3.2.80 \(\int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^8} \, dx\) [180]

Optimal. Leaf size=138 \[ \frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac {i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac {2 i \sec ^5(c+d x)}{231 a^2 d (a+i a \tan (c+d x))^6}+\frac {2 i \sec ^5(c+d x)}{1155 a^3 d (a+i a \tan (c+d x))^5} \]

[Out]

1/11*I*sec(d*x+c)^5/d/(a+I*a*tan(d*x+c))^8+1/33*I*sec(d*x+c)^5/a/d/(a+I*a*tan(d*x+c))^7+2/231*I*sec(d*x+c)^5/a
^2/d/(a+I*a*tan(d*x+c))^6+2/1155*I*sec(d*x+c)^5/a^3/d/(a+I*a*tan(d*x+c))^5

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Rubi [A]
time = 0.13, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3583, 3569} \begin {gather*} \frac {2 i \sec ^5(c+d x)}{1155 a^3 d (a+i a \tan (c+d x))^5}+\frac {2 i \sec ^5(c+d x)}{231 a^2 d (a+i a \tan (c+d x))^6}+\frac {i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^8,x]

[Out]

((I/11)*Sec[c + d*x]^5)/(d*(a + I*a*Tan[c + d*x])^8) + ((I/33)*Sec[c + d*x]^5)/(a*d*(a + I*a*Tan[c + d*x])^7)
+ (((2*I)/231)*Sec[c + d*x]^5)/(a^2*d*(a + I*a*Tan[c + d*x])^6) + (((2*I)/1155)*Sec[c + d*x]^5)/(a^3*d*(a + I*
a*Tan[c + d*x])^5)

Rule 3569

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 3583

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 ^5(c+d x)}{(a+i a \tan (c+d x))^8} \, dx &=\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac {3 \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^7} \, dx}{11 a}\\ &=\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac {i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac {2 \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^6} \, dx}{33 a^2}\\ &=\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac {i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac {2 i \sec ^5(c+d x)}{231 a^2 d (a+i a \tan (c+d x))^6}+\frac {2 \int \frac {\sec ^5(c+d x)}{(a+i a \tan (c+d x))^5} \, dx}{231 a^3}\\ &=\frac {i \sec ^5(c+d x)}{11 d (a+i a \tan (c+d x))^8}+\frac {i \sec ^5(c+d x)}{33 a d (a+i a \tan (c+d x))^7}+\frac {2 i \sec ^5(c+d x)}{231 a^2 d (a+i a \tan (c+d x))^6}+\frac {2 i \sec ^5(c+d x)}{1155 a^3 d (a+i a \tan (c+d x))^5}\\ \end {align*}

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Mathematica [A]
time = 0.37, size = 73, normalized size = 0.53 \begin {gather*} \frac {i \sec ^8(c+d x) (440 \cos (c+d x)+168 \cos (3 (c+d x))+55 i \sin (c+d x)+63 i \sin (3 (c+d x)))}{4620 a^8 d (-i+\tan (c+d x))^8} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^5/(a + I*a*Tan[c + d*x])^8,x]

[Out]

((I/4620)*Sec[c + d*x]^8*(440*Cos[c + d*x] + 168*Cos[3*(c + d*x)] + (55*I)*Sin[c + d*x] + (63*I)*Sin[3*(c + d*
x)]))/(a^8*d*(-I + Tan[c + d*x])^8)

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Maple [A]
time = 0.34, size = 189, normalized size = 1.37

method	result	size
risch	\(\frac {i {\mathrm e}^{-5 i \left (d x +c \right )}}{40 a^{8} d}+\frac {3 i {\mathrm e}^{-7 i \left (d x +c \right )}}{56 a^{8} d}+\frac {i {\mathrm e}^{-9 i \left (d x +c \right )}}{24 a^{8} d}+\frac {i {\mathrm e}^{-11 i \left (d x +c \right )}}{88 a^{8} d}\)	\(74\)
derivativedivides	\(\frac {-\frac {60}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {584 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {4752}{7 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {256}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {1024}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {176 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1864}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {576 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{a^{8} d}\)	\(189\)
default	\(\frac {-\frac {60}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}+\frac {584 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{6}}-\frac {4752}{7 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{7}}-\frac {256}{11 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{11}}+\frac {1024}{3 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{9}}+\frac {128 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{10}}+\frac {14 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {176 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {1864}{5 \left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {576 i}{\left (-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{8}}+\frac {2}{-i+\tan \left (\frac {d x}{2}+\frac {c}{2}\right )}}{a^{8} d}\)	\(189\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^5/(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

