3.74 \(\int \cos ^7(c+d x) (a+i a \tan (c+d x))^5 \, dx\)
Optimal. Leaf size=101 \[ -\frac{2 i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{105 d}-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d}-\frac{2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{35 d} \]
[Out]
(((-2*I)/105)*a^2*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/d - (((2*I)/35)*a*Cos[c + d*x]^5*(a + I*a*Tan[c + d
*x])^4)/d - ((I/7)*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^5)/d
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Rubi [A] time = 0.113655, antiderivative size = 101, 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 =
{3497, 3488} \[ -\frac{2 i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{105 d}-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d}-\frac{2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{35 d} \]
Antiderivative was successfully verified.
[In]
Int[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^5,x]
[Out]
(((-2*I)/105)*a^2*Cos[c + d*x]^3*(a + I*a*Tan[c + d*x])^3)/d - (((2*I)/35)*a*Cos[c + d*x]^5*(a + I*a*Tan[c + d
*x])^4)/d - ((I/7)*Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^5)/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 ^7(c+d x) (a+i a \tan (c+d x))^5 \, dx &=-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d}+\frac{1}{7} (2 a) \int \cos ^5(c+d x) (a+i a \tan (c+d x))^4 \, dx\\ &=-\frac{2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{35 d}-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d}+\frac{1}{35} \left (2 a^2\right ) \int \cos ^3(c+d x) (a+i a \tan (c+d x))^3 \, dx\\ &=-\frac{2 i a^2 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{105 d}-\frac{2 i a \cos ^5(c+d x) (a+i a \tan (c+d x))^4}{35 d}-\frac{i \cos ^7(c+d x) (a+i a \tan (c+d x))^5}{7 d}\\ \end{align*}
Mathematica [A] time = 0.771674, size = 55, normalized size = 0.54 \[ \frac{a^5 (-10 i \sin (2 (c+d x))+25 \cos (2 (c+d x))+21) (\sin (5 (c+d x))-i \cos (5 (c+d x)))}{210 d} \]
Antiderivative was successfully verified.
[In]
Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^5,x]
[Out]
(a^5*(21 + 25*Cos[2*(c + d*x)] - (10*I)*Sin[2*(c + d*x)])*((-I)*Cos[5*(c + d*x)] + Sin[5*(c + d*x)]))/(210*d)
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Maple [B] time = 0.091, size = 257, normalized size = 2.5 \begin{align*}{\frac{1}{d} \left ( i{a}^{5} \left ( -{\frac{ \left ( \sin \left ( dx+c \right ) \right ) ^{4} \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{7}}-{\frac{4\, \left ( \cos \left ( dx+c \right ) \right ) ^{3} \left ( \sin \left ( dx+c \right ) \right ) ^{2}}{35}}-{\frac{8\, \left ( \cos \left ( dx+c \right ) \right ) ^{3}}{105}} \right ) +5\,{a}^{5} \left ( -1/7\, \left ( \sin \left ( dx+c \right ) \right ) ^{3} \left ( \cos \left ( dx+c \right ) \right ) ^{4}-{\frac{3\,\sin \left ( dx+c \right ) \left ( \cos \left ( dx+c \right ) \right ) ^{4}}{35}}+1/35\, \left ( 2+ \left ( \cos \left ( dx+c \right ) \right ) ^{2} \right ) \sin \left ( dx+c \right ) \right ) -10\,i{a}^{5} \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 ) -10\,{a}^{5} \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{5\,i}{7}}{a}^{5} \left ( \cos \left ( dx+c \right ) \right ) ^{7}+{\frac{{a}^{5}\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))^5,x)
[Out]
1/d*(I*a^5*(-1/7*sin(d*x+c)^4*cos(d*x+c)^3-4/35*cos(d*x+c)^3*sin(d*x+c)^2-8/105*cos(d*x+c)^3)+5*a^5*(-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))-10*I*a^5*(-1/7*sin(d*x+c)
