∫ cos7(c + dx)(a + ia tan(c + dx))8 dx

3.94 \(\int \cos ^7(c+d x) (a+i a \tan (c+d x))^8 \, dx\)

Optimal. Leaf size=152 \[ \frac {a^8 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {2 i \cos (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{d}+\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a^2 \cos ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{3 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d} \]

[Out]

a^8*arctanh(sin(d*x+c))/d+2/5*I*a^3*cos(d*x+c)^5*(a+I*a*tan(d*x+c))^5/d-2/7*I*a*cos(d*x+c)^7*(a+I*a*tan(d*x+c)

)^7/d-2/3*I*a^2*cos(d*x+c)^3*(a^2+I*a^2*tan(d*x+c))^3/d+2*I*cos(d*x+c)*(a^8+I*a^8*tan(d*x+c))/d

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Rubi [A] time = 0.16, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3496, 3770} \[ \frac {a^8 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a^2 \cos ^3(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )^3}{3 d}+\frac {2 i \cos (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d} \]

Antiderivative was successfully veriﬁed.

[In]

Int[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*ArcTanh[Sin[c + d*x]])/d + (((2*I)/5)*a^3*Cos[c + d*x]^5*(a + I*a*Tan[c + d*x])^5)/d - (((2*I)/7)*a*Cos[c

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

*Cos[c + d*x]*(a^8 + I*a^8*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 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

\begin {align*} \int \cos ^7(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}-a^2 \int \cos ^5(c+d x) (a+i a \tan (c+d x))^6 \, dx\\ &=\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}+a^4 \int \cos ^3(c+d x) (a+i a \tan (c+d x))^4 \, dx\\ &=-\frac {2 i a^5 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}-a^6 \int \cos (c+d x) (a+i a \tan (c+d x))^2 \, dx\\ &=-\frac {2 i a^5 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}+\frac {2 i \cos (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{d}+a^8 \int \sec (c+d x) \, dx\\ &=\frac {a^8 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {2 i a^5 \cos ^3(c+d x) (a+i a \tan (c+d x))^3}{3 d}+\frac {2 i a^3 \cos ^5(c+d x) (a+i a \tan (c+d x))^5}{5 d}-\frac {2 i a \cos ^7(c+d x) (a+i a \tan (c+d x))^7}{7 d}+\frac {2 i \cos (c+d x) \left (a^8+i a^8 \tan (c+d x)\right )}{d}\\ \end {align*}

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Mathematica [B] time = 2.47, size = 305, normalized size = 2.01 \[ \frac {a^8 \left (\cos \left (\frac {1}{2} (7 c+23 d x)\right )+i \sin \left (\frac {1}{2} (7 c+23 d x)\right )\right ) \left (-70 \sin \left (\frac {1}{2} (c+d x)\right )-42 \sin \left (\frac {3}{2} (c+d x)\right )+210 \sin \left (\frac {5}{2} (c+d x)\right )+30 \sin \left (\frac {7}{2} (c+d x)\right )-70 i \cos \left (\frac {1}{2} (c+d x)\right )+42 i \cos \left (\frac {3}{2} (c+d x)\right )+210 i \cos \left (\frac {5}{2} (c+d x)\right )-30 i \cos \left (\frac {7}{2} (c+d x)\right )-105 \cos \left (\frac {7}{2} (c+d x)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+105 \cos \left (\frac {7}{2} (c+d x)\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )+105 i \sin \left (\frac {7}{2} (c+d x)\right ) \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-105 i \sin \left (\frac {7}{2} (c+d x)\right ) \log \left (\sin \left (\frac {1}{2} (c+d x)\right )+\cos \left (\frac {1}{2} (c+d x)\right )\right )\right )}{105 d (\cos (d x)+i \sin (d x))^8} \]

Antiderivative was successfully veriﬁed.

