3.35 \(\int \cos ^9(c+d x) (a+i a \tan (c+d x))^2 \, dx\)

Optimal. Leaf size=105 \[ -\frac{a^2 \sin ^7(c+d x)}{9 d}+\frac{7 a^2 \sin ^5(c+d x)}{15 d}-\frac{7 a^2 \sin ^3(c+d x)}{9 d}+\frac{7 a^2 \sin (c+d x)}{9 d}-\frac{2 i \cos ^9(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d} \]

[Out]

(7*a^2*Sin[c + d*x])/(9*d) - (7*a^2*Sin[c + d*x]^3)/(9*d) + (7*a^2*Sin[c + d*x]^5)/(15*d) - (a^2*Sin[c + d*x]^
7)/(9*d) - (((2*I)/9)*Cos[c + d*x]^9*(a^2 + I*a^2*Tan[c + d*x]))/d

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Rubi [A]  time = 0.0559433, antiderivative size = 105, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3496, 2633} \[ -\frac{a^2 \sin ^7(c+d x)}{9 d}+\frac{7 a^2 \sin ^5(c+d x)}{15 d}-\frac{7 a^2 \sin ^3(c+d x)}{9 d}+\frac{7 a^2 \sin (c+d x)}{9 d}-\frac{2 i \cos ^9(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^2,x]

[Out]

(7*a^2*Sin[c + d*x])/(9*d) - (7*a^2*Sin[c + d*x]^3)/(9*d) + (7*a^2*Sin[c + d*x]^5)/(15*d) - (a^2*Sin[c + d*x]^
7)/(9*d) - (((2*I)/9)*Cos[c + d*x]^9*(a^2 + I*a^2*Tan[c + d*x]))/d

Rule 3496

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]

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 ^9(c+d x) (a+i a \tan (c+d x))^2 \, dx &=-\frac{2 i \cos ^9(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d}+\frac{1}{9} \left (7 a^2\right ) \int \cos ^7(c+d x) \, dx\\ &=-\frac{2 i \cos ^9(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d}-\frac{\left (7 a^2\right ) \operatorname{Subst}\left (\int \left (1-3 x^2+3 x^4-x^6\right ) \, dx,x,-\sin (c+d x)\right )}{9 d}\\ &=\frac{7 a^2 \sin (c+d x)}{9 d}-\frac{7 a^2 \sin ^3(c+d x)}{9 d}+\frac{7 a^2 \sin ^5(c+d x)}{15 d}-\frac{a^2 \sin ^7(c+d x)}{9 d}-\frac{2 i \cos ^9(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{9 d}\\ \end{align*}

Mathematica [A]  time = 1.12329, size = 133, normalized size = 1.27 \[ \frac{a^2 (-525 \sin (c+d x)+567 \sin (3 (c+d x))+75 \sin (5 (c+d x))+7 \sin (7 (c+d x))-1050 i \cos (c+d x)+378 i \cos (3 (c+d x))+30 i \cos (5 (c+d x))+2 i \cos (7 (c+d x))) (\cos (2 (c+2 d x))+i \sin (2 (c+2 d x)))}{2880 d (\cos (d x)+i \sin (d x))^2} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^2,x]

[Out]

(a^2*((-1050*I)*Cos[c + d*x] + (378*I)*Cos[3*(c + d*x)] + (30*I)*Cos[5*(c + d*x)] + (2*I)*Cos[7*(c + d*x)] - 5
25*Sin[c + d*x] + 567*Sin[3*(c + d*x)] + 75*Sin[5*(c + d*x)] + 7*Sin[7*(c + d*x)])*(Cos[2*(c + 2*d*x)] + I*Sin
[2*(c + 2*d*x)]))/(2880*d*(Cos[d*x] + I*Sin[d*x])^2)

