3.1.79 \(\int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx\) [79]

Optimal. Leaf size=55 \[ -\frac {i (a+i a \tan (c+d x))^{10}}{5 a^2 d}+\frac {i (a+i a \tan (c+d x))^{11}}{11 a^3 d} \]

[Out]

-1/5*I*(a+I*a*tan(d*x+c))^10/a^2/d+1/11*I*(a+I*a*tan(d*x+c))^11/a^3/d

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Rubi [A]
time = 0.04, antiderivative size = 55, normalized size of antiderivative = 1.00, 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 = {3568, 45} \begin {gather*} \frac {i (a+i a \tan (c+d x))^{11}}{11 a^3 d}-\frac {i (a+i a \tan (c+d x))^{10}}{5 a^2 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^8,x]

[Out]

((-1/5*I)*(a + I*a*Tan[c + d*x])^10)/(a^2*d) + ((I/11)*(a + I*a*Tan[c + d*x])^11)/(a^3*d)

Rule 45

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])

Rule 3568

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]

Rubi steps

\begin {align*} \int \sec ^4(c+d x) (a+i a \tan (c+d x))^8 \, dx &=-\frac {i \text {Subst}\left (\int (a-x) (a+x)^9 \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {i \text {Subst}\left (\int \left (2 a (a+x)^9-(a+x)^{10}\right ) \, dx,x,i a \tan (c+d x)\right )}{a^3 d}\\ &=-\frac {i (a+i a \tan (c+d x))^{10}}{5 a^2 d}+\frac {i (a+i a \tan (c+d x))^{11}}{11 a^3 d}\\ \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. \(223\) vs. \(2(55)=110\).
time = 1.97, size = 223, normalized size = 4.05 \begin {gather*} \frac {a^8 \sec (c) \sec ^{11}(c+d x) (462 i \cos (d x)+462 i \cos (2 c+d x)+330 i \cos (2 c+3 d x)+330 i \cos (4 c+3 d x)+165 i \cos (4 c+5 d x)+165 i \cos (6 c+5 d x)+55 i \cos (6 c+7 d x)+55 i \cos (8 c+7 d x)+462 \sin (d x)-462 \sin (2 c+d x)+330 \sin (2 c+3 d x)-330 \sin (4 c+3 d x)+165 \sin (4 c+5 d x)-165 \sin (6 c+5 d x)+55 \sin (6 c+7 d x)-55 \sin (8 c+7 d x)+22 \sin (8 c+9 d x)+2 \sin (10 c+11 d x))}{220 d} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sec[c + d*x]^4*(a + I*a*Tan[c + d*x])^8,x]

[Out]

(a^8*Sec[c]*Sec[c + d*x]^11*((462*I)*Cos[d*x] + (462*I)*Cos[2*c + d*x] + (330*I)*Cos[2*c + 3*d*x] + (330*I)*Co
s[4*c + 3*d*x] + (165*I)*Cos[4*c + 5*d*x] + (165*I)*Cos[6*c + 5*d*x] + (55*I)*Cos[6*c + 7*d*x] + (55*I)*Cos[8*
c + 7*d*x] + 462*Sin[d*x] - 462*Sin[2*c + d*x] + 330*Sin[2*c + 3*d*x] - 330*Sin[4*c + 3*d*x] + 165*Sin[4*c + 5
*d*x] - 165*Sin[6*c + 5*d*x] + 55*Sin[6*c + 7*d*x] - 55*Sin[8*c + 7*d*x] + 22*Sin[8*c + 9*d*x] + 2*Sin[10*c +
11*d*x]))/(220*d)

