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

Optimal. Leaf size=269 \[ \frac{16 i \sec (c+d x)}{6435 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{16 i \sec (c+d x)}{6435 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac{8 i \sec (c+d x)}{2145 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac{8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8} \]

[Out]

((I/15)*Sec[c + d*x])/(d*(a + I*a*Tan[c + d*x])^8) + (((7*I)/195)*Sec[c + d*x])/(a*d*(a + I*a*Tan[c + d*x])^7)
+ (((14*I)/715)*Sec[c + d*x])/(a^2*d*(a + I*a*Tan[c + d*x])^6) + (((14*I)/1287)*Sec[c + d*x])/(a^3*d*(a + I*a
*Tan[c + d*x])^5) + (((8*I)/1287)*Sec[c + d*x])/(d*(a^2 + I*a^2*Tan[c + d*x])^4) + (((8*I)/2145)*Sec[c + d*x])
/(a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) + (((16*I)/6435)*Sec[c + d*x])/(d*(a^4 + I*a^4*Tan[c + d*x])^2) + (((16*
I)/6435)*Sec[c + d*x])/(d*(a^8 + I*a^8*Tan[c + d*x]))

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Rubi [A]  time = 0.259754, antiderivative size = 269, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {3502, 3488} \[ \frac{16 i \sec (c+d x)}{6435 d \left (a^8+i a^8 \tan (c+d x)\right )}+\frac{16 i \sec (c+d x)}{6435 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac{8 i \sec (c+d x)}{2145 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac{8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/15)*Sec[c + d*x])/(d*(a + I*a*Tan[c + d*x])^8) + (((7*I)/195)*Sec[c + d*x])/(a*d*(a + I*a*Tan[c + d*x])^7)
+ (((14*I)/715)*Sec[c + d*x])/(a^2*d*(a + I*a*Tan[c + d*x])^6) + (((14*I)/1287)*Sec[c + d*x])/(a^3*d*(a + I*a
*Tan[c + d*x])^5) + (((8*I)/1287)*Sec[c + d*x])/(d*(a^2 + I*a^2*Tan[c + d*x])^4) + (((8*I)/2145)*Sec[c + d*x])
/(a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) + (((16*I)/6435)*Sec[c + d*x])/(d*(a^4 + I*a^4*Tan[c + d*x])^2) + (((16*
I)/6435)*Sec[c + d*x])/(d*(a^8 + I*a^8*Tan[c + d*x]))

Rule 3502

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(a*(d
*Sec[e + f*x])^m*(a + b*Tan[e + f*x])^n)/(b*f*(m + 2*n)), x] + Dist[Simplify[m + n]/(a*(m + 2*n)), Int[(d*Sec[
e + f*x])^m*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a^2 + b^2, 0] && LtQ[n
, 0] && NeQ[m + 2*n, 0] && 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 \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{15 a}\\ &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{14 \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{65 a^2}\\ &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{14 \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^5} \, dx}{143 a^3}\\ &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{56 \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{1287 a^4}\\ &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{8 \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^3} \, dx}{429 a^5}\\ &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{8 i \sec (c+d x)}{2145 a^5 d (a+i a \tan (c+d x))^3}+\frac{8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{16 \int \frac{\sec (c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{2145 a^6}\\ &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{8 i \sec (c+d x)}{2145 a^5 d (a+i a \tan (c+d x))^3}+\frac{8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{16 i \sec (c+d x)}{6435 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac{16 \int \frac{\sec (c+d x)}{a+i a \tan (c+d x)} \, dx}{6435 a^7}\\ &=\frac{i \sec (c+d x)}{15 d (a+i a \tan (c+d x))^8}+\frac{7 i \sec (c+d x)}{195 a d (a+i a \tan (c+d x))^7}+\frac{14 i \sec (c+d x)}{715 a^2 d (a+i a \tan (c+d x))^6}+\frac{14 i \sec (c+d x)}{1287 a^3 d (a+i a \tan (c+d x))^5}+\frac{8 i \sec (c+d x)}{2145 a^5 d (a+i a \tan (c+d x))^3}+\frac{8 i \sec (c+d x)}{1287 d \left (a^2+i a^2 \tan (c+d x)\right )^4}+\frac{16 i \sec (c+d x)}{6435 d \left (a^4+i a^4 \tan (c+d x)\right )^2}+\frac{16 i \sec (c+d x)}{6435 d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end{align*}

