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

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

[Out]

1/15*I*sec(d*x+c)/d/(a+I*a*tan(d*x+c))^8+7/195*I*sec(d*x+c)/a/d/(a+I*a*tan(d*x+c))^7+14/715*I*sec(d*x+c)/a^2/d
/(a+I*a*tan(d*x+c))^6+14/1287*I*sec(d*x+c)/a^3/d/(a+I*a*tan(d*x+c))^5+8/1287*I*sec(d*x+c)/d/(a^2+I*a^2*tan(d*x
+c))^4+8/2145*I*sec(d*x+c)/a^2/d/(a^2+I*a^2*tan(d*x+c))^3+16/6435*I*sec(d*x+c)/d/(a^4+I*a^4*tan(d*x+c))^2+16/6
435*I*sec(d*x+c)/d/(a^8+I*a^8*tan(d*x+c))

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Rubi [A]  time = 0.26, antiderivative size = 269, normalized size of antiderivative = 1.00, 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 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]

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]

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*}

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Mathematica [A]  time = 0.64, 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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fricas [A]  time = 0.59, size = 96, normalized size = 0.36 \[ \frac {{\left (6435 i \, e^{\left (14 i \, d x + 14 i \, c\right )} + 15015 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 27027 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 32175 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 25025 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 12285 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3465 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 429 i\right )} e^{\left (-15 i \, d x - 15 i \, c\right )}}{823680 \, a^{8} d} \]

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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giac [A]  time = 3.58, size = 203, normalized size = 0.75 \[ \frac {2 \, {\left (6435 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{14} - 45045 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{13} - 210210 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{12} + 630630 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{11} + 1414413 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 2357355 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 3063060 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 3063060 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 2407405 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 1444443 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 668850 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 222950 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 54915 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 7845 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 952\right )}}{6435 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{15}} \]

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)

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maple [A]  time = 0.25, size = 255, normalized size = 0.95 \[ \frac {\frac {15008 i}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{10}}-\frac {2944 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{8}}+\frac {29792}{9 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{9}}+\frac {14 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{2}}-\frac {224 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{4}}+\frac {3752 i}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{6}}-\frac {23744}{11 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{11}}-\frac {2128}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{7}}-\frac {256}{15 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{15}}-\frac {196}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{3}}+\frac {2968}{5 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{5}}+\frac {2}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i}+\frac {128 i}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{14}}+\frac {6272}{13 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{13}}-\frac {3584 i}{3 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-i\right )^{12}}}{d \,a^{8}} \]

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

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maxima [A]  time = 0.65, size = 179, normalized size = 0.67 \[ \frac {429 i \, \cos \left (15 \, d x + 15 \, c\right ) + 3465 i \, \cos \left (13 \, d x + 13 \, c\right ) + 12285 i \, \cos \left (11 \, d x + 11 \, c\right ) + 25025 i \, \cos \left (9 \, d x + 9 \, c\right ) + 32175 i \, \cos \left (7 \, d x + 7 \, c\right ) + 27027 i \, \cos \left (5 \, d x + 5 \, c\right ) + 15015 i \, \cos \left (3 \, d x + 3 \, c\right ) + 6435 i \, \cos \left (d x + c\right ) + 429 \, \sin \left (15 \, d x + 15 \, c\right ) + 3465 \, \sin \left (13 \, d x + 13 \, c\right ) + 12285 \, \sin \left (11 \, d x + 11 \, c\right ) + 25025 \, \sin \left (9 \, d x + 9 \, c\right ) + 32175 \, \sin \left (7 \, d x + 7 \, c\right ) + 27027 \, \sin \left (5 \, d x + 5 \, c\right ) + 15015 \, \sin \left (3 \, d x + 3 \, c\right ) + 6435 \, \sin \left (d x + c\right )}{823680 \, a^{8} d} \]

