3.58 \(\int \cos ^9(c+d x) (a+i a \tan (c+d x))^4 \, dx\)

Optimal. Leaf size=120 \[ \frac{a^4 \sin ^5(c+d x)}{21 d}-\frac{10 a^4 \sin ^3(c+d x)}{63 d}+\frac{5 a^4 \sin (c+d x)}{21 d}-\frac{2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d} \]

[Out]

(5*a^4*Sin[c + d*x])/(21*d) - (10*a^4*Sin[c + d*x]^3)/(63*d) + (a^4*Sin[c + d*x]^5)/(21*d) - (((2*I)/9)*a*Cos[
c + d*x]^9*(a + I*a*Tan[c + d*x])^3)/d - (((2*I)/21)*Cos[c + d*x]^7*(a^4 + I*a^4*Tan[c + d*x]))/d

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Rubi [A]  time = 0.100777, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {3496, 2633} \[ \frac{a^4 \sin ^5(c+d x)}{21 d}-\frac{10 a^4 \sin ^3(c+d x)}{63 d}+\frac{5 a^4 \sin (c+d x)}{21 d}-\frac{2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d} \]

Antiderivative was successfully verified.

[In]

Int[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^4,x]

[Out]

(5*a^4*Sin[c + d*x])/(21*d) - (10*a^4*Sin[c + d*x]^3)/(63*d) + (a^4*Sin[c + d*x]^5)/(21*d) - (((2*I)/9)*a*Cos[
c + d*x]^9*(a + I*a*Tan[c + d*x])^3)/d - (((2*I)/21)*Cos[c + d*x]^7*(a^4 + I*a^4*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))^4 \, dx &=-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}+\frac{1}{3} a^2 \int \cos ^7(c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}-\frac{2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}+\frac{1}{21} \left (5 a^4\right ) \int \cos ^5(c+d x) \, dx\\ &=-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}-\frac{2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}-\frac{\left (5 a^4\right ) \operatorname{Subst}\left (\int \left (1-2 x^2+x^4\right ) \, dx,x,-\sin (c+d x)\right )}{21 d}\\ &=\frac{5 a^4 \sin (c+d x)}{21 d}-\frac{10 a^4 \sin ^3(c+d x)}{63 d}+\frac{a^4 \sin ^5(c+d x)}{21 d}-\frac{2 i a \cos ^9(c+d x) (a+i a \tan (c+d x))^3}{9 d}-\frac{2 i \cos ^7(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{21 d}\\ \end{align*}

Mathematica [A]  time = 0.706612, size = 111, normalized size = 0.92 \[ \frac{a^4 (-42 \sin (c+d x)-135 \sin (3 (c+d x))+35 \sin (5 (c+d x))-168 i \cos (c+d x)-180 i \cos (3 (c+d x))+28 i \cos (5 (c+d x))) (\cos (4 (c+2 d x))+i \sin (4 (c+2 d x)))}{1008 d (\cos (d x)+i \sin (d x))^4} \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[c + d*x]^9*(a + I*a*Tan[c + d*x])^4,x]

[Out]

(a^4*((-168*I)*Cos[c + d*x] - (180*I)*Cos[3*(c + d*x)] + (28*I)*Cos[5*(c + d*x)] - 42*Sin[c + d*x] - 135*Sin[3
*(c + d*x)] + 35*Sin[5*(c + d*x)])*(Cos[4*(c + 2*d*x)] + I*Sin[4*(c + 2*d*x)]))/(1008*d*(Cos[d*x] + I*Sin[d*x]
)^4)

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Maple [B]  time = 0.069, size = 233, normalized size = 1.9 \begin{align*}{\frac{1}{d} \left ({a}^{4} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{6}}{9}}-{\frac{ \left ( \cos \left ( dx+c \right ) \right ) ^{6}\sin \left ( dx+c \right ) }{21}}+{\frac{\sin \left ( dx+c \right ) }{105} \left ({\frac{8}{3}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{2}}{3}} \right ) } \right ) -4\,i{a}^{4} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{9}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{7}}{63}} \right ) -6\,{a}^{4} \left ( -1/9\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{8}+{\frac{\sin \left ( dx+c \right ) }{63} \left ({\frac{16}{5}}+ \left ( \cos \left ( dx+c \right ) \right ) ^{6}+6/5\, \left ( \cos \left ( dx+c \right ) \right ) ^{4}+8/5\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) } \right ) -{\frac{4\,i}{9}}{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{9}+{\frac{{a}^{4}\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))^4,x)

[Out]