[In]

Integrate[Cos[c + d*x]^7*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*((-70*I)*Cos[(c + d*x)/2] + (42*I)*Cos[(3*(c + d*x))/2] + (210*I)*Cos[(5*(c + d*x))/2] - (30*I)*Cos[(7*(c

+ d*x))/2] - 105*Cos[(7*(c + d*x))/2]*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]] + 105*Cos[(7*(c + d*x))/2]*Log

[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] - 70*Sin[(c + d*x)/2] - 42*Sin[(3*(c + d*x))/2] + 210*Sin[(5*(c + d*x))/

2] + 30*Sin[(7*(c + d*x))/2] + (105*I)*Log[Cos[(c + d*x)/2] - Sin[(c + d*x)/2]]*Sin[(7*(c + d*x))/2] - (105*I)

*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]]*Sin[(7*(c + d*x))/2])*(Cos[(7*c + 23*d*x)/2] + I*Sin[(7*c + 23*d*x)/

2]))/(105*d*(Cos[d*x] + I*Sin[d*x])^8)

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fricas [A] time = 0.68, size = 96, normalized size = 0.63 \[ \frac {-30 i \, a^{8} e^{\left (7 i \, d x + 7 i \, c\right )} + 42 i \, a^{8} e^{\left (5 i \, d x + 5 i \, c\right )} - 70 i \, a^{8} e^{\left (3 i \, d x + 3 i \, c\right )} + 210 i \, a^{8} e^{\left (i \, d x + i \, c\right )} + 105 \, a^{8} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 105 \, a^{8} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right )}{105 \, d} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/105*(-30*I*a^8*e^(7*I*d*x + 7*I*c) + 42*I*a^8*e^(5*I*d*x + 5*I*c) - 70*I*a^8*e^(3*I*d*x + 3*I*c) + 210*I*a^8

*e^(I*d*x + I*c) + 105*a^8*log(e^(I*d*x + I*c) + I) - 105*a^8*log(e^(I*d*x + I*c) - I))/d

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giac [B] time = 12.32, size = 2863, normalized size = 18.84 \[ \text {result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/55050240*(1635552135*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) + 1) + 22897729890*a^8*e^(26*I*d*x + 12

*I*c)*log(I*e^(I*d*x + I*c) + 1) + 148835244285*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 5953409

77140*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1637187687135*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*

d*x + I*c) + 1) + 3274375374270*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 4911563061405*a^8*e^(16*

I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) + 1) + 4911563061405*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) + 1)

+ 3274375374270*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) + 1) + 1637187687135*a^8*e^(8*I*d*x - 6*I*c)*lo

g(I*e^(I*d*x + I*c) + 1) + 595340977140*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) + 1) + 148835244285*a^8*

e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) + 1) + 22897729890*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) +

1) + 5613214927320*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I*c) + 1) + 1635552135*a^8*e^(-14*I*c)*log(I*e^(I*d*x +

I*c) + 1) + 1690450650*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x + I*c) - 1) + 23666309100*a^8*e^(26*I*d*x + 12

*I*c)*log(I*e^(I*d*x + I*c) - 1) + 153831009150*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 6153240

36600*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1692141100650*a^8*e^(20*I*d*x + 6*I*c)*log(I*e^(I*

d*x + I*c) - 1) + 3384282201300*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 5076423301950*a^8*e^(16*

I*d*x + 2*I*c)*log(I*e^(I*d*x + I*c) - 1) + 5076423301950*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x + I*c) - 1)

+ 3384282201300*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x + I*c) - 1) + 1692141100650*a^8*e^(8*I*d*x - 6*I*c)*lo

g(I*e^(I*d*x + I*c) - 1) + 615324036600*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x + I*c) - 1) + 153831009150*a^8*

e^(4*I*d*x - 10*I*c)*log(I*e^(I*d*x + I*c) - 1) + 23666309100*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x + I*c) -

1) + 5801626630800*a^8*e^(14*I*d*x)*log(I*e^(I*d*x + I*c) - 1) + 1690450650*a^8*e^(-14*I*c)*log(I*e^(I*d*x +

I*c) - 1) - 1635552135*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 22897729890*a^8*e^(26*I*d*x + 1

2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 148835244285*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 5953

