3.166 \(\int \frac{\sec ^{14}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\)

Optimal. Leaf size=134 \[ \frac{\tan ^5(c+d x)}{5 a^8 d}+\frac{2 i \tan ^4(c+d x)}{a^8 d}-\frac{10 \tan ^3(c+d x)}{a^8 d}-\frac{36 i \tan ^2(c+d x)}{a^8 d}+\frac{129 \tan (c+d x)}{a^8 d}+\frac{64 i}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac{192 i \log (\cos (c+d x))}{a^8 d}-\frac{192 x}{a^8} \]

[Out]

(-192*x)/a^8 - ((192*I)*Log[Cos[c + d*x]])/(a^8*d) + (129*Tan[c + d*x])/(a^8*d) - ((36*I)*Tan[c + d*x]^2)/(a^8
*d) - (10*Tan[c + d*x]^3)/(a^8*d) + ((2*I)*Tan[c + d*x]^4)/(a^8*d) + Tan[c + d*x]^5/(5*a^8*d) + (64*I)/(d*(a^8
+ I*a^8*Tan[c + d*x]))

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Rubi [A]  time = 0.0792108, antiderivative size = 134, 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 = {3487, 43} \[ \frac{\tan ^5(c+d x)}{5 a^8 d}+\frac{2 i \tan ^4(c+d x)}{a^8 d}-\frac{10 \tan ^3(c+d x)}{a^8 d}-\frac{36 i \tan ^2(c+d x)}{a^8 d}+\frac{129 \tan (c+d x)}{a^8 d}+\frac{64 i}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac{192 i \log (\cos (c+d x))}{a^8 d}-\frac{192 x}{a^8} \]

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^14/(a + I*a*Tan[c + d*x])^8,x]

[Out]

(-192*x)/a^8 - ((192*I)*Log[Cos[c + d*x]])/(a^8*d) + (129*Tan[c + d*x])/(a^8*d) - ((36*I)*Tan[c + d*x]^2)/(a^8
*d) - (10*Tan[c + d*x]^3)/(a^8*d) + ((2*I)*Tan[c + d*x]^4)/(a^8*d) + Tan[c + d*x]^5/(5*a^8*d) + (64*I)/(d*(a^8
+ I*a^8*Tan[c + d*x]))

Rule 3487

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Dist[1/(a^(m - 2)*b
*f), Subst[Int[(a - x)^(m/2 - 1)*(a + x)^(n + m/2 - 1), x], x, b*Tan[e + f*x]], x] /; FreeQ[{a, b, e, f, n}, x
] && EqQ[a^2 + b^2, 0] && IntegerQ[m/2]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{\sec ^{14}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=-\frac{i \operatorname{Subst}\left (\int \frac{(a-x)^6}{(a+x)^2} \, dx,x,i a \tan (c+d x)\right )}{a^{13} d}\\ &=-\frac{i \operatorname{Subst}\left (\int \left (129 a^4-72 a^3 x+30 a^2 x^2-8 a x^3+x^4+\frac{64 a^6}{(a+x)^2}-\frac{192 a^5}{a+x}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^{13} d}\\ &=-\frac{192 x}{a^8}-\frac{192 i \log (\cos (c+d x))}{a^8 d}+\frac{129 \tan (c+d x)}{a^8 d}-\frac{36 i \tan ^2(c+d x)}{a^8 d}-\frac{10 \tan ^3(c+d x)}{a^8 d}+\frac{2 i \tan ^4(c+d x)}{a^8 d}+\frac{\tan ^5(c+d x)}{5 a^8 d}+\frac{64 i}{d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [B]  time = 2.71195, size = 599, normalized size = 4.47 \[ \frac{\sec (c) \sec ^{13}(c+d x) (-\cos (7 (c+d x))-i \sin (7 (c+d x))) (300 i d x \sin (c+2 d x)-985 \sin (c+2 d x)+300 i d x \sin (3 c+2 d x)+320 \sin (3 c+2 d x)+240 i d x \sin (3 c+4 d x)-512 \sin (3 c+4 d x)+240 i d x \sin (5 c+4 d x)+10 \sin (5 c+4 d x)+60 i d x \sin (5 c+6 d x)-97 \sin (5 c+6 d x)+60 i d x \sin (7 c+6 d x)-10 \sin (7 c+6 d x)+900 d x \cos (3 c+2 d x)-220 i \cos (3 c+2 d x)+360 d x \cos (3 c+4 d x)+238 i \cos (3 c+4 d x)+360 d x \cos (5 c+4 d x)-110 i \cos (5 c+4 d x)+60 d x \cos (5 c+6 d x)+77 i \cos (5 c+6 d x)+60 d x \cos (7 c+6 d x)-10 i \cos (7 c+6 d x)+900 i \cos (3 c+2 d x) \log (\cos (c+d x))+10 \cos (c) (120 i \log (\cos (c+d x))+120 d x-7 i)+5 \cos (c+2 d x) (180 i \log (\cos (c+d x))+180 d x+43 i)+360 i \cos (3 c+4 d x) \log (\cos (c+d x))+360 i \cos (5 c+4 d x) \log (\cos (c+d x))+60 i \cos (5 c+6 d x) \log (\cos (c+d x))+60 i \cos (7 c+6 d x) \log (\cos (c+d x))-300 \sin (c+2 d x) \log (\cos (c+d x))-300 \sin (3 c+2 d x) \log (\cos (c+d x))-240 \sin (3 c+4 d x) \log (\cos (c+d x))-240 \sin (5 c+4 d x) \log (\cos (c+d x))-60 \sin (5 c+6 d x) \log (\cos (c+d x))-60 \sin (7 c+6 d x) \log (\cos (c+d x))+870 \sin (c))}{20 a^8 d (\tan (c+d x)-i)^8} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^14/(a + I*a*Tan[c + d*x])^8,x]