1/d*(a^4*(-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))-4*I*a^4*(-1/9*sin(d*x+c)^2*cos(d*x+c)^7-2/63*cos(d*x+c)^7)-6*a^4*(-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))-4/9*I*a^4*cos(d*x+c)^9+1/9*a^4*(128/35+c
os(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 [A]  time = 1.06109, size = 244, normalized size = 2.03 \begin{align*} -\frac{140 i \, a^{4} \cos \left (d x + c\right )^{9} + 20 i \,{\left (7 \, \cos \left (d x + c\right )^{9} - 9 \, \cos \left (d x + c\right )^{7}\right )} a^{4} -{\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^{4} - 6 \,{\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^{4} -{\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^{4}}{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))^4,x, algorithm="maxima")

[Out]

-1/315*(140*I*a^4*cos(d*x + c)^9 + 20*I*(7*cos(d*x + c)^9 - 9*cos(d*x + c)^7)*a^4 - (35*sin(d*x + c)^9 - 90*si
n(d*x + c)^7 + 63*sin(d*x + c)^5)*a^4 - 6*(35*sin(d*x + c)^9 - 135*sin(d*x + c)^7 + 189*sin(d*x + c)^5 - 105*s
in(d*x + c)^3)*a^4 - (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*s
in(d*x + c))*a^4)/d

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Fricas [A]  time = 1.09222, size = 267, normalized size = 2.22 \begin{align*} \frac{{\left (-7 i \, a^{4} e^{\left (10 i \, d x + 10 i \, c\right )} - 45 i \, a^{4} e^{\left (8 i \, d x + 8 i \, c\right )} - 126 i \, a^{4} e^{\left (6 i \, d x + 6 i \, c\right )} - 210 i \, a^{4} e^{\left (4 i \, d x + 4 i \, c\right )} - 315 i \, a^{4} e^{\left (2 i \, d x + 2 i \, c\right )} + 63 i \, a^{4}\right )} e^{\left (-i \, d x - i \, c\right )}}{2016 \, 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))^4,x, algorithm="fricas")

[Out]

1/2016*(-7*I*a^4*e^(10*I*d*x + 10*I*c) - 45*I*a^4*e^(8*I*d*x + 8*I*c) - 126*I*a^4*e^(6*I*d*x + 6*I*c) - 210*I*
a^4*e^(4*I*d*x + 4*I*c) - 315*I*a^4*e^(2*I*d*x + 2*I*c) + 63*I*a^4)*e^(-I*d*x - I*c)/d

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Sympy [A]  time = 1.26361, size = 230, normalized size = 1.92 \begin{align*} \begin{cases} \frac{\left (- 176160768 i a^{4} d^{5} e^{10 i c} e^{9 i d x} - 1132462080 i a^{4} d^{5} e^{8 i c} e^{7 i d x} - 3170893824 i a^{4} d^{5} e^{6 i c} e^{5 i d x} - 5284823040 i a^{4} d^{5} e^{4 i c} e^{3 i d x} - 7927234560 i a^{4} d^{5} e^{2 i c} e^{i d x} + 1585446912 i a^{4} d^{5} e^{- i d x}\right ) e^{- i c}}{50734301184 d^{6}} & \text{for}\: 50734301184 d^{6} e^{i c} \neq 0 \\\frac{x \left (a^{4} e^{10 i c} + 5 a^{4} e^{8 i c} + 10 a^{4} e^{6 i c} + 10 a^{4} e^{4 i c} + 5 a^{4} e^{2 i c} + a^{4}\right ) e^{- i c}}{32} & \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))**4,x)

[Out]

Piecewise(((-176160768*I*a**4*d**5*exp(10*I*c)*exp(9*I*d*x) - 1132462080*I*a**4*d**5*exp(8*I*c)*exp(7*I*d*x) -
3170893824*I*a**4*d**5*exp(6*I*c)*exp(5*I*d*x) - 5284823040*I*a**4*d**5*exp(4*I*c)*exp(3*I*d*x) - 7927234560*
I*a**4*d**5*exp(2*I*c)*exp(I*d*x) + 1585446912*I*a**4*d**5*exp(-I*d*x))*exp(-I*c)/(50734301184*d**6), Ne(50734
301184*d**6*exp(I*c), 0)), (x*(a**4*exp(10*I*c) + 5*a**4*exp(8*I*c) + 10*a**4*exp(6*I*c) + 10*a**4*exp(4*I*c)
+ 5*a**4*exp(2*I*c) + a**4)*exp(-I*c)/32, True))

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Giac [B]  time = 1.88701, size = 1902, normalized size = 15.85 \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))^4,x, algorithm="giac")

[Out]