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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mupad [B]  time = 5.36, size = 224, normalized size = 0.83 \[ \frac {2\,\left (2\,{\sin \left (\frac {c}{4}+\frac {d\,x}{4}\right )}^2-1\right )\,\left (-\frac {{\sin \left (c+d\,x\right )}^2\,44779{}\mathrm {i}}{32}+\frac {32175\,\sin \left (c+d\,x\right )}{128}-\frac {{\sin \left (2\,c+2\,d\,x\right )}^2\,26075{}\mathrm {i}}{16}-\frac {3575\,\sin \left (2\,c+2\,d\,x\right )}{8}+\frac {{\sin \left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,114583{}\mathrm {i}}{64}-\frac {{\sin \left (3\,c+3\,d\,x\right )}^2\,57925{}\mathrm {i}}{32}+\frac {84227\,\sin \left (3\,c+3\,d\,x\right )}{128}+\frac {{\sin \left (\frac {3\,c}{2}+\frac {3\,d\,x}{2}\right )}^2\,116585{}\mathrm {i}}{64}+\frac {{\sin \left (\frac {5\,c}{2}+\frac {5\,d\,x}{2}\right )}^2\,119315{}\mathrm {i}}{64}+\frac {{\sin \left (\frac {7\,c}{2}+\frac {7\,d\,x}{2}\right )}^2\,122285{}\mathrm {i}}{64}-754\,\sin \left (4\,c+4\,d\,x\right )+\frac {111527\,\sin \left (5\,c+5\,d\,x\right )}{128}-\frac {7187\,\sin \left (6\,c+6\,d\,x\right )}{8}+\frac {121427\,\sin \left (7\,c+7\,d\,x\right )}{128}-952{}\mathrm {i}\right )}{6435\,a^8\,d\,\left (-2\,{\sin \left (\frac {15\,c}{4}+\frac {15\,d\,x}{4}\right )}^2+\sin \left (\frac {15\,c}{2}+\frac {15\,d\,x}{2}\right )\,1{}\mathrm {i}+1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*(2*sin(c/4 + (d*x)/4)^2 - 1)*((32175*sin(c + d*x))/128 - (3575*sin(2*c + 2*d*x))/8 + (84227*sin(3*c + 3*d*x
))/128 - 754*sin(4*c + 4*d*x) + (111527*sin(5*c + 5*d*x))/128 - (7187*sin(6*c + 6*d*x))/8 + (121427*sin(7*c +
7*d*x))/128 - (sin(2*c + 2*d*x)^2*26075i)/16 + (sin(c/2 + (d*x)/2)^2*114583i)/64 - (sin(3*c + 3*d*x)^2*57925i)
/32 + (sin((3*c)/2 + (3*d*x)/2)^2*116585i)/64 + (sin((5*c)/2 + (5*d*x)/2)^2*119315i)/64 + (sin((7*c)/2 + (7*d*
x)/2)^2*122285i)/64 - (sin(c + d*x)^2*44779i)/32 - 952i))/(6435*a^8*d*(sin((15*c)/2 + (15*d*x)/2)*1i - 2*sin((
15*c)/4 + (15*d*x)/4)^2 + 1))

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sympy [A]  time = 36.32, size = 1221, normalized size = 4.54 \[ \text {result too large to display} \]

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]