[Out]

(Sec[c]*Sec[c + d*x]^13*(-Cos[7*(c + d*x)] - I*Sin[7*(c + d*x)])*((-220*I)*Cos[3*c + 2*d*x] + 900*d*x*Cos[3*c
+ 2*d*x] + (238*I)*Cos[3*c + 4*d*x] + 360*d*x*Cos[3*c + 4*d*x] - (110*I)*Cos[5*c + 4*d*x] + 360*d*x*Cos[5*c +
4*d*x] + (77*I)*Cos[5*c + 6*d*x] + 60*d*x*Cos[5*c + 6*d*x] - (10*I)*Cos[7*c + 6*d*x] + 60*d*x*Cos[7*c + 6*d*x]
+ 10*Cos[c]*(-7*I + 120*d*x + (120*I)*Log[Cos[c + d*x]]) + 5*Cos[c + 2*d*x]*(43*I + 180*d*x + (180*I)*Log[Cos
[c + d*x]]) + (900*I)*Cos[3*c + 2*d*x]*Log[Cos[c + d*x]] + (360*I)*Cos[3*c + 4*d*x]*Log[Cos[c + d*x]] + (360*I
)*Cos[5*c + 4*d*x]*Log[Cos[c + d*x]] + (60*I)*Cos[5*c + 6*d*x]*Log[Cos[c + d*x]] + (60*I)*Cos[7*c + 6*d*x]*Log
[Cos[c + d*x]] + 870*Sin[c] - 985*Sin[c + 2*d*x] + (300*I)*d*x*Sin[c + 2*d*x] - 300*Log[Cos[c + d*x]]*Sin[c +
2*d*x] + 320*Sin[3*c + 2*d*x] + (300*I)*d*x*Sin[3*c + 2*d*x] - 300*Log[Cos[c + d*x]]*Sin[3*c + 2*d*x] - 512*Si
n[3*c + 4*d*x] + (240*I)*d*x*Sin[3*c + 4*d*x] - 240*Log[Cos[c + d*x]]*Sin[3*c + 4*d*x] + 10*Sin[5*c + 4*d*x] +
(240*I)*d*x*Sin[5*c + 4*d*x] - 240*Log[Cos[c + d*x]]*Sin[5*c + 4*d*x] - 97*Sin[5*c + 6*d*x] + (60*I)*d*x*Sin[
5*c + 6*d*x] - 60*Log[Cos[c + d*x]]*Sin[5*c + 6*d*x] - 10*Sin[7*c + 6*d*x] + (60*I)*d*x*Sin[7*c + 6*d*x] - 60*
Log[Cos[c + d*x]]*Sin[7*c + 6*d*x]))/(20*a^8*d*(-I + Tan[c + d*x])^8)