^2*cos(d*x+c)^5-2/35*cos(d*x+c)^5)-10*a^5*(-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))-5/7*I*a^5*cos(d*x+c)^7+1/7*a^5*(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)
)
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Maxima [B] time = 1.15952, size = 252, normalized size = 2.5 \begin{align*} -\frac{75 i \, a^{5} \cos \left (d x + c\right )^{7} + i \,{\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{5} + 30 i \,{\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{5} + 10 \,{\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^{5} + 15 \,{\left (5 \, \sin \left (d x + c\right )^{7} - 7 \, \sin \left (d x + c\right )^{5}\right )} a^{5} + 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^{5}}{105 \, 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))^5,x, algorithm="maxima")
[Out]
-1/105*(75*I*a^5*cos(d*x + c)^7 + I*(15*cos(d*x + c)^7 - 42*cos(d*x + c)^5 + 35*cos(d*x + c)^3)*a^5 + 30*I*(5*
cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^5 + 10*(15*sin(d*x + c)^7 - 42*sin(d*x + c)^5 + 35*sin(d*x + c)^3)*a^5 +
15*(5*sin(d*x + c)^7 - 7*sin(d*x + c)^5)*a^5 + 3*(5*sin(d*x + c)^7 - 21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 3
5*sin(d*x + c))*a^5)/d
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Fricas [A] time = 1.39852, size = 139, normalized size = 1.38 \begin{align*} \frac{-15 i \, a^{5} e^{\left (7 i \, d x + 7 i \, c\right )} - 42 i \, a^{5} e^{\left (5 i \, d x + 5 i \, c\right )} - 35 i \, a^{5} e^{\left (3 i \, d x + 3 i \, c\right )}}{420 \, 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))^5,x, algorithm="fricas")
[Out]
1/420*(-15*I*a^5*e^(7*I*d*x + 7*I*c) - 42*I*a^5*e^(5*I*d*x + 5*I*c) - 35*I*a^5*e^(3*I*d*x + 3*I*c))/d
________________________________________________________________________________________
Sympy [A] time = 0.757066, size = 122, normalized size = 1.21 \begin{align*} \begin{cases} \frac{- 120 i a^{5} d^{2} e^{7 i c} e^{7 i d x} - 336 i a^{5} d^{2} e^{5 i c} e^{5 i d x} - 280 i a^{5} d^{2} e^{3 i c} e^{3 i d x}}{3360 d^{3}} & \text{for}\: 3360 d^{3} \neq 0 \\x \left (\frac{a^{5} e^{7 i c}}{4} + \frac{a^{5} e^{5 i c}}{2} + \frac{a^{5} e^{3 i c}}{4}\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))**5,x)
[Out]
Piecewise(((-120*I*a**5*d**2*exp(7*I*c)*exp(7*I*d*x) - 336*I*a**5*d**2*exp(5*I*c)*exp(5*I*d*x) - 280*I*a**5*d*
*2*exp(3*I*c)*exp(3*I*d*x))/(3360*d**3), Ne(3360*d**3, 0)), (x*(a**5*exp(7*I*c)/4 + a**5*exp(5*I*c)/2 + a**5*e
xp(3*I*c)/4), True))
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Giac [B] time = 2.1073, size = 2291, normalized size = 22.68 \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))^5,x, algorithm="giac")
[Out]
-1/3440640*(7357770*a^5*e^(16*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 58862160*a^5*e^(14*I*d*x + 6*I*c)*lo
g(I*e^(I*d*x + I*c) + 1) + 206017560*a^5*e^(12*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 412035120*a^5*e^(10
*I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 412035120*a^5*e^(6*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 20
6017560*a^5*e^(4*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 58862160*a^5*e^(2*I*d*x - 6*I*c)*log(I*e^(I*d*x +
I*c) + 1) + 515043900*a^5*e^(8*I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 7357770*a^5*e^(-8*I*c)*log(I*e^(I*d*x + I*
c) + 1) + 7390425*a^5*e^(16*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 59123400*a^5*e^(14*I*d*x + 6*I*c)*log(
I*e^(I*d*x + I*c) - 1) + 206931900*a^5*e^(12*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 413863800*a^5*e^(10*I