Piecewise((16*tan(c + d*x)**7*sec(c + d*x)/(6435*a**8*d*tan(c + d*x)**8 - 51480*I*a**8*d*tan(c + d*x)**7 - 180
180*a**8*d*tan(c + d*x)**6 + 360360*I*a**8*d*tan(c + d*x)**5 + 450450*a**8*d*tan(c + d*x)**4 - 360360*I*a**8*d
*tan(c + d*x)**3 - 180180*a**8*d*tan(c + d*x)**2 + 51480*I*a**8*d*tan(c + d*x) + 6435*a**8*d) - 128*I*tan(c +
d*x)**6*sec(c + d*x)/(6435*a**8*d*tan(c + d*x)**8 - 51480*I*a**8*d*tan(c + d*x)**7 - 180180*a**8*d*tan(c + d*x
)**6 + 360360*I*a**8*d*tan(c + d*x)**5 + 450450*a**8*d*tan(c + d*x)**4 - 360360*I*a**8*d*tan(c + d*x)**3 - 180
180*a**8*d*tan(c + d*x)**2 + 51480*I*a**8*d*tan(c + d*x) + 6435*a**8*d) - 456*tan(c + d*x)**5*sec(c + d*x)/(64
35*a**8*d*tan(c + d*x)**8 - 51480*I*a**8*d*tan(c + d*x)**7 - 180180*a**8*d*tan(c + d*x)**6 + 360360*I*a**8*d*t
an(c + d*x)**5 + 450450*a**8*d*tan(c + d*x)**4 - 360360*I*a**8*d*tan(c + d*x)**3 - 180180*a**8*d*tan(c + d*x)*
*2 + 51480*I*a**8*d*tan(c + d*x) + 6435*a**8*d) + 960*I*tan(c + d*x)**4*sec(c + d*x)/(6435*a**8*d*tan(c + d*x)
**8 - 51480*I*a**8*d*tan(c + d*x)**7 - 180180*a**8*d*tan(c + d*x)**6 + 360360*I*a**8*d*tan(c + d*x)**5 + 45045
0*a**8*d*tan(c + d*x)**4 - 360360*I*a**8*d*tan(c + d*x)**3 - 180180*a**8*d*tan(c + d*x)**2 + 51480*I*a**8*d*ta
n(c + d*x) + 6435*a**8*d) + 1350*tan(c + d*x)**3*sec(c + d*x)/(6435*a**8*d*tan(c + d*x)**8 - 51480*I*a**8*d*ta
n(c + d*x)**7 - 180180*a**8*d*tan(c + d*x)**6 + 360360*I*a**8*d*tan(c + d*x)**5 + 450450*a**8*d*tan(c + d*x)**
4 - 360360*I*a**8*d*tan(c + d*x)**3 - 180180*a**8*d*tan(c + d*x)**2 + 51480*I*a**8*d*tan(c + d*x) + 6435*a**8*
d) - 1392*I*tan(c + d*x)**2*sec(c + d*x)/(6435*a**8*d*tan(c + d*x)**8 - 51480*I*a**8*d*tan(c + d*x)**7 - 18018
0*a**8*d*tan(c + d*x)**6 + 360360*I*a**8*d*tan(c + d*x)**5 + 450450*a**8*d*tan(c + d*x)**4 - 360360*I*a**8*d*t
an(c + d*x)**3 - 180180*a**8*d*tan(c + d*x)**2 + 51480*I*a**8*d*tan(c + d*x) + 6435*a**8*d) - 1181*tan(c + d*x
)*sec(c + d*x)/(6435*a**8*d*tan(c + d*x)**8 - 51480*I*a**8*d*tan(c + d*x)**7 - 180180*a**8*d*tan(c + d*x)**6 +
360360*I*a**8*d*tan(c + d*x)**5 + 450450*a**8*d*tan(c + d*x)**4 - 360360*I*a**8*d*tan(c + d*x)**3 - 180180*a*
*8*d*tan(c + d*x)**2 + 51480*I*a**8*d*tan(c + d*x) + 6435*a**8*d) + 952*I*sec(c + d*x)/(6435*a**8*d*tan(c + d*
x)**8 - 51480*I*a**8*d*tan(c + d*x)**7 - 180180*a**8*d*tan(c + d*x)**6 + 360360*I*a**8*d*tan(c + d*x)**5 + 450
450*a**8*d*tan(c + d*x)**4 - 360360*I*a**8*d*tan(c + d*x)**3 - 180180*a**8*d*tan(c + d*x)**2 + 51480*I*a**8*d*
tan(c + d*x) + 6435*a**8*d), Ne(d, 0)), (x*sec(c)/(I*a*tan(c) + a)**8, True))

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