\(\int \frac {\sec ^{10}(c+d x)}{a+i a \tan (c+d x)} \, dx\) [99]

Optimal result

Integrand size = 24, antiderivative size = 107 \[ \int \frac {\sec ^{10}(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {8 i (a-i a \tan (c+d x))^5}{5 a^6 d}-\frac {2 i (a-i a \tan (c+d x))^6}{a^7 d}+\frac {6 i (a-i a \tan (c+d x))^7}{7 a^8 d}-\frac {i (a-i a \tan (c+d x))^8}{8 a^9 d} \]

[Out]

8/5*I*(a-I*a*tan(d*x+c))^5/a^6/d-2*I*(a-I*a*tan(d*x+c))^6/a^7/d+6/7*I*(a-I*a*tan(d*x+c))^7/a^8/d-1/8*I*(a-I*a*
tan(d*x+c))^8/a^9/d

Rubi [A] (verified)

Time = 0.09 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {3568, 45} \[ \int \frac {\sec ^{10}(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {i (a-i a \tan (c+d x))^8}{8 a^9 d}+\frac {6 i (a-i a \tan (c+d x))^7}{7 a^8 d}-\frac {2 i (a-i a \tan (c+d x))^6}{a^7 d}+\frac {8 i (a-i a \tan (c+d x))^5}{5 a^6 d} \]

[In]

Int[Sec[c + d*x]^10/(a + I*a*Tan[c + d*x]),x]

[Out]

(((8*I)/5)*(a - I*a*Tan[c + d*x])^5)/(a^6*d) - ((2*I)*(a - I*a*Tan[c + d*x])^6)/(a^7*d) + (((6*I)/7)*(a - I*a*
Tan[c + d*x])^7)/(a^8*d) - ((I/8)*(a - I*a*Tan[c + d*x])^8)/(a^9*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*} \text {integral}& = -\frac {i \text {Subst}\left (\int (a-x)^4 (a+x)^3 \, dx,x,i a \tan (c+d x)\right )}{a^9 d} \\ & = -\frac {i \text {Subst}\left (\int \left (8 a^3 (a-x)^4-12 a^2 (a-x)^5+6 a (a-x)^6-(a-x)^7\right ) \, dx,x,i a \tan (c+d x)\right )}{a^9 d} \\ & = \frac {8 i (a-i a \tan (c+d x))^5}{5 a^6 d}-\frac {2 i (a-i a \tan (c+d x))^6}{a^7 d}+\frac {6 i (a-i a \tan (c+d x))^7}{7 a^8 d}-\frac {i (a-i a \tan (c+d x))^8}{8 a^9 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.27 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.52 \[ \int \frac {\sec ^{10}(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {(i+\tan (c+d x))^5 \left (93+185 i \tan (c+d x)-135 \tan ^2(c+d x)-35 i \tan ^3(c+d x)\right )}{280 a d} \]

[In]

Integrate[Sec[c + d*x]^10/(a + I*a*Tan[c + d*x]),x]

[Out]

((I + Tan[c + d*x])^5*(93 + (185*I)*Tan[c + d*x] - 135*Tan[c + d*x]^2 - (35*I)*Tan[c + d*x]^3))/(280*a*d)

Maple [A] (verified)

Time = 0.45 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.54

method	result	size
risch	\(\frac {32 i \left (56 \,{\mathrm e}^{6 i \left (d x +c \right )}+28 \,{\mathrm e}^{4 i \left (d x +c \right )}+8 \,{\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{35 d a \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{8}}\)	\(58\)
derivativedivides	\(-\frac {-\tan \left (d x +c \right )+\frac {i \left (\tan ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}+\frac {i \left (\tan ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {3 i \left (\tan ^{4}\left (d x +c \right )\right )}{4}-\left (\tan ^{3}\left (d x +c \right )\right )+\frac {i \left (\tan ^{2}\left (d x +c \right )\right )}{2}}{a d}\)	\(92\)
default	\(-\frac {-\tan \left (d x +c \right )+\frac {i \left (\tan ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\tan ^{7}\left (d x +c \right )\right )}{7}+\frac {i \left (\tan ^{6}\left (d x +c \right )\right )}{2}-\frac {3 \left (\tan ^{5}\left (d x +c \right )\right )}{5}+\frac {3 i \left (\tan ^{4}\left (d x +c \right )\right )}{4}-\left (\tan ^{3}\left (d x +c \right )\right )+\frac {i \left (\tan ^{2}\left (d x +c \right )\right )}{2}}{a d}\)	\(92\)

[In]

int(sec(d*x+c)^10/(a+I*a*tan(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

32/35*I*(56*exp(6*I*(d*x+c))+28*exp(4*I*(d*x+c))+8*exp(2*I*(d*x+c))+1)/d/a/(exp(2*I*(d*x+c))+1)^8

