3.2.77 \(\int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\) [177]

Optimal. Leaf size=183 \[ -\frac {63 \tanh ^{-1}(\sin (c+d x))}{2 a^8 d}-\frac {63 \sec (c+d x) \tan (c+d x)}{2 a^8 d}+\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )} \]

[Out]

-63/2*arctanh(sin(d*x+c))/a^8/d-63/2*sec(d*x+c)*tan(d*x+c)/a^8/d+2/5*I*sec(d*x+c)^9/a/d/(a+I*a*tan(d*x+c))^7-6
/5*I*sec(d*x+c)^7/a^3/d/(a+I*a*tan(d*x+c))^5+42/5*I*sec(d*x+c)^5/a^2/d/(a^2+I*a^2*tan(d*x+c))^3+42*I*sec(d*x+c
)^3/d/(a^8+I*a^8*tan(d*x+c))

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Rubi [A]
time = 0.16, antiderivative size = 183, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {3581, 3853, 3855} \begin {gather*} -\frac {63 \tanh ^{-1}(\sin (c+d x))}{2 a^8 d}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac {63 \tan (c+d x) \sec (c+d x)}{2 a^8 d}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^2 d \left (a^2+i a^2 \tan (c+d x)\right )^3}+\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^11/(a + I*a*Tan[c + d*x])^8,x]

[Out]

(-63*ArcTanh[Sin[c + d*x]])/(2*a^8*d) - (63*Sec[c + d*x]*Tan[c + d*x])/(2*a^8*d) + (((2*I)/5)*Sec[c + d*x]^9)/
(a*d*(a + I*a*Tan[c + d*x])^7) - (((6*I)/5)*Sec[c + d*x]^7)/(a^3*d*(a + I*a*Tan[c + d*x])^5) + (((42*I)/5)*Sec
[c + d*x]^5)/(a^2*d*(a^2 + I*a^2*Tan[c + d*x])^3) + ((42*I)*Sec[c + d*x]^3)/(d*(a^8 + I*a^8*Tan[c + d*x]))

Rule 3581

Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[2*d^2*
(d*Sec[e + f*x])^(m - 2)*((a + b*Tan[e + f*x])^(n + 1)/(b*f*(m + 2*n))), x] - Dist[d^2*((m - 2)/(b^2*(m + 2*n)
)), Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 2), x], x] /; FreeQ[{a, b, d, e, f, m}, x] && EqQ[a
^2 + b^2, 0] && LtQ[n, -1] && ((ILtQ[n/2, 0] && IGtQ[m - 1/2, 0]) || EqQ[n, -2] || IGtQ[m + n, 0] || (Integers
Q[n, m + 1/2] && GtQ[2*m + n + 1, 0])) && IntegerQ[2*m]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

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

Rubi steps

\begin {align*} \int \frac {\sec ^{11}(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {9 \int \frac {\sec ^9(c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{5 a^2}\\ &=\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {21 \int \frac {\sec ^7(c+d x)}{(a+i a \tan (c+d x))^4} \, dx}{5 a^4}\\ &=\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^5 d (a+i a \tan (c+d x))^3}-\frac {21 \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^2} \, dx}{a^6}\\ &=\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^5 d (a+i a \tan (c+d x))^3}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac {63 \int \sec ^3(c+d x) \, dx}{a^8}\\ &=-\frac {63 \sec (c+d x) \tan (c+d x)}{2 a^8 d}+\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^5 d (a+i a \tan (c+d x))^3}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}-\frac {63 \int \sec (c+d x) \, dx}{2 a^8}\\ &=-\frac {63 \tanh ^{-1}(\sin (c+d x))}{2 a^8 d}-\frac {63 \sec (c+d x) \tan (c+d x)}{2 a^8 d}+\frac {2 i \sec ^9(c+d x)}{5 a d (a+i a \tan (c+d x))^7}-\frac {6 i \sec ^7(c+d x)}{5 a^3 d (a+i a \tan (c+d x))^5}+\frac {42 i \sec ^5(c+d x)}{5 a^5 d (a+i a \tan (c+d x))^3}+\frac {42 i \sec ^3(c+d x)}{d \left (a^8+i a^8 \tan (c+d x)\right )}\\ \end {align*}

