Integrand size = 26, antiderivative size = 125 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{8 \sqrt {2} a^{7/2} d}+\frac {i \sec (c+d x)}{2 a d (a+i a \tan (c+d x))^{5/2}}-\frac {i \sec (c+d x)}{8 a^2 d (a+i a \tan (c+d x))^{3/2}} \]
-1/16*I*arctanh(1/2*sec(d*x+c)*a^(1/2)*2^(1/2)/(a+I*a*tan(d*x+c))^(1/2))/a ^(7/2)/d*2^(1/2)+1/2*I*sec(d*x+c)/a/d/(a+I*a*tan(d*x+c))^(5/2)-1/8*I*sec(d *x+c)/a^2/d/(a+I*a*tan(d*x+c))^(3/2)
Time = 1.86 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.96 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {i \sec ^3(c+d x) \left (-3+e^{2 i (c+d x)} \sqrt {1+e^{2 i (c+d x)}} \text {arctanh}\left (\sqrt {1+e^{2 i (c+d x)}}\right )-3 \cos (2 (c+d x))+i \sin (2 (c+d x))\right )}{16 a^3 d (-i+\tan (c+d x))^2 \sqrt {a+i a \tan (c+d x)}} \]
((I/16)*Sec[c + d*x]^3*(-3 + E^((2*I)*(c + d*x))*Sqrt[1 + E^((2*I)*(c + d* x))]*ArcTanh[Sqrt[1 + E^((2*I)*(c + d*x))]] - 3*Cos[2*(c + d*x)] + I*Sin[2 *(c + d*x)]))/(a^3*d*(-I + Tan[c + d*x])^2*Sqrt[a + I*a*Tan[c + d*x]])
Time = 0.65 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.36, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {3042, 3982, 3042, 3983, 3042, 3983, 3042, 3970, 219}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\sec (c+d x)^3}{(a+i a \tan (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3982 |
| \(\displaystyle \frac {2 i \sec (c+d x)}{3 a d (a+i a \tan (c+d x))^{5/2}}-\frac {2 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^{5/2}}dx}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 i \sec (c+d x)}{3 a d (a+i a \tan (c+d x))^{5/2}}-\frac {2 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^{5/2}}dx}{3 a}\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {2 i \sec (c+d x)}{3 a d (a+i a \tan (c+d x))^{5/2}}-\frac {2 \left (\frac {3 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^{3/2}}dx}{8 a}+\frac {i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}\right )}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 i \sec (c+d x)}{3 a d (a+i a \tan (c+d x))^{5/2}}-\frac {2 \left (\frac {3 \int \frac {\sec (c+d x)}{(i \tan (c+d x) a+a)^{3/2}}dx}{8 a}+\frac {i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}\right )}{3 a}\) |
| \(\Big \downarrow \) 3983 |
| \(\displaystyle \frac {2 i \sec (c+d x)}{3 a d (a+i a \tan (c+d x))^{5/2}}-\frac {2 \left (\frac {3 \left (\frac {\int \frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx}{4 a}+\frac {i \sec (c+d x)}{2 d (a+i a \tan (c+d x))^{3/2}}\right )}{8 a}+\frac {i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}\right )}{3 a}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 i \sec (c+d x)}{3 a d (a+i a \tan (c+d x))^{5/2}}-\frac {2 \left (\frac {3 \left (\frac {\int \frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}dx}{4 a}+\frac {i \sec (c+d x)}{2 d (a+i a \tan (c+d x))^{3/2}}\right )}{8 a}+\frac {i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}\right )}{3 a}\) |
| \(\Big \downarrow \) 3970 |
| \(\displaystyle \frac {2 i \sec (c+d x)}{3 a d (a+i a \tan (c+d x))^{5/2}}-\frac {2 \left (\frac {3 \left (\frac {i \int \frac {1}{2-\frac {a \sec ^2(c+d x)}{i \tan (c+d x) a+a}}d\frac {\sec (c+d x)}{\sqrt {i \tan (c+d x) a+a}}}{2 a d}+\frac {i \sec (c+d x)}{2 d (a+i a \tan (c+d x))^{3/2}}\right )}{8 a}+\frac {i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}\right )}{3 a}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle \frac {2 i \sec (c+d x)}{3 a d (a+i a \tan (c+d x))^{5/2}}-\frac {2 \left (\frac {3 \left (\frac {i \text {arctanh}\left (\frac {\sqrt {a} \sec (c+d x)}{\sqrt {2} \sqrt {a+i a \tan (c+d x)}}\right )}{2 \sqrt {2} a^{3/2} d}+\frac {i \sec (c+d x)}{2 d (a+i a \tan (c+d x))^{3/2}}\right )}{8 a}+\frac {i \sec (c+d x)}{4 d (a+i a \tan (c+d x))^{5/2}}\right )}{3 a}\) |
(((2*I)/3)*Sec[c + d*x])/(a*d*(a + I*a*Tan[c + d*x])^(5/2)) - (2*(((I/4)*S ec[c + d*x])/(d*(a + I*a*Tan[c + d*x])^(5/2)) + (3*(((I/2)*ArcTanh[(Sqrt[a ]*Sec[c + d*x])/(Sqrt[2]*Sqrt[a + I*a*Tan[c + d*x]])])/(Sqrt[2]*a^(3/2)*d) + ((I/2)*Sec[c + d*x])/(d*(a + I*a*Tan[c + d*x])^(3/2))))/(8*a)))/(3*a)