40977140*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1637187687135*a^8*e^(20*I*d*x + 6*I*c)*log(-I*

e^(I*d*x + I*c) + 1) - 3274375374270*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 4911563061405*a^8*

e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 4911563061405*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x + I*

c) + 1) - 3274375374270*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 1637187687135*a^8*e^(8*I*d*x -

6*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 595340977140*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 148835

244285*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) + 1) - 22897729890*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I

*d*x + I*c) + 1) - 5613214927320*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x + I*c) + 1) - 1635552135*a^8*e^(-14*I*c)*log

(-I*e^(I*d*x + I*c) + 1) - 1690450650*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 23666309100*a^8*

e^(26*I*d*x + 12*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 153831009150*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x + I

*c) - 1) - 615324036600*a^8*e^(22*I*d*x + 8*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1692141100650*a^8*e^(20*I*d*x +

6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 3384282201300*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 507

6423301950*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 5076423301950*a^8*e^(12*I*d*x - 2*I*c)*log(-

I*e^(I*d*x + I*c) - 1) - 3384282201300*a^8*e^(10*I*d*x - 4*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 1692141100650*a^

8*e^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 615324036600*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x + I*c

) - 1) - 153831009150*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 23666309100*a^8*e^(2*I*d*x - 12*I

*c)*log(-I*e^(I*d*x + I*c) - 1) - 5801626630800*a^8*e^(14*I*d*x)*log(-I*e^(I*d*x + I*c) - 1) - 1690450650*a^8*

e^(-14*I*c)*log(-I*e^(I*d*x + I*c) - 1) - 151725*a^8*e^(28*I*d*x + 14*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 21241

50*a^8*e^(26*I*d*x + 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 13806975*a^8*e^(24*I*d*x + 10*I*c)*log(I*e^(I*d*x)

+ e^(-I*c)) - 55227900*a^8*e^(22*I*d*x + 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 151876725*a^8*e^(20*I*d*x + 6*I*

c)*log(I*e^(I*d*x) + e^(-I*c)) - 303753450*a^8*e^(18*I*d*x + 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 455630175*a^

8*e^(16*I*d*x + 2*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 455630175*a^8*e^(12*I*d*x - 2*I*c)*log(I*e^(I*d*x) + e^(-

I*c)) - 303753450*a^8*e^(10*I*d*x - 4*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 151876725*a^8*e^(8*I*d*x - 6*I*c)*log

(I*e^(I*d*x) + e^(-I*c)) - 55227900*a^8*e^(6*I*d*x - 8*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 13806975*a^8*e^(4*I*

d*x - 10*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 2124150*a^8*e^(2*I*d*x - 12*I*c)*log(I*e^(I*d*x) + e^(-I*c)) - 520

720200*a^8*e^(14*I*d*x)*log(I*e^(I*d*x) + e^(-I*c)) - 151725*a^8*e^(-14*I*c)*log(I*e^(I*d*x) + e^(-I*c)) + 151

725*a^8*e^(28*I*d*x + 14*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 2124150*a^8*e^(26*I*d*x + 12*I*c)*log(-I*e^(I*d*x

) + e^(-I*c)) + 13806975*a^8*e^(24*I*d*x + 10*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 55227900*a^8*e^(22*I*d*x + 8

*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 151876725*a^8*e^(20*I*d*x + 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 3037534

50*a^8*e^(18*I*d*x + 4*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 455630175*a^8*e^(16*I*d*x + 2*I*c)*log(-I*e^(I*d*x)

+ e^(-I*c)) + 455630175*a^8*e^(12*I*d*x - 2*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 303753450*a^8*e^(10*I*d*x - 4

*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 151876725*a^8*e^(8*I*d*x - 6*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 55227900

*a^8*e^(6*I*d*x - 8*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 13806975*a^8*e^(4*I*d*x - 10*I*c)*log(-I*e^(I*d*x) + e

^(-I*c)) + 2124150*a^8*e^(2*I*d*x - 12*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) + 520720200*a^8*e^(14*I*d*x)*log(-I*e