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Mathematica [B] Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(1244\) vs. \(2(183)=366\).
time = 6.31, size = 1244, normalized size = 6.80 \begin {gather*} \frac {63 \cos (8 c) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8}{2 d (a+i a \tan (c+d x))^8}-\frac {63 \cos (8 c) \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8}{2 d (a+i a \tan (c+d x))^8}+\frac {\cos (5 d x) \sec ^8(c+d x) \left (\frac {8}{5} i \cos (3 c)-\frac {8}{5} \sin (3 c)\right ) (\cos (d x)+i \sin (d x))^8}{d (a+i a \tan (c+d x))^8}+\frac {\cos (3 d x) \sec ^8(c+d x) (-8 i \cos (5 c)+8 \sin (5 c)) (\cos (d x)+i \sin (d x))^8}{d (a+i a \tan (c+d x))^8}+\frac {\cos (d x) \sec ^8(c+d x) (48 i \cos (7 c)-48 \sin (7 c)) (\cos (d x)+i \sin (d x))^8}{d (a+i a \tan (c+d x))^8}+\frac {\sec (c) \sec ^8(c+d x) (8 i \cos (8 c)-8 \sin (8 c)) (\cos (d x)+i \sin (d x))^8}{d (a+i a \tan (c+d x))^8}+\frac {63 i \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^8(c+d x) \sin (8 c) (\cos (d x)+i \sin (d x))^8}{2 d (a+i a \tan (c+d x))^8}-\frac {63 i \log \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) \sec ^8(c+d x) \sin (8 c) (\cos (d x)+i \sin (d x))^8}{2 d (a+i a \tan (c+d x))^8}+\frac {\sec ^8(c+d x) (48 \cos (7 c)+48 i \sin (7 c)) (\cos (d x)+i \sin (d x))^8 \sin (d x)}{d (a+i a \tan (c+d x))^8}+\frac {\sec ^8(c+d x) (-8 \cos (5 c)-8 i \sin (5 c)) (\cos (d x)+i \sin (d x))^8 \sin (3 d x)}{d (a+i a \tan (c+d x))^8}+\frac {\sec ^8(c+d x) \left (\frac {8}{5} \cos (3 c)+\frac {8}{5} i \sin (3 c)\right ) (\cos (d x)+i \sin (d x))^8 \sin (5 d x)}{d (a+i a \tan (c+d x))^8}+\frac {\sec ^8(c+d x) \left (\frac {1}{4} \cos (8 c)+\frac {1}{4} i \sin (8 c)\right ) (\cos (d x)+i \sin (d x))^8}{d \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2 (a+i a \tan (c+d x))^8}+\frac {\sec ^8(c+d x) \left (-\frac {1}{4} \cos (8 c)-\frac {1}{4} i \sin (8 c)\right ) (\cos (d x)+i \sin (d x))^8}{d \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right )^2 (a+i a \tan (c+d x))^8}+\frac {8 \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8 \left (\frac {1}{2} \cos \left (8 c-\frac {d x}{2}\right )-\frac {1}{2} \cos \left (8 c+\frac {d x}{2}\right )+\frac {1}{2} i \sin \left (8 c-\frac {d x}{2}\right )-\frac {1}{2} i \sin \left (8 c+\frac {d x}{2}\right )\right )}{d \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )+\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) (a+i a \tan (c+d x))^8}+\frac {8 \sec ^8(c+d x) (\cos (d x)+i \sin (d x))^8 \left (-\frac {1}{2} \cos \left (8 c-\frac {d x}{2}\right )+\frac {1}{2} \cos \left (8 c+\frac {d x}{2}\right )-\frac {1}{2} i \sin \left (8 c-\frac {d x}{2}\right )+\frac {1}{2} i \sin \left (8 c+\frac {d x}{2}\right )\right )}{d \left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}+\frac {d x}{2}\right )-\sin \left (\frac {c}{2}+\frac {d x}{2}\right )\right ) (a+i a \tan (c+d x))^8} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Integrate[Sec[c + d*x]^11/(a + I*a*Tan[c + d*x])^8,x]

[Out]