3.4.90.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[sec[(e_.) + (f_.)*(x_)]/Sqrt[(a_) + (b_.)*tan[(e_.) + (f_.)*(x_)]], x_S ymbol] :> Simp[-2*(a/(b*f)) Subst[Int[1/(2 - a*x^2), x], x, Sec[e + f*x]/ Sqrt[a + b*Tan[e + f*x]]], x] /; FreeQ[{a, b, e, f}, x] && EqQ[a^2 + b^2, 0 ]
Int[((d_.)*sec[(e_.) + (f_.)*(x_)])^(m_.)*((a_) + (b_.)*tan[(e_.) + (f_.)*( x_)])^(n_), x_Symbol] :> Simp[d^2*(d*Sec[e + f*x])^(m - 2)*((a + b*Tan[e + f*x])^(n + 1)/(b*f*(m + n - 1))), x] + Simp[d^2*((m - 2)/(a*(m + n - 1))) Int[(d*Sec[e + f*x])^(m - 2)*(a + b*Tan[e + f*x])^(n + 1), x], x] /; FreeQ [{a, b, d, e, f}, x] && EqQ[a^2 + b^2, 0] && LtQ[n, 0] && GtQ[m, 1] && !IL tQ[m + n, 0] && NeQ[m + n - 1, 0] && IntegersQ[2*m, 2*n]
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] + Simp[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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 794 vs. \(2 (100 ) = 200\).
Time = 10.65 (sec) , antiderivative size = 795, normalized size of antiderivative = 6.36
| method | result | size |
| default | \(\frac {8 i \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right ) \sin \left (d x +c \right )+4 i \tan \left (d x +c \right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-4 i \tan \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+8 \cos \left (d x +c \right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-4 i \tan \left (d x +c \right ) \sec \left (d x +c \right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-4 i \tan \left (d x +c \right ) \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+4 \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-4 \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-i \tan \left (d x +c \right ) \left (\sec ^{2}\left (d x +c \right )\right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-8 \sec \left (d x +c \right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-4 \sec \left (d x +c \right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-3 \left (\sec ^{2}\left (d x +c \right )\right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-2 \left (\sec ^{2}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}+\left (\sec ^{3}\left (d x +c \right )\right ) \arctan \left (\frac {i \sin \left (d x +c \right )-\cos \left (d x +c \right )-1}{2 \left (\cos \left (d x +c \right )+1\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}\right )-2 \left (\sec ^{3}\left (d x +c \right )\right ) \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}}{16 d \left (\tan \left (d x +c \right )-i\right )^{3} \sqrt {a \left (1+i \tan \left (d x +c \right )\right )}\, \sqrt {-\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, a^{3} \left (\cos \left (d x +c \right )+1\right )}\) | \(795\) |
1/16/d/(tan(d*x+c)-I)^3/(a*(1+I*tan(d*x+c)))^(1/2)/(-cos(d*x+c)/(cos(d*x+c )+1))^(1/2)/a^3/(cos(d*x+c)+1)*(8*I*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1) /(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*sin(d*x+c)+4*I*tan(d*x +c)*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(co s(d*x+c)+1))^(1/2))-4*I*tan(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+8*co s(d*x+c)*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c )/(cos(d*x+c)+1))^(1/2))-4*I*tan(d*x+c)*sec(d*x+c)*arctan(1/2*(I*sin(d*x+c )-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-4*I*tan (d*x+c)*sec(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+4*arctan(1/2*(I*sin( d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-4* (-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-I*tan(d*x+c)*sec(d*x+c)^2*arctan(1/2*(I *sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2 ))-8*sec(d*x+c)*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1)/(-co s(d*x+c)/(cos(d*x+c)+1))^(1/2))-4*sec(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^ (1/2)-3*sec(d*x+c)^2*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+c)+1) /(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-2*sec(d*x+c)^2*(-cos(d*x+c)/(cos(d*x+ c)+1))^(1/2)+sec(d*x+c)^3*arctan(1/2*(I*sin(d*x+c)-cos(d*x+c)-1)/(cos(d*x+ c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-2*sec(d*x+c)^3*(-cos(d*x+c)/(cos (d*x+c)+1))^(1/2))
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 267 vs. \(2 (94) = 188\).