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Maple [A]  time = 0.132, size = 120, normalized size = 0.9 \begin{align*} 129\,{\frac{\tan \left ( dx+c \right ) }{d{a}^{8}}}+{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{5}}{5\,d{a}^{8}}}+{\frac{2\,i \left ( \tan \left ( dx+c \right ) \right ) ^{4}}{d{a}^{8}}}-10\,{\frac{ \left ( \tan \left ( dx+c \right ) \right ) ^{3}}{d{a}^{8}}}-{\frac{36\,i \left ( \tan \left ( dx+c \right ) \right ) ^{2}}{d{a}^{8}}}+{\frac{192\,i\ln \left ( \tan \left ( dx+c \right ) -i \right ) }{d{a}^{8}}}+64\,{\frac{1}{d{a}^{8} \left ( \tan \left ( dx+c \right ) -i \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^14/(a+I*a*tan(d*x+c))^8,x)

[Out]

129*tan(d*x+c)/a^8/d+1/5*tan(d*x+c)^5/a^8/d+2*I*tan(d*x+c)^4/a^8/d-10*tan(d*x+c)^3/a^8/d-36*I*tan(d*x+c)^2/a^8
/d+192*I/d/a^8*ln(tan(d*x+c)-I)+64/d/a^8/(tan(d*x+c)-I)

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Maxima [A]  time = 1.19722, size = 313, normalized size = 2.34 \begin{align*} \frac{\frac{5 \,{\left (2240 \, \tan \left (d x + c\right )^{6} - 13440 i \, \tan \left (d x + c\right )^{5} - 33600 \, \tan \left (d x + c\right )^{4} + 44800 i \, \tan \left (d x + c\right )^{3} + 33600 \, \tan \left (d x + c\right )^{2} - 13440 i \, \tan \left (d x + c\right ) - 2240\right )}}{35 \, a^{8} \tan \left (d x + c\right )^{7} - 245 i \, a^{8} \tan \left (d x + c\right )^{6} - 735 \, a^{8} \tan \left (d x + c\right )^{5} + 1225 i \, a^{8} \tan \left (d x + c\right )^{4} + 1225 \, a^{8} \tan \left (d x + c\right )^{3} - 735 i \, a^{8} \tan \left (d x + c\right )^{2} - 245 \, a^{8} \tan \left (d x + c\right ) + 35 i \, a^{8}} + \frac{\tan \left (d x + c\right )^{5} + 10 i \, \tan \left (d x + c\right )^{4} - 50 \, \tan \left (d x + c\right )^{3} - 180 i \, \tan \left (d x + c\right )^{2} + 645 \, \tan \left (d x + c\right )}{a^{8}} + \frac{960 i \, \log \left (i \, \tan \left (d x + c\right ) + 1\right )}{a^{8}}}{5 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^14/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

1/5*(5*(2240*tan(d*x + c)^6 - 13440*I*tan(d*x + c)^5 - 33600*tan(d*x + c)^4 + 44800*I*tan(d*x + c)^3 + 33600*t
an(d*x + c)^2 - 13440*I*tan(d*x + c) - 2240)/(35*a^8*tan(d*x + c)^7 - 245*I*a^8*tan(d*x + c)^6 - 735*a^8*tan(d
*x + c)^5 + 1225*I*a^8*tan(d*x + c)^4 + 1225*a^8*tan(d*x + c)^3 - 735*I*a^8*tan(d*x + c)^2 - 245*a^8*tan(d*x +
c) + 35*I*a^8) + (tan(d*x + c)^5 + 10*I*tan(d*x + c)^4 - 50*tan(d*x + c)^3 - 180*I*tan(d*x + c)^2 + 645*tan(d
*x + c))/a^8 + 960*I*log(I*tan(d*x + c) + 1)/a^8)/d

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Fricas [B]  time = 3.33474, size = 878, normalized size = 6.55 \begin{align*} -\frac{1920 \, d x e^{\left (12 i \, d x + 12 i \, c\right )} +{\left (9600 \, d x - 960 i\right )} e^{\left (10 i \, d x + 10 i \, c\right )} +{\left (19200 \, d x - 4320 i\right )} e^{\left (8 i \, d x + 8 i \, c\right )} +{\left (19200 \, d x - 7520 i\right )} e^{\left (6 i \, d x + 6 i \, c\right )} +{\left (9600 \, d x - 6160 i\right )} e^{\left (4 i \, d x + 4 i \, c\right )} +{\left (1920 \, d x - 2192 i\right )} e^{\left (2 i \, d x + 2 i \, c\right )} -{\left (-960 i \, e^{\left (12 i \, d x + 12 i \, c\right )} - 4800 i \, e^{\left (10 i \, d x + 10 i \, c\right )} - 9600 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 9600 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 4800 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 960 i \, e^{\left (2 i \, d x + 2 i \, c\right )}\right )} \log \left (e^{\left (2 i \, d x + 2 i \, c\right )} + 1\right ) - 160 i}{5 \,{\left (a^{8} d e^{\left (12 i \, d x + 12 i \, c\right )} + 5 \, a^{8} d e^{\left (10 i \, d x + 10 i \, c\right )} + 10 \, a^{8} d e^{\left (8 i \, d x + 8 i \, c\right )} + 10 \, a^{8} d e^{\left (6 i \, d x + 6 i \, c\right )} + 5 \, a^{8} d e^{\left (4 i \, d x + 4 i \, c\right )} + a^{8} d e^{\left (2 i \, d x + 2 i \, c\right )}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^14/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