(63*Cos[8*c]*Log[Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8)/(2*d*(a +
I*a*Tan[c + d*x])^8) - (63*Cos[8*c]*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*(Cos[d*x] + I*
Sin[d*x])^8)/(2*d*(a + I*a*Tan[c + d*x])^8) + (Cos[5*d*x]*Sec[c + d*x]^8*(((8*I)/5)*Cos[3*c] - (8*Sin[3*c])/5)
*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (Cos[3*d*x]*Sec[c + d*x]^8*((-8*I)*Cos[5*c] + 8*Sin
[5*c])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (Cos[d*x]*Sec[c + d*x]^8*((48*I)*Cos[7*c] - 4
8*Sin[7*c])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (Sec[c]*Sec[c + d*x]^8*((8*I)*Cos[8*c] -
8*Sin[8*c])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) + (((63*I)/2)*Log[Cos[c/2 + (d*x)/2] - Si
n[c/2 + (d*x)/2]]*Sec[c + d*x]^8*Sin[8*c]*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*Tan[c + d*x])^8) - (((63*I)/2
)*Log[Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2]]*Sec[c + d*x]^8*Sin[8*c]*(Cos[d*x] + I*Sin[d*x])^8)/(d*(a + I*a*
Tan[c + d*x])^8) + (Sec[c + d*x]^8*(48*Cos[7*c] + (48*I)*Sin[7*c])*(Cos[d*x] + I*Sin[d*x])^8*Sin[d*x])/(d*(a +
I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*(-8*Cos[5*c] - (8*I)*Sin[5*c])*(Cos[d*x] + I*Sin[d*x])^8*Sin[3*d*x])/(
d*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*((8*Cos[3*c])/5 + ((8*I)/5)*Sin[3*c])*(Cos[d*x] + I*Sin[d*x])^8*
Sin[5*d*x])/(d*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*(Cos[8*c]/4 + (I/4)*Sin[8*c])*(Cos[d*x] + I*Sin[d*x
])^8)/(d*(Cos[c/2 + (d*x)/2] - Sin[c/2 + (d*x)/2])^2*(a + I*a*Tan[c + d*x])^8) + (Sec[c + d*x]^8*(-1/4*Cos[8*c
] - (I/4)*Sin[8*c])*(Cos[d*x] + I*Sin[d*x])^8)/(d*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/2])^2*(a + I*a*Tan[c +
d*x])^8) + (8*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8*(Cos[8*c - (d*x)/2]/2 - Cos[8*c + (d*x)/2]/2 + (I/2)*S
in[8*c - (d*x)/2] - (I/2)*Sin[8*c + (d*x)/2]))/(d*(Cos[c/2] + Sin[c/2])*(Cos[c/2 + (d*x)/2] + Sin[c/2 + (d*x)/
2])*(a + I*a*Tan[c + d*x])^8) + (8*Sec[c + d*x]^8*(Cos[d*x] + I*Sin[d*x])^8*(-1/2*Cos[8*c - (d*x)/2] + Cos[8*c
+ (d*x)/2]/2 - (I/2)*Sin[8*c - (d*x)/2] + (I/2)*Sin[8*c + (d*x)/2]))/(d*(Cos[c/2] - Sin[c/2])*(Cos[c/2 + (d*x
)/2] - Sin[c/2 + (d*x)/2])*(a + I*a*Tan[c + d*x])^8)

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Maple [A]
time = 0.34, size = 184, normalized size = 1.01

method	result	size
risch	\(\frac {48 i {\mathrm e}^{-i \left (d x +c \right )}}{a^{8} d}-\frac {8 i {\mathrm e}^{-3 i \left (d x +c \right )}}{a^{8} d}+\frac {8 i {\mathrm e}^{-5 i \left (d x +c \right )}}{5 a^{8} d}+\frac {i \left (15 \,{\mathrm e}^{3 i \left (d x +c \right )}+17 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{d \,a^{8} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{2}}-\frac {63 \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{2 a^{8} d}+\frac {63 \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 a^{8} d}\)	\(143\)
derivativedivides	\(\frac {\frac {2 \left (\frac {1}{4}-4 i\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {63 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}+\frac {2 \left (\frac {1}{4}+4 i\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {63 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}-\frac {32 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {256}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {64}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {64}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{a^{8} d}\)	\(184\)
default	\(\frac {\frac {2 \left (\frac {1}{4}-4 i\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1}+\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {63 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2}+\frac {2 \left (\frac {1}{4}+4 i\right )}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1}-\frac {1}{2 \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {63 \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2}-\frac {32 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {256}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {64}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {64}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{a^{8} d}\)	\(184\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^11/(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

