3.57 \(\int \cos ^7(c+d x) (a+i a \tan (c+d x))^4 \, dx\)
Optimal. Leaf size=102 \[ -\frac{a^4 \sin ^3(c+d x)}{35 d}+\frac{3 a^4 \sin (c+d x)}{35 d}-\frac{2 i \cos ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{35 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^3}{7 d} \]
[Out]
(3*a^4*Sin[c + d*x])/(35*d) - (a^4*Sin[c + d*x]^3)/(35*d) - (((2*I)/7)*a*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])
^3)/d - (((2*I)/35)*Cos[c + d*x]^5*(a^4 + I*a^4*Tan[c + d*x]))/d
________________________________________________________________________________________
Rubi [A] time = 0.0889054, antiderivative size = 102, 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} \[ -\frac{a^4 \sin ^3(c+d x)}{35 d}+\frac{3 a^4 \sin (c+d x)}{35 d}-\frac{2 i \cos ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{35 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^3}{7 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^4,x]
[Out]
(3*a^4*Sin[c + d*x])/(35*d) - (a^4*Sin[c + d*x]^3)/(35*d) - (((2*I)/7)*a*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])
^3)/d - (((2*I)/35)*Cos[c + d*x]^5*(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 ^7(c+d x) (a+i a \tan (c+d x))^4 \, dx &=-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^3}{7 d}+\frac{1}{7} a^2 \int \cos ^5(c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^3}{7 d}-\frac{2 i \cos ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{35 d}+\frac{1}{35} \left (3 a^4\right ) \int \cos ^3(c+d x) \, dx\\ &=-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^3}{7 d}-\frac{2 i \cos ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{35 d}-\frac{\left (3 a^4\right ) \operatorname{Subst}\left (\int \left (1-x^2\right ) \, dx,x,-\sin (c+d x)\right )}{35 d}\\ &=\frac{3 a^4 \sin (c+d x)}{35 d}-\frac{a^4 \sin ^3(c+d x)}{35 d}-\frac{2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^3}{7 d}-\frac{2 i \cos ^5(c+d x) \left (a^4+i a^4 \tan (c+d x)\right )}{35 d}\\ \end{align*}
Mathematica [A] time = 0.683372, size = 73, normalized size = 0.72 \[ \frac{a^4 (-i (7 \sin (c+d x)+15 \sin (3 (c+d x)))+28 \cos (c+d x)+20 \cos (3 (c+d x))) (\sin (4 (c+d x))-i \cos (4 (c+d x)))}{140 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^4,x]
[Out]
(a^4*(28*Cos[c + d*x] + 20*Cos[3*(c + d*x)] - I*(7*Sin[c + d*x] + 15*Sin[3*(c + d*x)]))*((-I)*Cos[4*(c + d*x)]
+ Sin[4*(c + d*x)]))/(140*d)
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Maple [B] time = 0.065, size = 203, normalized size = 2. \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 ) ^{4}}{7}}-{\frac{3\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{35}}+{\frac{ \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) }{35}} \right ) -4\,i{a}^{4} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{2} \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{7}}-{\frac{2\, \left ( \cos \left ( dx+c \right ) \right ) ^{5}}{35}} \right ) -6\,{a}^{4} \left ( -1/7\, \left ( \cos \left ( dx+c \right ) \right ) ^{6}\sin \left ( dx+c \right ) +1/35\, \left ( 8/3+ \left ( \cos \left ( dx+c \right ) \right ) ^{4}+4/3\, \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -{\frac{4\,i}{7}}{a}^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{7}+{\frac{{a}^{4}\sin \left ( dx+c \right ) }{7} \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 ) } \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^4,x)