-1/5*(1920*d*x*e^(12*I*d*x + 12*I*c) + (9600*d*x - 960*I)*e^(10*I*d*x + 10*I*c) + (19200*d*x - 4320*I)*e^(8*I*
d*x + 8*I*c) + (19200*d*x - 7520*I)*e^(6*I*d*x + 6*I*c) + (9600*d*x - 6160*I)*e^(4*I*d*x + 4*I*c) + (1920*d*x
- 2192*I)*e^(2*I*d*x + 2*I*c) - (-960*I*e^(12*I*d*x + 12*I*c) - 4800*I*e^(10*I*d*x + 10*I*c) - 9600*I*e^(8*I*d
*x + 8*I*c) - 9600*I*e^(6*I*d*x + 6*I*c) - 4800*I*e^(4*I*d*x + 4*I*c) - 960*I*e^(2*I*d*x + 2*I*c))*log(e^(2*I*
d*x + 2*I*c) + 1) - 160*I)/(a^8*d*e^(12*I*d*x + 12*I*c) + 5*a^8*d*e^(10*I*d*x + 10*I*c) + 10*a^8*d*e^(8*I*d*x
+ 8*I*c) + 10*a^8*d*e^(6*I*d*x + 6*I*c) + 5*a^8*d*e^(4*I*d*x + 4*I*c) + a^8*d*e^(2*I*d*x + 2*I*c))

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Sympy [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**14/(a+I*a*tan(d*x+c))**8,x)

[Out]

Exception raised: AttributeError

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Giac [B]  time = 1.25056, size = 340, normalized size = 2.54 \begin{align*} -\frac{2 \,{\left (-\frac{960 i \, \log \left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}{a^{8}} + \frac{480 i \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1 \right |}\right )}{a^{8}} + \frac{480 i \, \log \left ({\left | \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 1 \right |}\right )}{a^{8}} - \frac{5 \,{\left (-288 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 640 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 288 i\right )}}{a^{8}{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{2}} + \frac{-1096 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} + 645 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} + 5840 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} - 2780 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} - 12120 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} + 4286 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} + 12120 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} - 2780 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} - 5840 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} + 645 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) + 1096 i}{{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 1\right )}^{5} a^{8}}\right )}}{5 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^14/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

-2/5*(-960*I*log(tan(1/2*d*x + 1/2*c) - I)/a^8 + 480*I*log(abs(tan(1/2*d*x + 1/2*c) + 1))/a^8 + 480*I*log(abs(
tan(1/2*d*x + 1/2*c) - 1))/a^8 - 5*(-288*I*tan(1/2*d*x + 1/2*c)^2 - 640*tan(1/2*d*x + 1/2*c) + 288*I)/(a^8*(ta
n(1/2*d*x + 1/2*c) - I)^2) + (-1096*I*tan(1/2*d*x + 1/2*c)^10 + 645*tan(1/2*d*x + 1/2*c)^9 + 5840*I*tan(1/2*d*
x + 1/2*c)^8 - 2780*tan(1/2*d*x + 1/2*c)^7 - 12120*I*tan(1/2*d*x + 1/2*c)^6 + 4286*tan(1/2*d*x + 1/2*c)^5 + 12
120*I*tan(1/2*d*x + 1/2*c)^4 - 2780*tan(1/2*d*x + 1/2*c)^3 - 5840*I*tan(1/2*d*x + 1/2*c)^2 + 645*tan(1/2*d*x +
1/2*c) + 1096*I)/((tan(1/2*d*x + 1/2*c)^2 - 1)^5*a^8))/d