2/d/a^8*((1/4-4*I)/(tan(1/2*d*x+1/2*c)-1)+1/4/(tan(1/2*d*x+1/2*c)-1)^2+63/4*ln(tan(1/2*d*x+1/2*c)-1)+(1/4+4*I)
/(tan(1/2*d*x+1/2*c)+1)-1/4/(tan(1/2*d*x+1/2*c)+1)^2-63/4*ln(tan(1/2*d*x+1/2*c)+1)-16*I/(-I+tan(1/2*d*x+1/2*c)
)^2-64*I/(-I+tan(1/2*d*x+1/2*c))^4+128/5/(-I+tan(1/2*d*x+1/2*c))^5-32/(-I+tan(1/2*d*x+1/2*c))^3+32/(-I+tan(1/2
*d*x+1/2*c)))

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 531 vs. \(2 (157) = 314\).
time = 0.57, size = 531, normalized size = 2.90 \begin {gather*} \frac {630 \, {\left (\cos \left (9 \, d x + 9 \, c\right ) + 2 \, \cos \left (7 \, d x + 7 \, c\right ) + \cos \left (5 \, d x + 5 \, c\right ) + i \, \sin \left (9 \, d x + 9 \, c\right ) + 2 i \, \sin \left (7 \, d x + 7 \, c\right ) + i \, \sin \left (5 \, d x + 5 \, c\right )\right )} \arctan \left (\cos \left (d x + c\right ), \sin \left (d x + c\right ) + 1\right ) + 630 \, {\left (\cos \left (9 \, d x + 9 \, c\right ) + 2 \, \cos \left (7 \, d x + 7 \, c\right ) + \cos \left (5 \, d x + 5 \, c\right ) + i \, \sin \left (9 \, d x + 9 \, c\right ) + 2 i \, \sin \left (7 \, d x + 7 \, c\right ) + i \, \sin \left (5 \, d x + 5 \, c\right )\right )} \arctan \left (\cos \left (d x + c\right ), -\sin \left (d x + c\right ) + 1\right ) + 315 \, {\left (i \, \cos \left (9 \, d x + 9 \, c\right ) + 2 i \, \cos \left (7 \, d x + 7 \, c\right ) + i \, \cos \left (5 \, d x + 5 \, c\right ) - \sin \left (9 \, d x + 9 \, c\right ) - 2 \, \sin \left (7 \, d x + 7 \, c\right ) - \sin \left (5 \, d x + 5 \, c\right )\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) + 315 \, {\left (-i \, \cos \left (9 \, d x + 9 \, c\right ) - 2 i \, \cos \left (7 \, d x + 7 \, c\right ) - i \, \cos \left (5 \, d x + 5 \, c\right ) + \sin \left (9 \, d x + 9 \, c\right ) + 2 \, \sin \left (7 \, d x + 7 \, c\right ) + \sin \left (5 \, d x + 5 \, c\right )\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right ) + 1260 \, \cos \left (8 \, d x + 8 \, c\right ) + 2100 \, \cos \left (6 \, d x + 6 \, c\right ) + 672 \, \cos \left (4 \, d x + 4 \, c\right ) - 96 \, \cos \left (2 \, d x + 2 \, c\right ) + 1260 i \, \sin \left (8 \, d x + 8 \, c\right ) + 2100 i \, \sin \left (6 \, d x + 6 \, c\right ) + 672 i \, \sin \left (4 \, d x + 4 \, c\right ) - 96 i \, \sin \left (2 \, d x + 2 \, c\right ) + 32}{-20 \, {\left (i \, a^{8} \cos \left (9 \, d x + 9 \, c\right ) + 2 i \, a^{8} \cos \left (7 \, d x + 7 \, c\right ) + i \, a^{8} \cos \left (5 \, d x + 5 \, c\right ) - a^{8} \sin \left (9 \, d x + 9 \, c\right ) - 2 \, a^{8} \sin \left (7 \, d x + 7 \, c\right ) - a^{8} \sin \left (5 \, d x + 5 \, c\right )\right )} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^11/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