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (47 ) = 94\).
time = 0.30, size = 339, normalized size = 6.16

method	result	size
risch	\(\frac {1024 i a^{8} \left (55 \,{\mathrm e}^{18 i \left (d x +c \right )}+165 \,{\mathrm e}^{16 i \left (d x +c \right )}+330 \,{\mathrm e}^{14 i \left (d x +c \right )}+462 \,{\mathrm e}^{12 i \left (d x +c \right )}+462 \,{\mathrm e}^{10 i \left (d x +c \right )}+330 \,{\mathrm e}^{8 i \left (d x +c \right )}+165 \,{\mathrm e}^{6 i \left (d x +c \right )}+55 \,{\mathrm e}^{4 i \left (d x +c \right )}+11 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{55 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{11}}\)	\(124\)
derivativedivides	\(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{11 \cos \left (d x +c \right )^{11}}+\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{9}}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{8}\left (d x +c \right )}{40 \cos \left (d x +c \right )^{8}}\right )-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}\right )+\frac {2 i a^{8}}{\cos \left (d x +c \right )^{4}}+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )-56 i a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )-28 a^{8} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+56 i a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{6}}\right )-a^{8} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\)	\(339\)
default	\(\frac {a^{8} \left (\frac {\sin ^{9}\left (d x +c \right )}{11 \cos \left (d x +c \right )^{11}}+\frac {2 \left (\sin ^{9}\left (d x +c \right )\right )}{99 \cos \left (d x +c \right )^{9}}\right )-8 i a^{8} \left (\frac {\sin ^{8}\left (d x +c \right )}{10 \cos \left (d x +c \right )^{10}}+\frac {\sin ^{8}\left (d x +c \right )}{40 \cos \left (d x +c \right )^{8}}\right )-28 a^{8} \left (\frac {\sin ^{7}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}+\frac {2 \left (\sin ^{7}\left (d x +c \right )\right )}{63 \cos \left (d x +c \right )^{7}}\right )+\frac {2 i a^{8}}{\cos \left (d x +c \right )^{4}}+70 a^{8} \left (\frac {\sin ^{5}\left (d x +c \right )}{7 \cos \left (d x +c \right )^{7}}+\frac {2 \left (\sin ^{5}\left (d x +c \right )\right )}{35 \cos \left (d x +c \right )^{5}}\right )-56 i a^{8} \left (\frac {\sin ^{4}\left (d x +c \right )}{6 \cos \left (d x +c \right )^{6}}+\frac {\sin ^{4}\left (d x +c \right )}{12 \cos \left (d x +c \right )^{4}}\right )-28 a^{8} \left (\frac {\sin ^{3}\left (d x +c \right )}{5 \cos \left (d x +c \right )^{5}}+\frac {2 \left (\sin ^{3}\left (d x +c \right )\right )}{15 \cos \left (d x +c \right )^{3}}\right )+56 i a^{8} \left (\frac {\sin ^{6}\left (d x +c \right )}{8 \cos \left (d x +c \right )^{8}}+\frac {\sin ^{6}\left (d x +c \right )}{24 \cos \left (d x +c \right )^{6}}\right )-a^{8} \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )}{d}\)	\(339\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(d*x+c)^4*(a+I*a*tan(d*x+c))^8,x,method=_RETURNVERBOSE)

[Out]

1/d*(a^8*(1/11*sin(d*x+c)^9/cos(d*x+c)^11+2/99*sin(d*x+c)^9/cos(d*x+c)^9)-8*I*a^8*(1/10*sin(d*x+c)^8/cos(d*x+c
)^10+1/40*sin(d*x+c)^8/cos(d*x+c)^8)-28*a^8*(1/9*sin(d*x+c)^7/cos(d*x+c)^9+2/63*sin(d*x+c)^7/cos(d*x+c)^7)+2*I
*a^8/cos(d*x+c)^4+70*a^8*(1/7*sin(d*x+c)^5/cos(d*x+c)^7+2/35*sin(d*x+c)^5/cos(d*x+c)^5)-56*I*a^8*(1/6*sin(d*x+
c)^4/cos(d*x+c)^6+1/12*sin(d*x+c)^4/cos(d*x+c)^4)-28*a^8*(1/5*sin(d*x+c)^3/cos(d*x+c)^5+2/15*sin(d*x+c)^3/cos(
d*x+c)^3)+56*I*a^8*(1/8*sin(d*x+c)^6/cos(d*x+c)^8+1/24*sin(d*x+c)^6/cos(d*x+c)^6)-a^8*(-2/3-1/3*sec(d*x+c)^2)*
tan(d*x+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. 134 vs. \(2 (43) = 86\).
time = 0.30, size = 134, normalized size = 2.44 \begin {gather*} \frac {5 \, a^{8} \tan \left (d x + c\right )^{11} - 44 i \, a^{8} \tan \left (d x + c\right )^{10} - 165 \, a^{8} \tan \left (d x + c\right )^{9} + 330 i \, a^{8} \tan \left (d x + c\right )^{8} + 330 \, a^{8} \tan \left (d x + c\right )^{7} + 462 \, a^{8} \tan \left (d x + c\right )^{5} - 660 i \, a^{8} \tan \left (d x + c\right )^{4} - 495 \, a^{8} \tan \left (d x + c\right )^{3} + 220 i \, a^{8} \tan \left (d x + c\right )^{2} + 55 \, a^{8} \tan \left (d x + c\right )}{55 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+I*a*tan(d*x+c))^8,x, algorithm="maxima")