2/d/a^8*(-30/(-I+tan(1/2*d*x+1/2*c))^3+292*I/(-I+tan(1/2*d*x+1/2*c))^6-2376/7/(-I+tan(1/2*d*x+1/2*c))^7-128/11
/(-I+tan(1/2*d*x+1/2*c))^11+512/3/(-I+tan(1/2*d*x+1/2*c))^9+64*I/(-I+tan(1/2*d*x+1/2*c))^10+7*I/(-I+tan(1/2*d*
x+1/2*c))^2-88*I/(-I+tan(1/2*d*x+1/2*c))^4+932/5/(-I+tan(1/2*d*x+1/2*c))^5-288*I/(-I+tan(1/2*d*x+1/2*c))^8+1/(
-I+tan(1/2*d*x+1/2*c)))

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Maxima [A]
time = 0.32, size = 97, normalized size = 0.70 \begin {gather*} \frac {105 i \, \cos \left (11 \, d x + 11 \, c\right ) + 385 i \, \cos \left (9 \, d x + 9 \, c\right ) + 495 i \, \cos \left (7 \, d x + 7 \, c\right ) + 231 i \, \cos \left (5 \, d x + 5 \, c\right ) + 105 \, \sin \left (11 \, d x + 11 \, c\right ) + 385 \, \sin \left (9 \, d x + 9 \, c\right ) + 495 \, \sin \left (7 \, d x + 7 \, c\right ) + 231 \, \sin \left (5 \, d x + 5 \, c\right )}{9240 \, a^{8} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

1/9240*(105*I*cos(11*d*x + 11*c) + 385*I*cos(9*d*x + 9*c) + 495*I*cos(7*d*x + 7*c) + 231*I*cos(5*d*x + 5*c) +
105*sin(11*d*x + 11*c) + 385*sin(9*d*x + 9*c) + 495*sin(7*d*x + 7*c) + 231*sin(5*d*x + 5*c))/(a^8*d)

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Fricas [A]
time = 0.36, size = 52, normalized size = 0.38 \begin {gather*} \frac {{\left (231 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 495 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 385 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 105 i\right )} e^{\left (-11 i \, d x - 11 i \, c\right )}}{9240 \, a^{8} d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

1/9240*(231*I*e^(6*I*d*x + 6*I*c) + 495*I*e^(4*I*d*x + 4*I*c) + 385*I*e^(2*I*d*x + 2*I*c) + 105*I)*e^(-11*I*d*
x - 11*I*c)/(a^8*d)

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Sympy [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 620 vs. \(2 (119) = 238\).
time = 11.00, size = 620, normalized size = 4.49 \begin {gather*} \begin {cases} \frac {2 \tan ^{3}{\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}}{1155 a^{8} d \tan ^{8}{\left (c + d x \right )} - 9240 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{6}{\left (c + d x \right )} + 64680 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 80850 a^{8} d \tan ^{4}{\left (c + d x \right )} - 64680 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{2}{\left (c + d x \right )} + 9240 i a^{8} d \tan {\left (c + d x \right )} + 1155 a^{8} d} - \frac {16 i \tan ^{2}{\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}}{1155 a^{8} d \tan ^{8}{\left (c + d x \right )} - 9240 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{6}{\left (c + d x \right )} + 64680 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 80850 a^{8} d \tan ^{4}{\left (c + d x \right )} - 64680 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{2}{\left (c + d x \right )} + 9240 i a^{8} d \tan {\left (c + d x \right )} + 1155 a^{8} d} - \frac {61 \tan {\left (c + d x \right )} \sec ^{5}{\left (c + d x \right )}}{1155 a^{8} d \tan ^{8}{\left (c + d x \right )} - 9240 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{6}{\left (c + d x \right )} + 64680 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 80850 a^{8} d \tan ^{4}{\left (c + d x \right )} - 64680 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{2}{\left (c + d x \right )} + 9240 i a^{8} d \tan {\left (c + d x \right )} + 1155 a^{8} d} + \frac {152 i \sec ^{5}{\left (c + d x \right )}}{1155 a^{8} d \tan ^{8}{\left (c + d x \right )} - 9240 i a^{8} d \tan ^{7}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{6}{\left (c + d x \right )} + 64680 i a^{8} d \tan ^{5}{\left (c + d x \right )} + 80850 a^{8} d \tan ^{4}{\left (c + d x \right )} - 64680 i a^{8} d \tan ^{3}{\left (c + d x \right )} - 32340 a^{8} d \tan ^{2}{\left (c + d x \right )} + 9240 i a^{8} d \tan {\left (c + d x \right )} + 1155 a^{8} d} & \text {for}\: d \neq 0 \\\frac {x \sec ^{5}{\left (c \right )}}{\left (i a \tan {\left (c \right )} + a\right )^{8}} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**5/(a+I*a*tan(d*x+c))**8,x)

[Out]