[Out]
1/d*(a^4*(-1/7*sin(d*x+c)^3*cos(d*x+c)^4-3/35*sin(d*x+c)*cos(d*x+c)^4+1/35*(2+cos(d*x+c)^2)*sin(d*x+c))-4*I*a^
4*(-1/7*sin(d*x+c)^2*cos(d*x+c)^5-2/35*cos(d*x+c)^5)-6*a^4*(-1/7*cos(d*x+c)^6*sin(d*x+c)+1/35*(8/3+cos(d*x+c)^
4+4/3*cos(d*x+c)^2)*sin(d*x+c))-4/7*I*a^4*cos(d*x+c)^7+1/7*a^4*(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))
________________________________________________________________________________________
Maxima [A] time = 1.10339, size = 201, normalized size = 1.97 \begin{align*} -\frac{20 i \, a^{4} \cos \left (d x + c\right )^{7} + 4 i \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{4} + 2 \,{\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^{4} +{\left (5 \, \sin \left (d x + c\right )^{7} - 7 \, \sin \left (d x + c\right )^{5}\right )} a^{4} +{\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^{4}}{35 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^4,x, algorithm="maxima")
[Out]
-1/35*(20*I*a^4*cos(d*x + c)^7 + 4*I*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^4 + 2*(15*sin(d*x + c)^7 - 42*sin
(d*x + c)^5 + 35*sin(d*x + c)^3)*a^4 + (5*sin(d*x + c)^7 - 7*sin(d*x + c)^5)*a^4 + (5*sin(d*x + c)^7 - 21*sin(
d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*a^4)/d
________________________________________________________________________________________
Fricas [A] time = 1.24938, size = 174, normalized size = 1.71 \begin{align*} \frac{-5 i \, a^{4} e^{\left (7 i \, d x + 7 i \, c\right )} - 21 i \, a^{4} e^{\left (5 i \, d x + 5 i \, c\right )} - 35 i \, a^{4} e^{\left (3 i \, d x + 3 i \, c\right )} - 35 i \, a^{4} e^{\left (i \, d x + i \, c\right )}}{280 \, d} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^4,x, algorithm="fricas")
[Out]
1/280*(-5*I*a^4*e^(7*I*d*x + 7*I*c) - 21*I*a^4*e^(5*I*d*x + 5*I*c) - 35*I*a^4*e^(3*I*d*x + 3*I*c) - 35*I*a^4*e
^(I*d*x + I*c))/d
________________________________________________________________________________________
Sympy [A] time = 1.06817, size = 158, normalized size = 1.55 \begin{align*} \begin{cases} \frac{- 2560 i a^{4} d^{3} e^{7 i c} e^{7 i d x} - 10752 i a^{4} d^{3} e^{5 i c} e^{5 i d x} - 17920 i a^{4} d^{3} e^{3 i c} e^{3 i d x} - 17920 i a^{4} d^{3} e^{i c} e^{i d x}}{143360 d^{4}} & \text{for}\: 143360 d^{4} \neq 0 \\x \left (\frac{a^{4} e^{7 i c}}{8} + \frac{3 a^{4} e^{5 i c}}{8} + \frac{3 a^{4} e^{3 i c}}{8} + \frac{a^{4} e^{i c}}{8}\right ) & \text{otherwise} \end{cases} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate(cos(d*x+c)**7*(a+I*a*tan(d*x+c))**4,x)
[Out]
Piecewise(((-2560*I*a**4*d**3*exp(7*I*c)*exp(7*I*d*x) - 10752*I*a**4*d**3*exp(5*I*c)*exp(5*I*d*x) - 17920*I*a*
*4*d**3*exp(3*I*c)*exp(3*I*d*x) - 17920*I*a**4*d**3*exp(I*c)*exp(I*d*x))/(143360*d**4), Ne(143360*d**4, 0)), (
x*(a**4*exp(7*I*c)/8 + 3*a**4*exp(5*I*c)/8 + 3*a**4*exp(3*I*c)/8 + a**4*exp(I*c)/8), True))
________________________________________________________________________________________
Giac [B] time = 1.81071, size = 1791, normalized size = 17.56 \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)^7*(a+I*a*tan(d*x+c))^4,x, algorithm="giac")
[Out]
1/143360*(89950*a^4*e^(12*I*d*x + 6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 539700*a^4*e^(10*I*d*x + 4*I*c)*log(I*e^
