3.95 \(\int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx\)
Optimal. Leaf size=66 \[ -\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d}-\frac{i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{63 d} \]
[Out]
((-I/63)*a*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^7)/d - ((I/9)*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^8)/d
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Rubi [A] time = 0.073501, antiderivative size = 66, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used =
{3497, 3488} \[ -\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d}-\frac{i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{63 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^8,x]
[Out]
((-I/63)*a*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^7)/d - ((I/9)*Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^8)/d
Rule 3497
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(d*
Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(a*f*m), x] + Dist[(a*(m + n))/(m*d^2), Int[(d*Sec[e + f*x])^(m + 2)*(
a + b*Tan[e + f*x])^(n - 1), x], x] /; FreeQ[{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && GtQ[n, 0] && LtQ[m, -
1] && IntegersQ[2*m, 2*n]
Rule 3488
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(a*f*m), x] /; FreeQ[{a, b, d, e, f, m, n}, x] && EqQ[a^2 + b^2, 0] &
& EqQ[Simplify[m + n], 0]
Rubi steps
\begin{align*} \int \cos ^9(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d}+\frac{1}{9} a \int \cos ^7(c+d x) (a+i a \tan (c+d x))^7 \, dx\\ &=-\frac{i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{63 d}-\frac{i \cos ^9(c+d x) (a+i a \tan (c+d x))^8}{9 d}\\ \end{align*}
Mathematica [A] time = 0.565298, size = 50, normalized size = 0.76 \[ \frac{a^8 (8 \cos (c+d x)-i \sin (c+d x)) (\sin (8 (c+d x))-i \cos (8 (c+d x)))}{63 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^8,x]
[Out]
(a^8*(8*Cos[c + d*x] - I*Sin[c + d*x])*((-I)*Cos[8*(c + d*x)] + Sin[8*(c + d*x)]))/(63*d)
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Maple [B] time = 0.094, size = 447, normalized size = 6.8 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^8,x)
[Out]
1/d*(1/9*a^8*sin(d*x+c)^9-8*I*a^8*(-1/9*sin(d*x+c)^6*cos(d*x+c)^3-2/21*sin(d*x+c)^4*cos(d*x+c)^3-8/105*cos(d*x
+c)^3*sin(d*x+c)^2-16/315*cos(d*x+c)^3)-28*a^8*(-1/9*sin(d*x+c)^5*cos(d*x+c)^4-5/63*sin(d*x+c)^3*cos(d*x+c)^4-
1/21*sin(d*x+c)*cos(d*x+c)^4+1/63*(2+cos(d*x+c)^2)*sin(d*x+c))+56*I*a^8*(-1/9*sin(d*x+c)^4*cos(d*x+c)^5-4/63*s
in(d*x+c)^2*cos(d*x+c)^5-8/315*cos(d*x+c)^5)+70*a^8*(-1/9*sin(d*x+c)^3*cos(d*x+c)^6-1/21*cos(d*x+c)^6*sin(d*x+
c)+1/105*(8/3+cos(d*x+c)^4+4/3*cos(d*x+c)^2)*sin(d*x+c))-56*I*a^8*(-1/9*sin(d*x+c)^2*cos(d*x+c)^7-2/63*cos(d*x
+c)^7)-28*a^8*(-1/9*sin(d*x+c)*cos(d*x+c)^8+1/63*(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))-8/9*I*a^8*cos(d*x+c)^9+1/9*a^8*(128/35+cos(d*x+c)^8+8/7*cos(d*x+c)^6+48/35*cos(d*x+c)^4+64/35*cos(d*x+c)^
2)*sin(d*x+c))