1/516096*(435267*a^4*e^(13*I*d*x + 7*I*c)*log(I*e^(I*d*x + I*c) + 1) + 2611602*a^4*e^(11*I*d*x + 5*I*c)*log(I*
e^(I*d*x + I*c) + 1) + 6529005*a^4*e^(9*I*d*x + 3*I*c)*log(I*e^(I*d*x + I*c) + 1) + 8705340*a^4*e^(7*I*d*x + I
*c)*log(I*e^(I*d*x + I*c) + 1) + 6529005*a^4*e^(5*I*d*x - I*c)*log(I*e^(I*d*x + I*c) + 1) + 2611602*a^4*e^(3*I
*d*x - 3*I*c)*log(I*e^(I*d*x + I*c) + 1) + 435267*a^4*e^(I*d*x - 5*I*c)*log(I*e^(I*d*x + I*c) + 1) + 427896*a^
4*e^(13*I*d*x + 7*I*c)*log(I*e^(I*d*x + I*c) - 1) + 2567376*a^4*e^(11*I*d*x + 5*I*c)*log(I*e^(I*d*x + I*c) - 1
) + 6418440*a^4*e^(9*I*d*x + 3*I*c)*log(I*e^(I*d*x + I*c) - 1) + 8557920*a^4*e^(7*I*d*x + I*c)*log(I*e^(I*d*x
+ I*c) - 1) + 6418440*a^4*e^(5*I*d*x - I*c)*log(I*e^(I*d*x + I*c) - 1) + 2567376*a^4*e^(3*I*d*x - 3*I*c)*log(I
*e^(I*d*x + I*c) - 1) + 427896*a^4*e^(I*d*x - 5*I*c)*log(I*e^(I*d*x + I*c) - 1) - 435267*a^4*e^(13*I*d*x + 7*I
*c)*log(-I*e^(I*d*x + I*c) + 1) - 2611602*a^4*e^(11*I*d*x + 5*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 6529005*a^4*e
^(9*I*d*x + 3*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 8705340*a^4*e^(7*I*d*x + I*c)*log(-I*e^(I*d*x + I*c) + 1) - 6
529005*a^4*e^(5*I*d*x - I*c)*log(-I*e^(I*d*x + I*c) + 1) - 2611602*a^4*e^(3*I*d*x - 3*I*c)*log(-I*e^(I*d*x + I
*c) + 1) - 435267*a^4*e^(I*d*x - 5*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 427896*a^4*e^(13*I*d*x + 7*I*c)*log(-I*e
^(I*d*x + I*c) - 1) - 2567376*a^4*e^(11*I*d*x + 5*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 6418440*a^4*e^(9*I*d*x +
3*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 8557920*a^4*e^(7*I*d*x + I*c)*log(-I*e^(I*d*x + I*c) - 1) - 6418440*a^4*e
^(5*I*d*x - I*c)*log(-I*e^(I*d*x + I*c) - 1) - 2567376*a^4*e^(3*I*d*x - 3*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 4
27896*a^4*e^(I*d*x - 5*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 7371*a^4*e^(13*I*d*x + 7*I*c)*log(I*e^(I*d*x) + e^(-
I*c)) - 44226*a^4*e^(11*I*d*x + 5*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 110565*a^4*e^(9*I*d*x + 3*I*c)*log(I*e^(I
*d*x) + e^(-I*c)) - 147420*a^4*e^(7*I*d*x + I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 110565*a^4*e^(5*I*d*x - I*c)*lo
g(I*e^(I*d*x) + e^(-I*c)) - 44226*a^4*e^(3*I*d*x - 3*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 7371*a^4*e^(I*d*x - 5*
I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 7371*a^4*e^(13*I*d*x + 7*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 44226*a^4*e^(1
1*I*d*x + 5*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 110565*a^4*e^(9*I*d*x + 3*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) +
147420*a^4*e^(7*I*d*x + I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 110565*a^4*e^(5*I*d*x - I*c)*log(-I*e^(I*d*x) + e^
(-I*c)) + 44226*a^4*e^(3*I*d*x - 3*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 7371*a^4*e^(I*d*x - 5*I*c)*log(-I*e^(I*
d*x) + e^(-I*c)) - 1792*I*a^4*e^(22*I*d*x + 16*I*c) - 22272*I*a^4*e^(20*I*d*x + 14*I*c) - 128256*I*a^4*e^(18*I
*d*x + 12*I*c) - 455936*I*a^4*e^(16*I*d*x + 10*I*c) - 1144320*I*a^4*e^(14*I*d*x + 8*I*c) - 2102784*I*a^4*e^(12
*I*d*x + 6*I*c) - 2742784*I*a^4*e^(10*I*d*x + 4*I*c) - 2382336*I*a^4*e^(8*I*d*x + 2*I*c) - 295680*I*a^4*e^(4*I
*d*x - 2*I*c) + 16128*I*a^4*e^(2*I*d*x - 4*I*c) - 1241856*I*a^4*e^(6*I*d*x) + 16128*I*a^4*e^(-6*I*c))/(d*e^(13
*I*d*x + 7*I*c) + 6*d*e^(11*I*d*x + 5*I*c) + 15*d*e^(9*I*d*x + 3*I*c) + 20*d*e^(7*I*d*x + I*c) + 15*d*e^(5*I*d
*x - I*c) + 6*d*e^(3*I*d*x - 3*I*c) + d*e^(I*d*x - 5*I*c))