[Out]

1/55*(5*a^8*tan(d*x + c)^11 - 44*I*a^8*tan(d*x + c)^10 - 165*a^8*tan(d*x + c)^9 + 330*I*a^8*tan(d*x + c)^8 + 3
30*a^8*tan(d*x + c)^7 + 462*a^8*tan(d*x + c)^5 - 660*I*a^8*tan(d*x + c)^4 - 495*a^8*tan(d*x + c)^3 + 220*I*a^8
*tan(d*x + c)^2 + 55*a^8*tan(d*x + c))/d

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 269 vs. \(2 (43) = 86\).
time = 0.36, size = 269, normalized size = 4.89 \begin {gather*} -\frac {1024 \, {\left (-55 i \, a^{8} e^{\left (18 i \, d x + 18 i \, c\right )} - 165 i \, a^{8} e^{\left (16 i \, d x + 16 i \, c\right )} - 330 i \, a^{8} e^{\left (14 i \, d x + 14 i \, c\right )} - 462 i \, a^{8} e^{\left (12 i \, d x + 12 i \, c\right )} - 462 i \, a^{8} e^{\left (10 i \, d x + 10 i \, c\right )} - 330 i \, a^{8} e^{\left (8 i \, d x + 8 i \, c\right )} - 165 i \, a^{8} e^{\left (6 i \, d x + 6 i \, c\right )} - 55 i \, a^{8} e^{\left (4 i \, d x + 4 i \, c\right )} - 11 i \, a^{8} e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{8}\right )}}{55 \, {\left (d e^{\left (22 i \, d x + 22 i \, c\right )} + 11 \, d e^{\left (20 i \, d x + 20 i \, c\right )} + 55 \, d e^{\left (18 i \, d x + 18 i \, c\right )} + 165 \, d e^{\left (16 i \, d x + 16 i \, c\right )} + 330 \, d e^{\left (14 i \, d x + 14 i \, c\right )} + 462 \, d e^{\left (12 i \, d x + 12 i \, c\right )} + 462 \, d e^{\left (10 i \, d x + 10 i \, c\right )} + 330 \, d e^{\left (8 i \, d x + 8 i \, c\right )} + 165 \, d e^{\left (6 i \, d x + 6 i \, c\right )} + 55 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 11 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+I*a*tan(d*x+c))^8,x, algorithm="fricas")

[Out]

-1024/55*(-55*I*a^8*e^(18*I*d*x + 18*I*c) - 165*I*a^8*e^(16*I*d*x + 16*I*c) - 330*I*a^8*e^(14*I*d*x + 14*I*c)
- 462*I*a^8*e^(12*I*d*x + 12*I*c) - 462*I*a^8*e^(10*I*d*x + 10*I*c) - 330*I*a^8*e^(8*I*d*x + 8*I*c) - 165*I*a^
8*e^(6*I*d*x + 6*I*c) - 55*I*a^8*e^(4*I*d*x + 4*I*c) - 11*I*a^8*e^(2*I*d*x + 2*I*c) - I*a^8)/(d*e^(22*I*d*x +
22*I*c) + 11*d*e^(20*I*d*x + 20*I*c) + 55*d*e^(18*I*d*x + 18*I*c) + 165*d*e^(16*I*d*x + 16*I*c) + 330*d*e^(14*
I*d*x + 14*I*c) + 462*d*e^(12*I*d*x + 12*I*c) + 462*d*e^(10*I*d*x + 10*I*c) + 330*d*e^(8*I*d*x + 8*I*c) + 165*
d*e^(6*I*d*x + 6*I*c) + 55*d*e^(4*I*d*x + 4*I*c) + 11*d*e^(2*I*d*x + 2*I*c) + d)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)**4*(a+I*a*tan(d*x+c))**8,x)