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Maxima [B] time = 1.14898, size = 408, normalized size = 6.18 \begin{align*} -\frac{280 i \, a^{8} \cos \left (d x + c\right )^{9} - 35 \, a^{8} \sin \left (d x + c\right )^{9} + 56 i \,{\left (35 \, \cos \left (d x + c\right )^{9} - 90 \, \cos \left (d x + c\right )^{7} + 63 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 8 i \,{\left (35 \, \cos \left (d x + c\right )^{9} - 135 \, \cos \left (d x + c\right )^{7} + 189 \, \cos \left (d x + c\right )^{5} - 105 \, \cos \left (d x + c\right )^{3}\right )} a^{8} + 280 i \,{\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{8} - 70 \,{\left (35 \, \sin \left (d x + c\right )^{9} - 90 \, \sin \left (d x + c\right )^{7} + 63 \, \sin \left (d x + c\right )^{5}\right )} a^{8} - 28 \,{\left (35 \, \sin \left (d x + c\right )^{9} - 135 \, \sin \left (d x + c\right )^{7} + 189 \, \sin \left (d x + c\right )^{5} - 105 \, \sin \left (d x + c\right )^{3}\right )} a^{8} -{\left (35 \, \sin \left (d x + c\right )^{9} - 180 \, \sin \left (d x + c\right )^{7} + 378 \, \sin \left (d x + c\right )^{5} - 420 \, \sin \left (d x + c\right )^{3} + 315 \, \sin \left (d x + c\right )\right )} a^{8} - 140 \,{\left (7 \, \sin \left (d x + c\right )^{9} - 9 \, \sin \left (d x + c\right )^{7}\right )} a^{8}}{315 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")
[Out]
-1/315*(280*I*a^8*cos(d*x + c)^9 - 35*a^8*sin(d*x + c)^9 + 56*I*(35*cos(d*x + c)^9 - 90*cos(d*x + c)^7 + 63*co
s(d*x + c)^5)*a^8 + 8*I*(35*cos(d*x + c)^9 - 135*cos(d*x + c)^7 + 189*cos(d*x + c)^5 - 105*cos(d*x + c)^3)*a^8
+ 280*I*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a^8 - 70*(35*sin(d*x + c)^9 - 90*sin(d*x + c)^7 + 63*sin(d*x +
c)^5)*a^8 - 28*(35*sin(d*x + c)^9 - 135*sin(d*x + c)^7 + 189*sin(d*x + c)^5 - 105*sin(d*x + c)^3)*a^8 - (35*si
n(d*x + c)^9 - 180*sin(d*x + c)^7 + 378*sin(d*x + c)^5 - 420*sin(d*x + c)^3 + 315*sin(d*x + c))*a^8 - 140*(7*s
in(d*x + c)^9 - 9*sin(d*x + c)^7)*a^8)/d
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Fricas [A] time = 1.90622, size = 95, normalized size = 1.44 \begin{align*} \frac{-7 i \, a^{8} e^{\left (9 i \, d x + 9 i \, c\right )} - 9 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )}}{126 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")
[Out]
1/126*(-7*I*a^8*e^(9*I*d*x + 9*I*c) - 9*I*a^8*e^(7*I*d*x + 7*I*c))/d
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Sympy [A] time = 1.21285, size = 82, normalized size = 1.24 \begin{align*} \begin{cases} \frac{- 14 i a^{8} d e^{9 i c} e^{9 i d x} - 18 i a^{8} d e^{7 i c} e^{7 i d x}}{252 d^{2}} & \text{for}\: 252 d^{2} \neq 0 \\x \left (\frac{a^{8} e^{9 i c}}{2} + \frac{a^{8} e^{7 i c}}{2}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**9*(a+I*a*tan(d*x+c))**8,x)
[Out]
Piecewise(((-14*I*a**8*d*exp(9*I*c)*exp(9*I*d*x) - 18*I*a**8*d*exp(7*I*c)*exp(7*I*d*x))/(252*d**2), Ne(252*d**
2, 0)), (x*(a**8*exp(9*I*c)/2 + a**8*exp(7*I*c)/2), True))
________________________________________________________________________________________
Giac [B] time = 3.54493, size = 3309, normalized size = 50.14 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^9*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")
[Out]
1/330301440*(7096716585*a^8*e^(24*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 85160599020*a^8*e^(22*I*d*x + 1
0*I*c)*log(I*e^(I*d*x + I*c) + 1) + 468383294610*a^8*e^(20*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1561277