*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 413863800*a^5*e^(6*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 2069
31900*a^5*e^(4*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 59123400*a^5*e^(2*I*d*x - 6*I*c)*log(I*e^(I*d*x + I
*c) - 1) + 517329750*a^5*e^(8*I*d*x)*log(I*e^(I*d*x + I*c) - 1) + 7390425*a^5*e^(-8*I*c)*log(I*e^(I*d*x + I*c)
- 1) - 7357770*a^5*e^(16*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 58862160*a^5*e^(14*I*d*x + 6*I*c)*log(-
I*e^(I*d*x + I*c) + 1) - 206017560*a^5*e^(12*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 412035120*a^5*e^(10*
I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 412035120*a^5*e^(6*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 2
06017560*a^5*e^(4*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 58862160*a^5*e^(2*I*d*x - 6*I*c)*log(-I*e^(I*d*
x + I*c) + 1) - 515043900*a^5*e^(8*I*d*x)*log(-I*e^(I*d*x + I*c) + 1) - 7357770*a^5*e^(-8*I*c)*log(-I*e^(I*d*x
+ I*c) + 1) - 7390425*a^5*e^(16*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 59123400*a^5*e^(14*I*d*x + 6*I*c
)*log(-I*e^(I*d*x + I*c) - 1) - 206931900*a^5*e^(12*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 413863800*a^5
*e^(10*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 413863800*a^5*e^(6*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*c) -
1) - 206931900*a^5*e^(4*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 59123400*a^5*e^(2*I*d*x - 6*I*c)*log(-I*
e^(I*d*x + I*c) - 1) - 517329750*a^5*e^(8*I*d*x)*log(-I*e^(I*d*x + I*c) - 1) - 7390425*a^5*e^(-8*I*c)*log(-I*e
^(I*d*x + I*c) - 1) + 32655*a^5*e^(16*I*d*x + 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 261240*a^5*e^(14*I*d*x + 6*
I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 914340*a^5*e^(12*I*d*x + 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 1828680*a^5*e
^(10*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 1828680*a^5*e^(6*I*d*x - 2*I*c)*log(I*e^(I*d*x) + e^(-I*c))
+ 914340*a^5*e^(4*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 261240*a^5*e^(2*I*d*x - 6*I*c)*log(I*e^(I*d*x)
+ e^(-I*c)) + 2285850*a^5*e^(8*I*d*x)*log(I*e^(I*d*x) + e^(-I*c)) + 32655*a^5*e^(-8*I*c)*log(I*e^(I*d*x) + e^(
-I*c)) - 32655*a^5*e^(16*I*d*x + 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 261240*a^5*e^(14*I*d*x + 6*I*c)*log(-I*
e^(I*d*x) + e^(-I*c)) - 914340*a^5*e^(12*I*d*x + 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 1828680*a^5*e^(10*I*d*x
+ 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 1828680*a^5*e^(6*I*d*x - 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 914340
*a^5*e^(4*I*d*x - 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 261240*a^5*e^(2*I*d*x - 6*I*c)*log(-I*e^(I*d*x) + e^(-
I*c)) - 2285850*a^5*e^(8*I*d*x)*log(-I*e^(I*d*x) + e^(-I*c)) - 32655*a^5*e^(-8*I*c)*log(-I*e^(I*d*x) + e^(-I*c
)) + 122880*I*a^5*e^(23*I*d*x + 15*I*c) + 1327104*I*a^5*e^(21*I*d*x + 13*I*c) + 6479872*I*a^5*e^(19*I*d*x + 11
*I*c) + 18808832*I*a^5*e^(17*I*d*x + 9*I*c) + 35897344*I*a^5*e^(15*I*d*x + 7*I*c) + 47022080*I*a^5*e^(13*I*d*x
+ 5*I*c) + 42778624*I*a^5*e^(11*I*d*x + 3*I*c) + 26673152*I*a^5*e^(9*I*d*x + I*c) + 10903552*I*a^5*e^(7*I*d*x
- I*c) + 2637824*I*a^5*e^(5*I*d*x - 3*I*c) + 286720*I*a^5*e^(3*I*d*x - 5*I*c))/(d*e^(16*I*d*x + 8*I*c) + 8*d*
e^(14*I*d*x + 6*I*c) + 28*d*e^(12*I*d*x + 4*I*c) + 56*d*e^(10*I*d*x + 2*I*c) + 56*d*e^(6*I*d*x - 2*I*c) + 28*d
*e^(4*I*d*x - 4*I*c) + 8*d*e^(2*I*d*x - 6*I*c) + 70*d*e^(8*I*d*x) + d*e^(-8*I*c))