Mathematica [A]  time = 0.428746, size = 117, normalized size = 0.43 \[ \frac{i \sec ^8(c+d x) (3575 i \sin (c+d x)+7371 i \sin (3 (c+d x))+5775 i \sin (5 (c+d x))+3003 i \sin (7 (c+d x))+28600 \cos (c+d x)+19656 \cos (3 (c+d x))+9240 \cos (5 (c+d x))+3432 \cos (7 (c+d x)))}{411840 a^8 d (\tan (c+d x)-i)^8} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((I/411840)*Sec[c + d*x]^8*(28600*Cos[c + d*x] + 19656*Cos[3*(c + d*x)] + 9240*Cos[5*(c + d*x)] + 3432*Cos[7*(
c + d*x)] + (3575*I)*Sin[c + d*x] + (7371*I)*Sin[3*(c + d*x)] + (5775*I)*Sin[5*(c + d*x)] + (3003*I)*Sin[7*(c
+ d*x)]))/(a^8*d*(-I + Tan[c + d*x])^8)

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Maple [A]  time = 0.059, size = 255, normalized size = 1. \begin{align*} 2\,{\frac{1}{d{a}^{8}} \left ({\frac{-{\frac{1792\,i}{3}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{12}}}+{\frac{{\frac{1876\,i}{3}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{6}}}-{\frac{1472\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{8}}}+ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-1}+{\frac{14896}{9\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{9}}}-{\frac{112\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{4}}}+{\frac{64\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{14}}}-{\frac{11872}{11\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{11}}}+{\frac{7\,i}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{2}}}+{\frac{{\frac{7504\,i}{5}}}{ \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{10}}}-1064\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{-7}-{\frac{98}{3\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{3}}}+{\frac{3136}{13\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{13}}}+{\frac{1484}{5\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{5}}}-{\frac{128}{15\, \left ( \tan \left ( 1/2\,dx+c/2 \right ) -i \right ) ^{15}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/d/a^8*(-1792/3*I/(tan(1/2*d*x+1/2*c)-I)^12+1876/3*I/(tan(1/2*d*x+1/2*c)-I)^6-1472*I/(tan(1/2*d*x+1/2*c)-I)^8
+1/(tan(1/2*d*x+1/2*c)-I)+14896/9/(tan(1/2*d*x+1/2*c)-I)^9-112*I/(tan(1/2*d*x+1/2*c)-I)^4+64*I/(tan(1/2*d*x+1/
2*c)-I)^14-11872/11/(tan(1/2*d*x+1/2*c)-I)^11+7*I/(tan(1/2*d*x+1/2*c)-I)^2+7504/5*I/(tan(1/2*d*x+1/2*c)-I)^10-
1064/(tan(1/2*d*x+1/2*c)-I)^7-98/3/(tan(1/2*d*x+1/2*c)-I)^3+3136/13/(tan(1/2*d*x+1/2*c)-I)^13+1484/5/(tan(1/2*
d*x+1/2*c)-I)^5-128/15/(tan(1/2*d*x+1/2*c)-I)^15)