648700*a^8*e^(18*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 3512874709575*a^8*e^(16*I*d*x + 4*I*c)*log(I*e^(I
*d*x + I*c) + 1) + 5620599535320*a^8*e^(14*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 5620599535320*a^8*e^(10
*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 3512874709575*a^8*e^(8*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1)
+ 1561277648700*a^8*e^(6*I*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 468383294610*a^8*e^(4*I*d*x - 8*I*c)*log(
I*e^(I*d*x + I*c) + 1) + 85160599020*a^8*e^(2*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 6557366124540*a^8*e
^(12*I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 7096716585*a^8*e^(-12*I*c)*log(I*e^(I*d*x + I*c) + 1) + 7095485250*a^
8*e^(24*I*d*x + 12*I*c)*log(I*e^(I*d*x + I*c) - 1) + 85145823000*a^8*e^(22*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*
c) - 1) + 468302026500*a^8*e^(20*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1561006755000*a^8*e^(18*I*d*x + 6
*I*c)*log(I*e^(I*d*x + I*c) - 1) + 3512265198750*a^8*e^(16*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 5619624
318000*a^8*e^(14*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 5619624318000*a^8*e^(10*I*d*x - 2*I*c)*log(I*e^(I
*d*x + I*c) - 1) + 3512265198750*a^8*e^(8*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1561006755000*a^8*e^(6*I
*d*x - 6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 468302026500*a^8*e^(4*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 8
5145823000*a^8*e^(2*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 6556228371000*a^8*e^(12*I*d*x)*log(I*e^(I*d*x
+ I*c) - 1) + 7095485250*a^8*e^(-12*I*c)*log(I*e^(I*d*x + I*c) - 1) - 7096716585*a^8*e^(24*I*d*x + 12*I*c)*lo
g(-I*e^(I*d*x + I*c) + 1) - 85160599020*a^8*e^(22*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 468383294610*a
^8*e^(20*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1561277648700*a^8*e^(18*I*d*x + 6*I*c)*log(-I*e^(I*d*x +
I*c) + 1) - 3512874709575*a^8*e^(16*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 5620599535320*a^8*e^(14*I*d*
x + 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 5620599535320*a^8*e^(10*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) -
3512874709575*a^8*e^(8*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1561277648700*a^8*e^(6*I*d*x - 6*I*c)*log(
-I*e^(I*d*x + I*c) + 1) - 468383294610*a^8*e^(4*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 85160599020*a^8*e
^(2*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 6557366124540*a^8*e^(12*I*d*x)*log(-I*e^(I*d*x + I*c) + 1) -
7096716585*a^8*e^(-12*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 7095485250*a^8*e^(24*I*d*x + 12*I*c)*log(-I*e^(I*d*x
+ I*c) - 1) - 85145823000*a^8*e^(22*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 468302026500*a^8*e^(20*I*d*
x + 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1561006755000*a^8*e^(18*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) -
3512265198750*a^8*e^(16*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 5619624318000*a^8*e^(14*I*d*x + 2*I*c)*lo
g(-I*e^(I*d*x + I*c) - 1) - 5619624318000*a^8*e^(10*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 3512265198750
*a^8*e^(8*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1561006755000*a^8*e^(6*I*d*x - 6*I*c)*log(-I*e^(I*d*x +
