sin[c + d*x]^5/5))/(7*d) - (((2*I)/7)*Cos[c + d*x]^7*(a^2 + I*a^2*Tan[c + d*x]))/d
Int[sin[(c_.) + (d_.)*(x_)]^(n_), x_Symbol] :> Simp[-d^(-1) Subst[Int[Exp and[(1 - x^2)^((n - 1)/2), x], x], x, Cos[c + d*x]], x] /; FreeQ[{c, d}, x] && IGtQ[(n - 1)/2, 0]
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] - Simp[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]) || (ILtQ[m, 0] & & LtQ[m/2 + n - 1, 0] && IntegerQ[n]) || (EqQ[n, 3/2] && EqQ[m, -2^(-1)])) && IntegerQ[2*m]
Time = 44.04 (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 | \(\frac {-a^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{7}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {2 i a^{2} \cos \left (d x +c \right )^{7}}{7}+\frac {a^{2} \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\) | \(111\) |
default | \(\frac {-a^{2} \left (-\frac {\sin \left (d x +c \right ) \cos \left (d x +c \right )^{6}}{7}+\frac {\left (\frac {8}{3}+\cos \left (d x +c \right )^{4}+\frac {4 \cos \left (d x +c \right )^{2}}{3}\right ) \sin \left (d x +c \right )}{35}\right )-\frac {2 i a^{2} \cos \left (d x +c \right )^{7}}{7}+\frac {a^{2} \left (\frac {16}{5}+\cos \left (d x +c \right )^{6}+\frac {6 \cos \left (d x +c \right )^{4}}{5}+\frac {8 \cos \left (d x +c \right )^{2}}{5}\right ) \sin \left (d x +c \right )}{7}}{d}\) | \(111\) |
Input:
int(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^2,x,method=_RETURNVERBOSE)
Output:
-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)
Time = 0.08 (sec) , antiderivative size = 90, normalized size of antiderivative = 1.03 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {{\left (-3 i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} - 21 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 70 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} - 210 i \, a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + 105 i \, a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + 7 i \, a^{2}\right )} e^{\left (-3 i \, d x - 3 i \, c\right )}}{672 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^2,x, algorithm="fricas")
Output:
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
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 = 0.34 (sec) , antiderivative size = 238, normalized size of antiderivative = 2.74 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=\begin {cases} \frac {\left (- 75497472 i a^{2} d^{5} e^{11 i c} e^{7 i d x} - 528482304 i a^{2} d^{5} e^{9 i c} e^{5 i d x} - 1761607680 i a^{2} d^{5} e^{7 i c} e^{3 i d x} - 5284823040 i a^{2} d^{5} e^{5 i c} e^{i d x} + 2642411520 i a^{2} d^{5} e^{3 i c} e^{- i d x} + 176160768 i a^{2} d^{5} e^{i c} e^{- 3 i d x}\right ) e^{- 4 i c}}{16911433728 d^{6}} & \text {for}\: d^{6} e^{4 i c} \neq 0 \\\frac {x \left (a^{2} e^{10 i c} + 5 a^{2} e^{8 i c} + 10 a^{2} e^{6 i c} + 10 a^{2} e^{4 i c} + 5 a^{2} e^{2 i c} + a^{2}\right ) e^{- 3 i c}}{32} & \text {otherwise} \end {cases} \] Input:
integrate(cos(d*x+c)**7*(a+I*a*tan(d*x+c))**2,x)
Output:
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) - 1761607680*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(1 0*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))
Time = 0.04 (sec) , antiderivative size = 98, normalized size of antiderivative = 1.13 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=-\frac {30 i \, a^{2} \cos \left (d x + c\right )^{7} + {\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} a^{2} + 3 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{2}}{105 \, d} \] Input:
integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^2,x, algorithm="maxima")
Output:
-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
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 = 0.32 (sec) , antiderivative size = 641, normalized size of antiderivative = 7.37 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx =\text {Too large to display} \] Input:
integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^2,x, algorithm="giac")
Output:
-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) + 21 21*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))
Time = 0.78 (sec) , antiderivative size = 256, normalized size of antiderivative = 2.94 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {2\,a^2\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-2{}\mathrm {i}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}+\frac {256\,a^2\,\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-\mathrm {i}\right )}{7\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^7}-\frac {8\,a^2\,\left (4\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-9{}\mathrm {i}\right )}{3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^2}-\frac {128\,a^2\,\left (6\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-7{}\mathrm {i}\right )}{7\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^6}+\frac {16\,a^2\,\left (8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-15{}\mathrm {i}\right )}{3\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^3}-\frac {32\,a^2\,\left (22\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-35{}\mathrm {i}\right )}{7\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^4}+\frac {32\,a^2\,\left (31\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-42{}\mathrm {i}\right )}{7\,d\,{\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )}^5} \] Input:
int(cos(c + d*x)^7*(a + a*tan(c + d*x)*1i)^2,x)
Output:
(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*(tan(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)
Time = 0.17 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.26 \[ \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx=\frac {a^{2} \left (6 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{6} i -18 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{4} i +18 \cos \left (d x +c \right ) \sin \left (d x +c \right )^{2} i -6 \cos \left (d x +c \right ) i -6 \sin \left (d x +c \right )^{7}+21 \sin \left (d x +c \right )^{5}-28 \sin \left (d x +c \right )^{3}+21 \sin \left (d x +c \right )+6 i \right )}{21 d} \] Input:
int(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^2,x)
Output:
(a**2*(6*cos(c + d*x)*sin(c + d*x)**6*i - 18*cos(c + d*x)*sin(c + d*x)**4* i + 18*cos(c + d*x)*sin(c + d*x)**2*i - 6*cos(c + d*x)*i - 6*sin(c + d*x)* *7 + 21*sin(c + d*x)**5 - 28*sin(c + d*x)**3 + 21*sin(c + d*x) + 6*i))/(21 *d)