Time = 0.26 (sec) , antiderivative size = 267, normalized size of antiderivative = 2.14 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {{\left (-i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} + i\right )} e^{\left (-i \, d x - i \, c\right )}}{4 \, a^{3} d}\right ) + i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {{\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (-i \, a^{3} d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, a^{3} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} + i\right )} e^{\left (-i \, d x - i \, c\right )}}{4 \, a^{3} d}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 3 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 2 i\right )}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{16 \, a^{4} d} \]
1/16*(-I*sqrt(1/2)*a^4*d*sqrt(1/(a^7*d^2))*e^(4*I*d*x + 4*I*c)*log(-1/4*(s qrt(2)*sqrt(1/2)*(I*a^3*d*e^(2*I*d*x + 2*I*c) + I*a^3*d)*sqrt(a/(e^(2*I*d* x + 2*I*c) + 1))*sqrt(1/(a^7*d^2)) + I)*e^(-I*d*x - I*c)/(a^3*d)) + I*sqrt (1/2)*a^4*d*sqrt(1/(a^7*d^2))*e^(4*I*d*x + 4*I*c)*log(-1/4*(sqrt(2)*sqrt(1 /2)*(-I*a^3*d*e^(2*I*d*x + 2*I*c) - I*a^3*d)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*sqrt(1/(a^7*d^2)) + I)*e^(-I*d*x - I*c)/(a^3*d)) + sqrt(2)*sqrt(a/(e^ (2*I*d*x + 2*I*c) + 1))*(I*e^(4*I*d*x + 4*I*c) + 3*I*e^(2*I*d*x + 2*I*c) + 2*I))*e^(-4*I*d*x - 4*I*c)/(a^4*d)
\[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\int \frac {\sec ^{3}{\left (c + d x \right )}}{\left (i a \left (\tan {\left (c + d x \right )} - i\right )\right )^{\frac {7}{2}}}\, dx \]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 977 vs. \(2 (94) = 188\).
Time = 0.46 (sec) , antiderivative size = 977, normalized size of antiderivative = 7.82 \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\text {Too large to display} \]
-1/64*(4*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1 )^(3/4)*((-I*sqrt(2)*cos(4*d*x + 4*c) - sqrt(2)*sin(4*d*x + 4*c))*cos(3/2* arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + (sqrt(2)*cos(4*d*x + 4* c) - I*sqrt(2)*sin(4*d*x + 4*c))*sin(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d *x + 2*c) + 1)))*sqrt(a) + 4*(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2* cos(2*d*x + 2*c) + 1)^(1/4)*((-I*sqrt(2)*cos(4*d*x + 4*c) - sqrt(2)*sin(4* d*x + 4*c))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + (sq rt(2)*cos(4*d*x + 4*c) - I*sqrt(2)*sin(4*d*x + 4*c))*sin(1/2*arctan2(sin(2 *d*x + 2*c), cos(2*d*x + 2*c) + 1)))*sqrt(a) - (2*sqrt(2)*arctan2((cos(2*d *x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*a rctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)), (cos(2*d*x + 2*c)^2 + sin (2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c) + 1)) + 1) - 2*sqrt(2)*arctan2((cos(2*d*x + 2*c)^ 2 + sin(2*d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*sin(1/2*arctan2(sin (2*d*x + 2*c), cos(2*d*x + 2*c) + 1)), (cos(2*d*x + 2*c)^2 + sin(2*d*x + 2 *c)^2 + 2*cos(2*d*x + 2*c) + 1)^(1/4)*cos(1/2*arctan2(sin(2*d*x + 2*c), co s(2*d*x + 2*c) + 1)) - 1) - I*sqrt(2)*log(sqrt(cos(2*d*x + 2*c)^2 + sin(2* d*x + 2*c)^2 + 2*cos(2*d*x + 2*c) + 1)*cos(1/2*arctan2(sin(2*d*x + 2*c), c os(2*d*x + 2*c) + 1))^2 + sqrt(cos(2*d*x + 2*c)^2 + sin(2*d*x + 2*c)^2 + 2 *cos(2*d*x + 2*c) + 1)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*...
\[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\int { \frac {\sec \left (d x + c\right )^{3}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\sec ^3(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^3\,{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2}} \,d x \]