(630*(cos(9*d*x + 9*c) + 2*cos(7*d*x + 7*c) + cos(5*d*x + 5*c) + I*sin(9*d*x + 9*c) + 2*I*sin(7*d*x + 7*c) + I
*sin(5*d*x + 5*c))*arctan2(cos(d*x + c), sin(d*x + c) + 1) + 630*(cos(9*d*x + 9*c) + 2*cos(7*d*x + 7*c) + cos(
5*d*x + 5*c) + I*sin(9*d*x + 9*c) + 2*I*sin(7*d*x + 7*c) + I*sin(5*d*x + 5*c))*arctan2(cos(d*x + c), -sin(d*x
+ c) + 1) + 315*(I*cos(9*d*x + 9*c) + 2*I*cos(7*d*x + 7*c) + I*cos(5*d*x + 5*c) - sin(9*d*x + 9*c) - 2*sin(7*d
*x + 7*c) - sin(5*d*x + 5*c))*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1) + 315*(-I*cos(9*d*x +
9*c) - 2*I*cos(7*d*x + 7*c) - I*cos(5*d*x + 5*c) + sin(9*d*x + 9*c) + 2*sin(7*d*x + 7*c) + sin(5*d*x + 5*c))*l
og(cos(d*x + c)^2 + sin(d*x + c)^2 - 2*sin(d*x + c) + 1) + 1260*cos(8*d*x + 8*c) + 2100*cos(6*d*x + 6*c) + 672
*cos(4*d*x + 4*c) - 96*cos(2*d*x + 2*c) + 1260*I*sin(8*d*x + 8*c) + 2100*I*sin(6*d*x + 6*c) + 672*I*sin(4*d*x
+ 4*c) - 96*I*sin(2*d*x + 2*c) + 32)/((-20*I*a^8*cos(9*d*x + 9*c) - 40*I*a^8*cos(7*d*x + 7*c) - 20*I*a^8*cos(5
*d*x + 5*c) + 20*a^8*sin(9*d*x + 9*c) + 40*a^8*sin(7*d*x + 7*c) + 20*a^8*sin(5*d*x + 5*c))*d)

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Fricas [A]
time = 0.41, size = 182, normalized size = 0.99 \begin {gather*} -\frac {315 \, {\left (e^{\left (9 i \, d x + 9 i \, c\right )} + 2 \, e^{\left (7 i \, d x + 7 i \, c\right )} + e^{\left (5 i \, d x + 5 i \, c\right )}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} + i\right ) - 315 \, {\left (e^{\left (9 i \, d x + 9 i \, c\right )} + 2 \, e^{\left (7 i \, d x + 7 i \, c\right )} + e^{\left (5 i \, d x + 5 i \, c\right )}\right )} \log \left (e^{\left (i \, d x + i \, c\right )} - i\right ) - 630 i \, e^{\left (8 i \, d x + 8 i \, c\right )} - 1050 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 336 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 48 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - 16 i}{10 \, {\left (a^{8} d e^{\left (9 i \, d x + 9 i \, c\right )} + 2 \, a^{8} d e^{\left (7 i \, d x + 7 i \, c\right )} + a^{8} d e^{\left (5 i \, d x + 5 i \, c\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^11/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

-1/10*(315*(e^(9*I*d*x + 9*I*c) + 2*e^(7*I*d*x + 7*I*c) + e^(5*I*d*x + 5*I*c))*log(e^(I*d*x + I*c) + I) - 315*
(e^(9*I*d*x + 9*I*c) + 2*e^(7*I*d*x + 7*I*c) + e^(5*I*d*x + 5*I*c))*log(e^(I*d*x + I*c) - I) - 630*I*e^(8*I*d*
x + 8*I*c) - 1050*I*e^(6*I*d*x + 6*I*c) - 336*I*e^(4*I*d*x + 4*I*c) + 48*I*e^(2*I*d*x + 2*I*c) - 16*I)/(a^8*d*
e^(9*I*d*x + 9*I*c) + 2*a^8*d*e^(7*I*d*x + 7*I*c) + a^8*d*e^(5*I*d*x + 5*I*c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {\sec ^{11}{\left (c + d x \right )}}{\tan ^{8}{\left (c + d x \right )} - 8 i \tan ^{7}{\left (c + d x \right )} - 28 \tan ^{6}{\left (c + d x \right )} + 56 i \tan ^{5}{\left (c + d x \right )} + 70 \tan ^{4}{\left (c + d x \right )} - 56 i \tan ^{3}{\left (c + d x \right )} - 28 \tan ^{2}{\left (c + d x \right )} + 8 i \tan {\left (c + d x \right )} + 1}\, dx}{a^{8}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**11/(a+I*a*tan(d*x+c))**8,x)