^(I*d*x) + e^(-I*c)) + 151725*a^8*e^(-14*I*c)*log(-I*e^(I*d*x) + e^(-I*c)) - 15728640*I*a^8*e^(35*I*d*x + 21*I

*c) - 198180864*I*a^8*e^(33*I*d*x + 19*I*c) - 1159725056*I*a^8*e^(31*I*d*x + 17*I*c) - 4125097984*I*a^8*e^(29*

I*d*x + 15*I*c) - 9527361536*I*a^8*e^(27*I*d*x + 13*I*c) - 12786335744*I*a^8*e^(25*I*d*x + 11*I*c) + 190840832

*I*a^8*e^(23*I*d*x + 9*I*c) + 48882515968*I*a^8*e^(21*I*d*x + 7*I*c) + 138550444032*I*a^8*e^(19*I*d*x + 5*I*c)

+ 239314403328*I*a^8*e^(17*I*d*x + 3*I*c) + 295994130432*I*a^8*e^(15*I*d*x + I*c) + 273474912256*I*a^8*e^(13*

I*d*x - I*c) + 190268309504*I*a^8*e^(11*I*d*x - 3*I*c) + 98635350016*I*a^8*e^(9*I*d*x - 5*I*c) + 37029412864*I

*a^8*e^(7*I*d*x - 7*I*c) + 9527361536*I*a^8*e^(5*I*d*x - 9*I*c) + 1504706560*I*a^8*e^(3*I*d*x - 11*I*c) + 1101

00480*I*a^8*e^(I*d*x - 13*I*c))/(d*e^(28*I*d*x + 14*I*c) + 14*d*e^(26*I*d*x + 12*I*c) + 91*d*e^(24*I*d*x + 10*

I*c) + 364*d*e^(22*I*d*x + 8*I*c) + 1001*d*e^(20*I*d*x + 6*I*c) + 2002*d*e^(18*I*d*x + 4*I*c) + 3003*d*e^(16*I

*d*x + 2*I*c) + 3003*d*e^(12*I*d*x - 2*I*c) + 2002*d*e^(10*I*d*x - 4*I*c) + 1001*d*e^(8*I*d*x - 6*I*c) + 364*d

*e^(6*I*d*x - 8*I*c) + 91*d*e^(4*I*d*x - 10*I*c) + 14*d*e^(2*I*d*x - 12*I*c) + 3432*d*e^(14*I*d*x) + d*e^(-14*

I*c))

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maple [B] time = 0.61, size = 385, normalized size = 2.53 \[ -\frac {29 a^{8} \left (\sin ^{7}\left (d x +c \right )\right )}{7 d}-\frac {a^{8} \left (\sin ^{5}\left (d x +c \right )\right )}{5 d}-\frac {a^{8} \left (\sin ^{3}\left (d x +c \right )\right )}{3 d}+\frac {139 a^{8} \sin \left (d x +c \right )}{105 d}+\frac {a^{8} \ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{d}+\frac {128 i a^{8} \cos \left (d x +c \right )}{35 d}-\frac {32 i a^{8} \left (\cos ^{3}\left (d x +c \right )\right ) \left (\sin ^{2}\left (d x +c \right )\right )}{5 d}-\frac {10 a^{8} \left (\sin ^{3}\left (d x +c \right )\right ) \left (\cos ^{4}\left (d x +c \right )\right )}{d}-\frac {232 a^{8} \left (\cos ^{4}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{35 d}+\frac {122 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right ) a^{8}}{105 d}+\frac {64 i a^{8} \cos \left (d x +c \right ) \left (\sin ^{2}\left (d x +c \right )\right )}{35 d}-\frac {8 i a^{8} \left (\cos ^{7}\left (d x +c \right )\right )}{7 d}+\frac {48 i a^{8} \cos \left (d x +c \right ) \left (\sin ^{4}\left (d x +c \right )\right )}{35 d}+\frac {29 a^{8} \left (\cos ^{6}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{7 d}+\frac {16 i a^{8} \left (\cos ^{5}\left (d x +c \right )\right )}{5 d}-\frac {8 i a^{8} \left (\sin ^{4}\left (d x +c \right )\right ) \left (\cos ^{3}\left (d x +c \right )\right )}{d}+\frac {8 i a^{8} \cos \left (d x +c \right ) \left (\sin ^{6}\left (d x +c \right )\right )}{7 d}-\frac {64 i a^{8} \left (\cos ^{3}\left (d x +c \right )\right )}{15 d}+\frac {8 i a^{8} \left (\sin ^{2}\left (d x +c \right )\right ) \left (\cos ^{5}\left (d x +c \right )\right )}{d} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^8,x)