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Maxima [A]  time = 1.25304, size = 242, normalized size = 0.9 \begin{align*} \frac{429 i \, \cos \left (15 \, d x + 15 \, c\right ) + 3465 i \, \cos \left (13 \, d x + 13 \, c\right ) + 12285 i \, \cos \left (11 \, d x + 11 \, c\right ) + 25025 i \, \cos \left (9 \, d x + 9 \, c\right ) + 32175 i \, \cos \left (7 \, d x + 7 \, c\right ) + 27027 i \, \cos \left (5 \, d x + 5 \, c\right ) + 15015 i \, \cos \left (3 \, d x + 3 \, c\right ) + 6435 i \, \cos \left (d x + c\right ) + 429 \, \sin \left (15 \, d x + 15 \, c\right ) + 3465 \, \sin \left (13 \, d x + 13 \, c\right ) + 12285 \, \sin \left (11 \, d x + 11 \, c\right ) + 25025 \, \sin \left (9 \, d x + 9 \, c\right ) + 32175 \, \sin \left (7 \, d x + 7 \, c\right ) + 27027 \, \sin \left (5 \, d x + 5 \, c\right ) + 15015 \, \sin \left (3 \, d x + 3 \, c\right ) + 6435 \, \sin \left (d x + c\right )}{823680 \, a^{8} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/823680*(429*I*cos(15*d*x + 15*c) + 3465*I*cos(13*d*x + 13*c) + 12285*I*cos(11*d*x + 11*c) + 25025*I*cos(9*d*
x + 9*c) + 32175*I*cos(7*d*x + 7*c) + 27027*I*cos(5*d*x + 5*c) + 15015*I*cos(3*d*x + 3*c) + 6435*I*cos(d*x + c
) + 429*sin(15*d*x + 15*c) + 3465*sin(13*d*x + 13*c) + 12285*sin(11*d*x + 11*c) + 25025*sin(9*d*x + 9*c) + 321
75*sin(7*d*x + 7*c) + 27027*sin(5*d*x + 5*c) + 15015*sin(3*d*x + 3*c) + 6435*sin(d*x + c))/(a^8*d)

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Fricas [A]  time = 2.53364, size = 355, normalized size = 1.32 \begin{align*} \frac{{\left (6435 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 15015 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 27027 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 32175 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 25025 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 12285 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3465 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 429 i\right )} e^{\left (-15 i \, d x - 15 i \, c\right )}}{823680 \, a^{8} d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/823680*(6435*I*e^(14*I*d*x + 14*I*c) + 15015*I*e^(12*I*d*x + 12*I*c) + 27027*I*e^(10*I*d*x + 10*I*c) + 32175
*I*e^(8*I*d*x + 8*I*c) + 25025*I*e^(6*I*d*x + 6*I*c) + 12285*I*e^(4*I*d*x + 4*I*c) + 3465*I*e^(2*I*d*x + 2*I*c
) + 429*I)*e^(-15*I*d*x - 15*I*c)/(a^8*d)

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

[Out]

Exception raised: AttributeError

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Giac [A]  time = 1.16317, size = 274, normalized size = 1.02 \begin{align*} \frac{2 \,{\left (6435 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{14} - 45045 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{13} - 210210 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{12} + 630630 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{11} + 1414413 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{10} - 2357355 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{9} - 3063060 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{8} + 3063060 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{7} + 2407405 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{6} - 1444443 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{5} - 668850 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{4} + 222950 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{3} + 54915 \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right )^{2} - 7845 i \, \tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - 952\right )}}{6435 \, a^{8} d{\left (\tan \left (\frac{1}{2} \, d x + \frac{1}{2} \, c\right ) - i\right )}^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/6435*(6435*tan(1/2*d*x + 1/2*c)^14 - 45045*I*tan(1/2*d*x + 1/2*c)^13 - 210210*tan(1/2*d*x + 1/2*c)^12 + 6306
30*I*tan(1/2*d*x + 1/2*c)^11 + 1414413*tan(1/2*d*x + 1/2*c)^10 - 2357355*I*tan(1/2*d*x + 1/2*c)^9 - 3063060*ta
n(1/2*d*x + 1/2*c)^8 + 3063060*I*tan(1/2*d*x + 1/2*c)^7 + 2407405*tan(1/2*d*x + 1/2*c)^6 - 1444443*I*tan(1/2*d
*x + 1/2*c)^5 - 668850*tan(1/2*d*x + 1/2*c)^4 + 222950*I*tan(1/2*d*x + 1/2*c)^3 + 54915*tan(1/2*d*x + 1/2*c)^2
- 7845*I*tan(1/2*d*x + 1/2*c) - 952)/(a^8*d*(tan(1/2*d*x + 1/2*c) - I)^15)