I*c) - 1) - 468302026500*a^8*e^(4*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 85145823000*a^8*e^(2*I*d*x - 1
0*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 6556228371000*a^8*e^(12*I*d*x)*log(-I*e^(I*d*x + I*c) - 1) - 7095485250*a
^8*e^(-12*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1231335*a^8*e^(24*I*d*x + 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 1
4776020*a^8*e^(22*I*d*x + 10*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 81268110*a^8*e^(20*I*d*x + 8*I*c)*log(I*e^(I*d
*x) + e^(-I*c)) - 270893700*a^8*e^(18*I*d*x + 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 609510825*a^8*e^(16*I*d*x +
4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 975217320*a^8*e^(14*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 9752173
20*a^8*e^(10*I*d*x - 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 609510825*a^8*e^(8*I*d*x - 4*I*c)*log(I*e^(I*d*x) +
e^(-I*c)) - 270893700*a^8*e^(6*I*d*x - 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 81268110*a^8*e^(4*I*d*x - 8*I*c)*l
og(I*e^(I*d*x) + e^(-I*c)) - 14776020*a^8*e^(2*I*d*x - 10*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 1137753540*a^8*e^
(12*I*d*x)*log(I*e^(I*d*x) + e^(-I*c)) - 1231335*a^8*e^(-12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 1231335*a^8*e^(
24*I*d*x + 12*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 14776020*a^8*e^(22*I*d*x + 10*I*c)*log(-I*e^(I*d*x) + e^(-I*
c)) + 81268110*a^8*e^(20*I*d*x + 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 270893700*a^8*e^(18*I*d*x + 6*I*c)*log(
-I*e^(I*d*x) + e^(-I*c)) + 609510825*a^8*e^(16*I*d*x + 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 975217320*a^8*e^(
14*I*d*x + 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 975217320*a^8*e^(10*I*d*x - 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c
)) + 609510825*a^8*e^(8*I*d*x - 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 270893700*a^8*e^(6*I*d*x - 6*I*c)*log(-I
*e^(I*d*x) + e^(-I*c)) + 81268110*a^8*e^(4*I*d*x - 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 14776020*a^8*e^(2*I*d
*x - 10*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 1137753540*a^8*e^(12*I*d*x)*log(-I*e^(I*d*x) + e^(-I*c)) + 1231335
*a^8*e^(-12*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 18350080*I*a^8*e^(33*I*d*x + 21*I*c) - 243793920*I*a^8*e^(31*I
*d*x + 19*I*c) - 1494220800*I*a^8*e^(29*I*d*x + 17*I*c) - 5594152960*I*a^8*e^(27*I*d*x + 15*I*c) - 14273740800
*I*a^8*e^(25*I*d*x + 13*I*c) - 26211778560*I*a^8*e^(23*I*d*x + 11*I*c) - 35641098240*I*a^8*e^(21*I*d*x + 9*I*c
) - 36333158400*I*a^8*e^(19*I*d*x + 7*I*c) - 27768913920*I*a^8*e^(17*I*d*x + 5*I*c) - 15715532800*I*a^8*e^(15*
I*d*x + 3*I*c) - 6401556480*I*a^8*e^(13*I*d*x + I*c) - 1777336320*I*a^8*e^(11*I*d*x - I*c) - 301465600*I*a^8*e
^(9*I*d*x - 3*I*c) - 23592960*I*a^8*e^(7*I*d*x - 5*I*c))/(d*e^(24*I*d*x + 12*I*c) + 12*d*e^(22*I*d*x + 10*I*c)
+ 66*d*e^(20*I*d*x + 8*I*c) + 220*d*e^(18*I*d*x + 6*I*c) + 495*d*e^(16*I*d*x + 4*I*c) + 792*d*e^(14*I*d*x + 2
*I*c) + 792*d*e^(10*I*d*x - 2*I*c) + 495*d*e^(8*I*d*x - 4*I*c) + 220*d*e^(6*I*d*x - 6*I*c) + 66*d*e^(4*I*d*x -
8*I*c) + 12*d*e^(2*I*d*x - 10*I*c) + 924*d*e^(12*I*d*x) + d*e^(-12*I*c))