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Maple [A]  time = 0.09, size = 131, normalized size = 1.3 \begin{align*}{\frac{1}{d} \left ( -{a}^{2} \left ( -{\frac{\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{8}}{9}}+{\frac{\sin \left ( dx+c \right ) }{63} \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 ) -{\frac{2\,i}{9}}{a}^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{9}+{\frac{{a}^{2}\sin \left ( dx+c \right ) }{9} \left ({\frac{128}{35}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{8}+{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{7}}+{\frac{48\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{64\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{35}} \right ) } \right ) } \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))^2,x)

[Out]

1/d*(-a^2*(-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))
-2/9*I*a^2*cos(d*x+c)^9+1/9*a^2*(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)*s
in(d*x+c))

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Maxima [A]  time = 1.11045, size = 161, normalized size = 1.53 \begin{align*} -\frac{70 i \, a^{2} \cos \left (d x + c\right )^{9} -{\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^{2} -{\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^{2}}{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))^2,x, algorithm="maxima")

[Out]

-1/315*(70*I*a^2*cos(d*x + c)^9 - (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^2 - (35*sin(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^2)/d

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Fricas [A]  time = 1.17465, size = 365, normalized size = 3.48 \begin{align*} \frac{{\left (-5 i \, a^{2} e^{\left (14 i \, d x + 14 i \, c\right )} - 45 i \, a^{2} e^{\left (12 i \, d x + 12 i \, c\right )} - 189 i \, a^{2} e^{\left (10 i \, d x + 10 i \, c\right )} - 525 i \, a^{2} e^{\left (8 i \, d x + 8 i \, c\right )} - 1575 i \, a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + 945 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 )} + 9 i \, a^{2}\right )} e^{\left (-5 i \, d x - 5 i \, c\right )}}{5760 \, 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))^2,x, algorithm="fricas")

[Out]

1/5760*(-5*I*a^2*e^(14*I*d*x + 14*I*c) - 45*I*a^2*e^(12*I*d*x + 12*I*c) - 189*I*a^2*e^(10*I*d*x + 10*I*c) - 52
5*I*a^2*e^(8*I*d*x + 8*I*c) - 1575*I*a^2*e^(6*I*d*x + 6*I*c) + 945*I*a^2*e^(4*I*d*x + 4*I*c) + 105*I*a^2*e^(2*
I*d*x + 2*I*c) + 9*I*a^2)*e^(-5*I*d*x - 5*I*c)/d

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Sympy [A]  time = 1.35423, size = 316, normalized size = 3.01 \begin{align*} \begin{cases} \frac{\left (- 126663739519795200 i a^{2} d^{7} e^{18 i c} e^{9 i d x} - 1139973655678156800 i a^{2} d^{7} e^{16 i c} e^{7 i d x} - 4787889353848258560 i a^{2} d^{7} e^{14 i c} e^{5 i d x} - 13299692649578496000 i a^{2} d^{7} e^{12 i c} e^{3 i d x} - 39899077948735488000 i a^{2} d^{7} e^{10 i c} e^{i d x} + 23939446769241292800 i a^{2} d^{7} e^{8 i c} e^{- i d x} + 2659938529915699200 i a^{2} d^{7} e^{6 i c} e^{- 3 i d x} + 227994731135631360 i a^{2} d^{7} e^{4 i c} e^{- 5 i d x}\right ) e^{- 9 i c}}{145916627926804070400 d^{8}} & \text{for}\: 145916627926804070400 d^{8} e^{9 i c} \neq 0 \\\frac{x \left (a^{2} e^{14 i c} + 7 a^{2} e^{12 i c} + 21 a^{2} e^{10 i c} + 35 a^{2} e^{8 i c} + 35 a^{2} e^{6 i c} + 21 a^{2} e^{4 i c} + 7 a^{2} e^{2 i c} + a^{2}\right ) e^{- 5 i c}}{128} & \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))**2,x)

[Out]