[Out]

a**8*(Integral(-28*tan(c + d*x)**2*sec(c + d*x)**4, x) + Integral(70*tan(c + d*x)**4*sec(c + d*x)**4, x) + Int
egral(-28*tan(c + d*x)**6*sec(c + d*x)**4, x) + Integral(tan(c + d*x)**8*sec(c + d*x)**4, x) + Integral(8*I*ta
n(c + d*x)*sec(c + d*x)**4, x) + Integral(-56*I*tan(c + d*x)**3*sec(c + d*x)**4, x) + Integral(56*I*tan(c + d*
x)**5*sec(c + d*x)**4, x) + Integral(-8*I*tan(c + d*x)**7*sec(c + d*x)**4, x) + Integral(sec(c + d*x)**4, x))

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Giac [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 134 vs. \(2 (43) = 86\).
time = 1.13, size = 134, normalized size = 2.44 \begin {gather*} \frac {5 \, a^{8} \tan \left (d x + c\right )^{11} - 44 i \, a^{8} \tan \left (d x + c\right )^{10} - 165 \, a^{8} \tan \left (d x + c\right )^{9} + 330 i \, a^{8} \tan \left (d x + c\right )^{8} + 330 \, a^{8} \tan \left (d x + c\right )^{7} + 462 \, a^{8} \tan \left (d x + c\right )^{5} - 660 i \, a^{8} \tan \left (d x + c\right )^{4} - 495 \, a^{8} \tan \left (d x + c\right )^{3} + 220 i \, a^{8} \tan \left (d x + c\right )^{2} + 55 \, a^{8} \tan \left (d x + c\right )}{55 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(d*x+c)^4*(a+I*a*tan(d*x+c))^8,x, algorithm="giac")

[Out]

1/55*(5*a^8*tan(d*x + c)^11 - 44*I*a^8*tan(d*x + c)^10 - 165*a^8*tan(d*x + c)^9 + 330*I*a^8*tan(d*x + c)^8 + 3
30*a^8*tan(d*x + c)^7 + 462*a^8*tan(d*x + c)^5 - 660*I*a^8*tan(d*x + c)^4 - 495*a^8*tan(d*x + c)^3 + 220*I*a^8
*tan(d*x + c)^2 + 55*a^8*tan(d*x + c))/d

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Mupad [B]
time = 4.22, size = 107, normalized size = 1.95 \begin {gather*} \frac {a^8\,\left (\frac {\sin \left (9\,c+9\,d\,x\right )}{10}+\frac {\sin \left (11\,c+11\,d\,x\right )}{110}+\frac {\cos \left (c+d\,x\right )\,63{}\mathrm {i}}{1280}+\frac {\cos \left (3\,c+3\,d\,x\right )\,9{}\mathrm {i}}{256}+\frac {\cos \left (5\,c+5\,d\,x\right )\,9{}\mathrm {i}}{512}+\frac {\cos \left (7\,c+7\,d\,x\right )\,3{}\mathrm {i}}{512}-\frac {\cos \left (9\,c+9\,d\,x\right )\,253{}\mathrm {i}}{2560}-\frac {\cos \left (11\,c+11\,d\,x\right )\,23{}\mathrm {i}}{2560}\right )}{d\,{\cos \left (c+d\,x\right )}^{11}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + a*tan(c + d*x)*1i)^8/cos(c + d*x)^4,x)

[Out]

(a^8*((cos(c + d*x)*63i)/1280 + (cos(3*c + 3*d*x)*9i)/256 + (cos(5*c + 5*d*x)*9i)/512 + (cos(7*c + 7*d*x)*3i)/
512 - (cos(9*c + 9*d*x)*253i)/2560 - (cos(11*c + 11*d*x)*23i)/2560 + sin(9*c + 9*d*x)/10 + sin(11*c + 11*d*x)/
110))/(d*cos(c + d*x)^11)

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