Piecewise((2*tan(c + d*x)**3*sec(c + d*x)**5/(1155*a**8*d*tan(c + d*x)**8 - 9240*I*a**8*d*tan(c + d*x)**7 - 32
340*a**8*d*tan(c + d*x)**6 + 64680*I*a**8*d*tan(c + d*x)**5 + 80850*a**8*d*tan(c + d*x)**4 - 64680*I*a**8*d*ta
n(c + d*x)**3 - 32340*a**8*d*tan(c + d*x)**2 + 9240*I*a**8*d*tan(c + d*x) + 1155*a**8*d) - 16*I*tan(c + d*x)**
2*sec(c + d*x)**5/(1155*a**8*d*tan(c + d*x)**8 - 9240*I*a**8*d*tan(c + d*x)**7 - 32340*a**8*d*tan(c + d*x)**6
+ 64680*I*a**8*d*tan(c + d*x)**5 + 80850*a**8*d*tan(c + d*x)**4 - 64680*I*a**8*d*tan(c + d*x)**3 - 32340*a**8*
d*tan(c + d*x)**2 + 9240*I*a**8*d*tan(c + d*x) + 1155*a**8*d) - 61*tan(c + d*x)*sec(c + d*x)**5/(1155*a**8*d*t
an(c + d*x)**8 - 9240*I*a**8*d*tan(c + d*x)**7 - 32340*a**8*d*tan(c + d*x)**6 + 64680*I*a**8*d*tan(c + d*x)**5
+ 80850*a**8*d*tan(c + d*x)**4 - 64680*I*a**8*d*tan(c + d*x)**3 - 32340*a**8*d*tan(c + d*x)**2 + 9240*I*a**8*
d*tan(c + d*x) + 1155*a**8*d) + 152*I*sec(c + d*x)**5/(1155*a**8*d*tan(c + d*x)**8 - 9240*I*a**8*d*tan(c + d*x
)**7 - 32340*a**8*d*tan(c + d*x)**6 + 64680*I*a**8*d*tan(c + d*x)**5 + 80850*a**8*d*tan(c + d*x)**4 - 64680*I*
a**8*d*tan(c + d*x)**3 - 32340*a**8*d*tan(c + d*x)**2 + 9240*I*a**8*d*tan(c + d*x) + 1155*a**8*d), Ne(d, 0)),
(x*sec(c)**5/(I*a*tan(c) + a)**8, True))

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Giac [A]
time = 1.43, size = 151, normalized size = 1.09 \begin {gather*} \frac {2 \, {\left (1155 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{10} - 3465 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{9} - 13860 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{8} + 23100 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{7} + 37422 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{6} - 32802 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{5} - 27060 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{4} + 11220 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3} + 4895 \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 517 i \, \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 152\right )}}{1155 \, a^{8} d {\left (\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - i\right )}^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^5/(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

2/1155*(1155*tan(1/2*d*x + 1/2*c)^10 - 3465*I*tan(1/2*d*x + 1/2*c)^9 - 13860*tan(1/2*d*x + 1/2*c)^8 + 23100*I*
tan(1/2*d*x + 1/2*c)^7 + 37422*tan(1/2*d*x + 1/2*c)^6 - 32802*I*tan(1/2*d*x + 1/2*c)^5 - 27060*tan(1/2*d*x + 1
/2*c)^4 + 11220*I*tan(1/2*d*x + 1/2*c)^3 + 4895*tan(1/2*d*x + 1/2*c)^2 - 517*I*tan(1/2*d*x + 1/2*c) - 152)/(a^
8*d*(tan(1/2*d*x + 1/2*c) - I)^11)

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Mupad [B]
time = 3.91, size = 64, normalized size = 0.46 \begin {gather*} \frac {\frac {{\mathrm {e}}^{-c\,5{}\mathrm {i}-d\,x\,5{}\mathrm {i}}\,1{}\mathrm {i}}{40}+\frac {{\mathrm {e}}^{-c\,7{}\mathrm {i}-d\,x\,7{}\mathrm {i}}\,3{}\mathrm {i}}{56}+\frac {{\mathrm {e}}^{-c\,9{}\mathrm {i}-d\,x\,9{}\mathrm {i}}\,1{}\mathrm {i}}{24}+\frac {{\mathrm {e}}^{-c\,11{}\mathrm {i}-d\,x\,11{}\mathrm {i}}\,1{}\mathrm {i}}{88}}{a^8\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(cos(c + d*x)^5*(a + a*tan(c + d*x)*1i)^8),x)

[Out]

((exp(- c*5i - d*x*5i)*1i)/40 + (exp(- c*7i - d*x*7i)*3i)/56 + (exp(- c*9i - d*x*9i)*1i)/24 + (exp(- c*11i - d
*x*11i)*1i)/88)/(a^8*d)

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