(I*d*x + I*c) + 1) + 1349250*a^4*e^(8*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1349250*a^4*e^(4*I*d*x - 2*I
*c)*log(I*e^(I*d*x + I*c) + 1) + 539700*a^4*e^(2*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1799000*a^4*e^(6*
I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 89950*a^4*e^(-6*I*c)*log(I*e^(I*d*x + I*c) + 1) + 86065*a^4*e^(12*I*d*x +
6*I*c)*log(I*e^(I*d*x + I*c) - 1) + 516390*a^4*e^(10*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1290975*a^4*e
^(8*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1290975*a^4*e^(4*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 5
16390*a^4*e^(2*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1721300*a^4*e^(6*I*d*x)*log(I*e^(I*d*x + I*c) - 1)
+ 86065*a^4*e^(-6*I*c)*log(I*e^(I*d*x + I*c) - 1) - 89950*a^4*e^(12*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) + 1)
- 539700*a^4*e^(10*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1349250*a^4*e^(8*I*d*x + 2*I*c)*log(-I*e^(I*d
*x + I*c) + 1) - 1349250*a^4*e^(4*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 539700*a^4*e^(2*I*d*x - 4*I*c)*
log(-I*e^(I*d*x + I*c) + 1) - 1799000*a^4*e^(6*I*d*x)*log(-I*e^(I*d*x + I*c) + 1) - 89950*a^4*e^(-6*I*c)*log(-
I*e^(I*d*x + I*c) + 1) - 86065*a^4*e^(12*I*d*x + 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 516390*a^4*e^(10*I*d*x +
4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1290975*a^4*e^(8*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1290975*a^
4*e^(4*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 516390*a^4*e^(2*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) - 1)
- 1721300*a^4*e^(6*I*d*x)*log(-I*e^(I*d*x + I*c) - 1) - 86065*a^4*e^(-6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 38
85*a^4*e^(12*I*d*x + 6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 23310*a^4*e^(10*I*d*x + 4*I*c)*log(I*e^(I*d*x) + e^(
-I*c)) - 58275*a^4*e^(8*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 58275*a^4*e^(4*I*d*x - 2*I*c)*log(I*e^(I*
d*x) + e^(-I*c)) - 23310*a^4*e^(2*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 77700*a^4*e^(6*I*d*x)*log(I*e^(
I*d*x) + e^(-I*c)) - 3885*a^4*e^(-6*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 3885*a^4*e^(12*I*d*x + 6*I*c)*log(-I*e^
(I*d*x) + e^(-I*c)) + 23310*a^4*e^(10*I*d*x + 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 58275*a^4*e^(8*I*d*x + 2*I
*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 58275*a^4*e^(4*I*d*x - 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 23310*a^4*e^(2
*I*d*x - 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 77700*a^4*e^(6*I*d*x)*log(-I*e^(I*d*x) + e^(-I*c)) + 3885*a^4*e
^(-6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 2560*I*a^4*e^(19*I*d*x + 13*I*c) - 26112*I*a^4*e^(17*I*d*x + 11*I*c)
- 120832*I*a^4*e^(15*I*d*x + 9*I*c) - 337920*I*a^4*e^(13*I*d*x + 7*I*c) - 629760*I*a^4*e^(11*I*d*x + 5*I*c) -
803840*I*a^4*e^(9*I*d*x + 3*I*c) - 694272*I*a^4*e^(7*I*d*x + I*c) - 387072*I*a^4*e^(5*I*d*x - I*c) - 125440*I*
a^4*e^(3*I*d*x - 3*I*c) - 17920*I*a^4*e^(I*d*x - 5*I*c))/(d*e^(12*I*d*x + 6*I*c) + 6*d*e^(10*I*d*x + 4*I*c) +
15*d*e^(8*I*d*x + 2*I*c) + 15*d*e^(4*I*d*x - 2*I*c) + 6*d*e^(2*I*d*x - 4*I*c) + 20*d*e^(6*I*d*x) + d*e^(-6*I*c
))