[Out]

-29/7*a^8*sin(d*x+c)^7/d-1/5*a^8*sin(d*x+c)^5/d-1/3*a^8*sin(d*x+c)^3/d+139/105*a^8*sin(d*x+c)/d+1/d*a^8*ln(sec

(d*x+c)+tan(d*x+c))+128/35*I/d*a^8*cos(d*x+c)-32/5*I/d*a^8*cos(d*x+c)^3*sin(d*x+c)^2-10/d*a^8*sin(d*x+c)^3*cos

(d*x+c)^4-232/35/d*a^8*cos(d*x+c)^4*sin(d*x+c)+122/105/d*cos(d*x+c)^2*sin(d*x+c)*a^8+64/35*I/d*a^8*cos(d*x+c)*

sin(d*x+c)^2-8/7*I/d*a^8*cos(d*x+c)^7+48/35*I/d*a^8*cos(d*x+c)*sin(d*x+c)^4+29/7/d*a^8*cos(d*x+c)^6*sin(d*x+c)

+16/5*I/d*a^8*cos(d*x+c)^5-8*I/d*a^8*sin(d*x+c)^4*cos(d*x+c)^3+8/7*I/d*a^8*cos(d*x+c)*sin(d*x+c)^6-64/15*I/d*a

^8*cos(d*x+c)^3+8*I/d*a^8*sin(d*x+c)^2*cos(d*x+c)^5

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maxima [B] time = 0.36, size = 309, normalized size = 2.03 \[ -\frac {240 i \, a^{8} \cos \left (d x + c\right )^{7} + 840 \, a^{8} \sin \left (d x + c\right )^{7} + 112 i \, {\left (15 \, \cos \left (d x + c\right )^{7} - 42 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3}\right )} a^{8} + 336 i \, {\left (5 \, \cos \left (d x + c\right )^{7} - 7 \, \cos \left (d x + c\right )^{5}\right )} a^{8} + 48 i \, {\left (5 \, \cos \left (d x + c\right )^{7} - 21 \, \cos \left (d x + c\right )^{5} + 35 \, \cos \left (d x + c\right )^{3} - 35 \, \cos \left (d x + c\right )\right )} a^{8} + {\left (30 \, \sin \left (d x + c\right )^{7} + 42 \, \sin \left (d x + c\right )^{5} + 70 \, \sin \left (d x + c\right )^{3} - 105 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 105 \, \log \left (\sin \left (d x + c\right ) - 1\right ) + 210 \, \sin \left (d x + c\right )\right )} a^{8} + 56 \, {\left (15 \, \sin \left (d x + c\right )^{7} - 42 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3}\right )} a^{8} + 420 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 7 \, \sin \left (d x + c\right )^{5}\right )} a^{8} + 6 \, {\left (5 \, \sin \left (d x + c\right )^{7} - 21 \, \sin \left (d x + c\right )^{5} + 35 \, \sin \left (d x + c\right )^{3} - 35 \, \sin \left (d x + c\right )\right )} a^{8}}{210 \, d} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^7*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

-1/210*(240*I*a^8*cos(d*x + c)^7 + 840*a^8*sin(d*x + c)^7 + 112*I*(15*cos(d*x + c)^7 - 42*cos(d*x + c)^5 + 35*

cos(d*x + c)^3)*a^8 + 336*I*(5*cos(d*x + c)^7 - 7*cos(d*x + c)^5)*a^8 + 48*I*(5*cos(d*x + c)^7 - 21*cos(d*x +

c)^5 + 35*cos(d*x + c)^3 - 35*cos(d*x + c))*a^8 + (30*sin(d*x + c)^7 + 42*sin(d*x + c)^5 + 70*sin(d*x + c)^3 -