Piecewise(((-126663739519795200*I*a**2*d**7*exp(18*I*c)*exp(9*I*d*x) - 1139973655678156800*I*a**2*d**7*exp(16*
I*c)*exp(7*I*d*x) - 4787889353848258560*I*a**2*d**7*exp(14*I*c)*exp(5*I*d*x) - 13299692649578496000*I*a**2*d**
7*exp(12*I*c)*exp(3*I*d*x) - 39899077948735488000*I*a**2*d**7*exp(10*I*c)*exp(I*d*x) + 23939446769241292800*I*
a**2*d**7*exp(8*I*c)*exp(-I*d*x) + 2659938529915699200*I*a**2*d**7*exp(6*I*c)*exp(-3*I*d*x) + 2279947311356313
60*I*a**2*d**7*exp(4*I*c)*exp(-5*I*d*x))*exp(-9*I*c)/(145916627926804070400*d**8), Ne(145916627926804070400*d*
*8*exp(9*I*c), 0)), (x*(a**2*exp(14*I*c) + 7*a**2*exp(12*I*c) + 21*a**2*exp(10*I*c) + 35*a**2*exp(8*I*c) + 35*
a**2*exp(6*I*c) + 21*a**2*exp(4*I*c) + 7*a**2*exp(2*I*c) + a**2)*exp(-5*I*c)/128, True))

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Giac [B]  time = 1.37149, size = 903, normalized size = 8.6 \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))^2,x, algorithm="giac")

[Out]

-1/92160*(18585*a^2*e^(9*I*d*x + 3*I*c)*log(I*e^(I*d*x + I*c) + 1) + 37170*a^2*e^(7*I*d*x + I*c)*log(I*e^(I*d*
x + I*c) + 1) + 18585*a^2*e^(5*I*d*x - I*c)*log(I*e^(I*d*x + I*c) + 1) + 14625*a^2*e^(9*I*d*x + 3*I*c)*log(I*e
^(I*d*x + I*c) - 1) + 29250*a^2*e^(7*I*d*x + I*c)*log(I*e^(I*d*x + I*c) - 1) + 14625*a^2*e^(5*I*d*x - I*c)*log
(I*e^(I*d*x + I*c) - 1) - 18585*a^2*e^(9*I*d*x + 3*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 37170*a^2*e^(7*I*d*x + I
*c)*log(-I*e^(I*d*x + I*c) + 1) - 18585*a^2*e^(5*I*d*x - I*c)*log(-I*e^(I*d*x + I*c) + 1) - 14625*a^2*e^(9*I*d
*x + 3*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 29250*a^2*e^(7*I*d*x + I*c)*log(-I*e^(I*d*x + I*c) - 1) - 14625*a^2*
e^(5*I*d*x - I*c)*log(-I*e^(I*d*x + I*c) - 1) - 3960*a^2*e^(9*I*d*x + 3*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 792
0*a^2*e^(7*I*d*x + I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 3960*a^2*e^(5*I*d*x - I*c)*log(I*e^(I*d*x) + e^(-I*c)) +
3960*a^2*e^(9*I*d*x + 3*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 7920*a^2*e^(7*I*d*x + I*c)*log(-I*e^(I*d*x) + e^(
-I*c)) + 3960*a^2*e^(5*I*d*x - I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 80*I*a^2*e^(18*I*d*x + 12*I*c) + 880*I*a^2*
e^(16*I*d*x + 10*I*c) + 4544*I*a^2*e^(14*I*d*x + 8*I*c) + 15168*I*a^2*e^(12*I*d*x + 6*I*c) + 45024*I*a^2*e^(10
*I*d*x + 4*I*c) + 43680*I*a^2*e^(8*I*d*x + 2*I*c) - 18624*I*a^2*e^(4*I*d*x - 2*I*c) - 1968*I*a^2*e^(2*I*d*x -
4*I*c) - 6720*I*a^2*e^(6*I*d*x) - 144*I*a^2*e^(-6*I*c))/(d*e^(9*I*d*x + 3*I*c) + 2*d*e^(7*I*d*x + I*c) + d*e^(
5*I*d*x - I*c))