[Out]

Integral(sec(c + d*x)**11/(tan(c + d*x)**8 - 8*I*tan(c + d*x)**7 - 28*tan(c + d*x)**6 + 56*I*tan(c + d*x)**5 +
70*tan(c + d*x)**4 - 56*I*tan(c + d*x)**3 - 28*tan(c + d*x)**2 + 8*I*tan(c + d*x) + 1), x)/a**8

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Giac [A]
time = 2.36, size = 165, normalized size = 0.90 \begin {gather*} -\frac {\frac {315 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1\right )}{a^{8}} - \frac {315 \, \log \left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1\right )}{a^{8}} - \frac {10 \, {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 16 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 16 i\right )}}{{\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1\right )}^{2} a^{8}} - \frac {64 \, {\left (10 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} - 45 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} - 85 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + 55 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 13\right )}}{a^{8} {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{5}}}{10 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^11/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

-1/10*(315*log(tan(1/2*d*x + 1/2*c) + 1)/a^8 - 315*log(tan(1/2*d*x + 1/2*c) - 1)/a^8 - 10*(tan(1/2*d*x + 1/2*c
)^3 - 16*I*tan(1/2*d*x + 1/2*c)^2 + tan(1/2*d*x + 1/2*c) + 16*I)/((tan(1/2*d*x + 1/2*c)^2 - 1)^2*a^8) - 64*(10
*tan(1/2*d*x + 1/2*c)^4 - 45*I*tan(1/2*d*x + 1/2*c)^3 - 85*tan(1/2*d*x + 1/2*c)^2 + 55*I*tan(1/2*d*x + 1/2*c)
+ 13)/(a^8*(tan(1/2*d*x + 1/2*c) - I)^5))/d

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Mupad [B]
time = 7.44, size = 284, normalized size = 1.55 \begin {gather*} -\frac {63\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{a^8\,d}+\frac {\frac {1223\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3}{a^8}-\frac {1109\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5}{a^8}+\frac {309\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7}{a^8}-\frac {431\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{a^8}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,4407{}\mathrm {i}}{5\,a^8}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,7351{}\mathrm {i}}{5\,a^8}-\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,761{}\mathrm {i}}{a^8}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,65{}\mathrm {i}}{a^8}+\frac {496{}\mathrm {i}}{5\,a^8}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,1{}\mathrm {i}+5\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,12{}\mathrm {i}-20\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,26{}\mathrm {i}+26\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,20{}\mathrm {i}-12\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,5{}\mathrm {i}+1\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^11*(a + a*tan(c + d*x)*1i)^8),x)

[Out]

((1223*tan(c/2 + (d*x)/2)^3)/a^8 - (tan(c/2 + (d*x)/2)^2*4407i)/(5*a^8) + (tan(c/2 + (d*x)/2)^4*7351i)/(5*a^8)
- (1109*tan(c/2 + (d*x)/2)^5)/a^8 - (tan(c/2 + (d*x)/2)^6*761i)/a^8 + (309*tan(c/2 + (d*x)/2)^7)/a^8 + (tan(c
/2 + (d*x)/2)^8*65i)/a^8 + 496i/(5*a^8) - (431*tan(c/2 + (d*x)/2))/a^8)/(d*(tan(c/2 + (d*x)/2)*5i - 12*tan(c/2
+ (d*x)/2)^2 - tan(c/2 + (d*x)/2)^3*20i + 26*tan(c/2 + (d*x)/2)^4 + tan(c/2 + (d*x)/2)^5*26i - 20*tan(c/2 + (
d*x)/2)^6 - tan(c/2 + (d*x)/2)^7*12i + 5*tan(c/2 + (d*x)/2)^8 + tan(c/2 + (d*x)/2)^9*1i + 1)) - (63*atanh(tan(
c/2 + (d*x)/2)))/(a^8*d)

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