105*log(sin(d*x + c) + 1) + 105*log(sin(d*x + c) - 1) + 210*sin(d*x + c))*a^8 + 56*(15*sin(d*x + c)^7 - 42*si

n(d*x + c)^5 + 35*sin(d*x + c)^3)*a^8 + 420*(5*sin(d*x + c)^7 - 7*sin(d*x + c)^5)*a^8 + 6*(5*sin(d*x + c)^7 -

21*sin(d*x + c)^5 + 35*sin(d*x + c)^3 - 35*sin(d*x + c))*a^8)/d

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mupad [B] time = 6.69, size = 207, normalized size = 1.36 \[ \frac {2\,a^8\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}+\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,16{}\mathrm {i}-\frac {80\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{3}-\frac {a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,224{}\mathrm {i}}{3}+\frac {224\,a^8\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{5}+\frac {a^8\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,304{}\mathrm {i}}{15}-\frac {304\,a^8}{105}}{d\,\left (-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,7{}\mathrm {i}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,35{}\mathrm {i}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,21{}\mathrm {i}+7\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1{}\mathrm {i}\right )} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^7*(a + a*tan(c + d*x)*1i)^8,x)

[Out]

(2*a^8*atanh(tan(c/2 + (d*x)/2)))/d + ((224*a^8*tan(c/2 + (d*x)/2)^2)/5 - (a^8*tan(c/2 + (d*x)/2)^3*224i)/3 -

(80*a^8*tan(c/2 + (d*x)/2)^4)/3 + a^8*tan(c/2 + (d*x)/2)^5*16i - (304*a^8)/105 + (a^8*tan(c/2 + (d*x)/2)*304i)

/15)/(d*(7*tan(c/2 + (d*x)/2) - tan(c/2 + (d*x)/2)^2*21i - 35*tan(c/2 + (d*x)/2)^3 + tan(c/2 + (d*x)/2)^4*35i

+ 21*tan(c/2 + (d*x)/2)^5 - tan(c/2 + (d*x)/2)^6*7i - tan(c/2 + (d*x)/2)^7 + 1i))

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sympy [A] time = 1.01, size = 189, normalized size = 1.24 \[ \frac {a^{8} \left (- \log {\left (e^{i d x} - i e^{- i c} \right )} + \log {\left (e^{i d x} + i e^{- i c} \right )}\right )}{d} + \begin {cases} \frac {- 30 i a^{8} d^{3} e^{7 i c} e^{7 i d x} + 42 i a^{8} d^{3} e^{5 i c} e^{5 i d x} - 70 i a^{8} d^{3} e^{3 i c} e^{3 i d x} + 210 i a^{8} d^{3} e^{i c} e^{i d x}}{105 d^{4}} & \text {for}\: 105 d^{4} \neq 0 \\x \left (2 a^{8} e^{7 i c} - 2 a^{8} e^{5 i c} + 2 a^{8} e^{3 i c} - 2 a^{8} e^{i c}\right ) & \text {otherwise} \end {cases} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**7*(a+I*a*tan(d*x+c))**8,x)

[Out]

a**8*(-log(exp(I*d*x) - I*exp(-I*c)) + log(exp(I*d*x) + I*exp(-I*c)))/d + Piecewise(((-30*I*a**8*d**3*exp(7*I*

c)*exp(7*I*d*x) + 42*I*a**8*d**3*exp(5*I*c)*exp(5*I*d*x) - 70*I*a**8*d**3*exp(3*I*c)*exp(3*I*d*x) + 210*I*a**8

*d**3*exp(I*c)*exp(I*d*x))/(105*d**4), Ne(105*d**4, 0)), (x*(2*a**8*exp(7*I*c) - 2*a**8*exp(5*I*c) + 2*a**8*ex

p(3*I*c) - 2*a**8*exp(I*c)), True))

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