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.36 \[ \int \frac {\sec ^{10}(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {32 \, {\left (-56 i \, e^{\left (6 i \, d x + 6 i \, c\right )} - 28 i \, e^{\left (4 i \, d x + 4 i \, c\right )} - 8 i \, e^{\left (2 i \, d x + 2 i \, c\right )} - i\right )}}{35 \, {\left (a d e^{\left (16 i \, d x + 16 i \, c\right )} + 8 \, a d e^{\left (14 i \, d x + 14 i \, c\right )} + 28 \, a d e^{\left (12 i \, d x + 12 i \, c\right )} + 56 \, a d e^{\left (10 i \, d x + 10 i \, c\right )} + 70 \, a d e^{\left (8 i \, d x + 8 i \, c\right )} + 56 \, a d e^{\left (6 i \, d x + 6 i \, c\right )} + 28 \, a d e^{\left (4 i \, d x + 4 i \, c\right )} + 8 \, a d e^{\left (2 i \, d x + 2 i \, c\right )} + a d\right )}} \]

[In]

integrate(sec(d*x+c)^10/(a+I*a*tan(d*x+c)),x, algorithm="fricas")

[Out]

-32/35*(-56*I*e^(6*I*d*x + 6*I*c) - 28*I*e^(4*I*d*x + 4*I*c) - 8*I*e^(2*I*d*x + 2*I*c) - I)/(a*d*e^(16*I*d*x +
16*I*c) + 8*a*d*e^(14*I*d*x + 14*I*c) + 28*a*d*e^(12*I*d*x + 12*I*c) + 56*a*d*e^(10*I*d*x + 10*I*c) + 70*a*d*
e^(8*I*d*x + 8*I*c) + 56*a*d*e^(6*I*d*x + 6*I*c) + 28*a*d*e^(4*I*d*x + 4*I*c) + 8*a*d*e^(2*I*d*x + 2*I*c) + a*
d)

Sympy [F]

\[ \int \frac {\sec ^{10}(c+d x)}{a+i a \tan (c+d x)} \, dx=- \frac {i \int \frac {\sec ^{10}{\left (c + d x \right )}}{\tan {\left (c + d x \right )} - i}\, dx}{a} \]

[In]

integrate(sec(d*x+c)**10/(a+I*a*tan(d*x+c)),x)

[Out]

-I*Integral(sec(c + d*x)**10/(tan(c + d*x) - I), x)/a

Maxima [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.81 \[ \int \frac {\sec ^{10}(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {35 i \, \tan \left (d x + c\right )^{8} - 40 \, \tan \left (d x + c\right )^{7} + 140 i \, \tan \left (d x + c\right )^{6} - 168 \, \tan \left (d x + c\right )^{5} + 210 i \, \tan \left (d x + c\right )^{4} - 280 \, \tan \left (d x + c\right )^{3} + 140 i \, \tan \left (d x + c\right )^{2} - 280 \, \tan \left (d x + c\right )}{280 \, a d} \]

[In]

integrate(sec(d*x+c)^10/(a+I*a*tan(d*x+c)),x, algorithm="maxima")

[Out]

-1/280*(35*I*tan(d*x + c)^8 - 40*tan(d*x + c)^7 + 140*I*tan(d*x + c)^6 - 168*tan(d*x + c)^5 + 210*I*tan(d*x +
c)^4 - 280*tan(d*x + c)^3 + 140*I*tan(d*x + c)^2 - 280*tan(d*x + c))/(a*d)

Giac [A] (verification not implemented)

none

Time = 0.40 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.81 \[ \int \frac {\sec ^{10}(c+d x)}{a+i a \tan (c+d x)} \, dx=-\frac {35 i \, \tan \left (d x + c\right )^{8} - 40 \, \tan \left (d x + c\right )^{7} + 140 i \, \tan \left (d x + c\right )^{6} - 168 \, \tan \left (d x + c\right )^{5} + 210 i \, \tan \left (d x + c\right )^{4} - 280 \, \tan \left (d x + c\right )^{3} + 140 i \, \tan \left (d x + c\right )^{2} - 280 \, \tan \left (d x + c\right )}{280 \, a d} \]

[In]

integrate(sec(d*x+c)^10/(a+I*a*tan(d*x+c)),x, algorithm="giac")

[Out]

-1/280*(35*I*tan(d*x + c)^8 - 40*tan(d*x + c)^7 + 140*I*tan(d*x + c)^6 - 168*tan(d*x + c)^5 + 210*I*tan(d*x +
c)^4 - 280*tan(d*x + c)^3 + 140*I*tan(d*x + c)^2 - 280*tan(d*x + c))/(a*d)

Mupad [B] (verification not implemented)

Time = 4.35 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.86 \[ \int \frac {\sec ^{10}(c+d x)}{a+i a \tan (c+d x)} \, dx=\frac {{\cos \left (c+d\,x\right )}^8\,35{}\mathrm {i}+128\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^7+64\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^5+48\,\sin \left (c+d\,x\right )\,{\cos \left (c+d\,x\right )}^3+40\,\sin \left (c+d\,x\right )\,\cos \left (c+d\,x\right )-35{}\mathrm {i}}{280\,a\,d\,{\cos \left (c+d\,x\right )}^8} \]

[In]

int(1/(cos(c + d*x)^10*(a + a*tan(c + d*x)*1i)),x)

[Out]

(40*cos(c + d*x)*sin(c + d*x) + 48*cos(c + d*x)^3*sin(c + d*x) + 64*cos(c + d*x)^5*sin(c + d*x) + 128*cos(c +
d*x)^7*sin(c + d*x) + cos(c + d*x)^8*35i - 35i)/(280*a*d*cos(c + d*x)^8)