Integrand size = 26, antiderivative size = 233 \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=-\frac {11 i \text {arctanh}\left (\frac {\sqrt {a+i a \tan (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{64 \sqrt {2} a^{7/2} d}+\frac {11 i a}{36 d (a+i a \tan (c+d x))^{9/2}}-\frac {i a^2}{2 d (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}+\frac {11 i}{56 d (a+i a \tan (c+d x))^{7/2}}+\frac {11 i}{80 a d (a+i a \tan (c+d x))^{5/2}}+\frac {11 i}{96 a^2 d (a+i a \tan (c+d x))^{3/2}}+\frac {11 i}{64 a^3 d \sqrt {a+i a \tan (c+d x)}} \]
-11/128*I*arctanh(1/2*(a+I*a*tan(d*x+c))^(1/2)*2^(1/2)/a^(1/2))/a^(7/2)/d* 2^(1/2)+11/64*I/a^3/d/(a+I*a*tan(d*x+c))^(1/2)+11/36*I*a/d/(a+I*a*tan(d*x+ c))^(9/2)-1/2*I*a^2/d/(a-I*a*tan(d*x+c))/(a+I*a*tan(d*x+c))^(9/2)+11/56*I/ d/(a+I*a*tan(d*x+c))^(7/2)+11/80*I/a/d/(a+I*a*tan(d*x+c))^(5/2)+11/96*I/a^ 2/d/(a+I*a*tan(d*x+c))^(3/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 0.44 (sec) , antiderivative size = 51, normalized size of antiderivative = 0.22 \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {i a \operatorname {Hypergeometric2F1}\left (-\frac {9}{2},2,-\frac {7}{2},\frac {1}{2} (1+i \tan (c+d x))\right )}{18 d (a+i a \tan (c+d x))^{9/2}} \]
((I/18)*a*Hypergeometric2F1[-9/2, 2, -7/2, (1 + I*Tan[c + d*x])/2])/(d*(a + I*a*Tan[c + d*x])^(9/2))
Time = 0.34 (sec) , antiderivative size = 240, normalized size of antiderivative = 1.03, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {3042, 3968, 52, 61, 61, 61, 61, 61, 73, 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.
| \(\displaystyle \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sec (c+d x)^2 (a+i a \tan (c+d x))^{7/2}}dx\) |
| \(\Big \downarrow \) 3968 |
| \(\displaystyle -\frac {i a^3 \int \frac {1}{(a-i a \tan (c+d x))^2 (i \tan (c+d x) a+a)^{11/2}}d(i a \tan (c+d x))}{d}\) |
| \(\Big \downarrow \) 52 |
| \(\displaystyle -\frac {i a^3 \left (\frac {11 \int \frac {1}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{11/2}}d(i a \tan (c+d x))}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {i a^3 \left (\frac {11 \left (\frac {\int \frac {1}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{9/2}}d(i a \tan (c+d x))}{2 a}-\frac {1}{9 a (a+i a \tan (c+d x))^{9/2}}\right )}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {i a^3 \left (\frac {11 \left (\frac {\frac {\int \frac {1}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{7/2}}d(i a \tan (c+d x))}{2 a}-\frac {1}{7 a (a+i a \tan (c+d x))^{7/2}}}{2 a}-\frac {1}{9 a (a+i a \tan (c+d x))^{9/2}}\right )}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {i a^3 \left (\frac {11 \left (\frac {\frac {\frac {\int \frac {1}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{5/2}}d(i a \tan (c+d x))}{2 a}-\frac {1}{5 a (a+i a \tan (c+d x))^{5/2}}}{2 a}-\frac {1}{7 a (a+i a \tan (c+d x))^{7/2}}}{2 a}-\frac {1}{9 a (a+i a \tan (c+d x))^{9/2}}\right )}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {i a^3 \left (\frac {11 \left (\frac {\frac {\frac {\frac {\int \frac {1}{(a-i a \tan (c+d x)) (i \tan (c+d x) a+a)^{3/2}}d(i a \tan (c+d x))}{2 a}-\frac {1}{3 a (a+i a \tan (c+d x))^{3/2}}}{2 a}-\frac {1}{5 a (a+i a \tan (c+d x))^{5/2}}}{2 a}-\frac {1}{7 a (a+i a \tan (c+d x))^{7/2}}}{2 a}-\frac {1}{9 a (a+i a \tan (c+d x))^{9/2}}\right )}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 61 |
| \(\displaystyle -\frac {i a^3 \left (\frac {11 \left (\frac {\frac {\frac {\frac {\frac {\int \frac {1}{(a-i a \tan (c+d x)) \sqrt {i \tan (c+d x) a+a}}d(i a \tan (c+d x))}{2 a}-\frac {1}{a \sqrt {a+i a \tan (c+d x)}}}{2 a}-\frac {1}{3 a (a+i a \tan (c+d x))^{3/2}}}{2 a}-\frac {1}{5 a (a+i a \tan (c+d x))^{5/2}}}{2 a}-\frac {1}{7 a (a+i a \tan (c+d x))^{7/2}}}{2 a}-\frac {1}{9 a (a+i a \tan (c+d x))^{9/2}}\right )}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {i a^3 \left (\frac {11 \left (\frac {\frac {\frac {\frac {\frac {\int \frac {1}{a^2 \tan ^2(c+d x)+2 a}d\sqrt {i \tan (c+d x) a+a}}{a}-\frac {1}{a \sqrt {a+i a \tan (c+d x)}}}{2 a}-\frac {1}{3 a (a+i a \tan (c+d x))^{3/2}}}{2 a}-\frac {1}{5 a (a+i a \tan (c+d x))^{5/2}}}{2 a}-\frac {1}{7 a (a+i a \tan (c+d x))^{7/2}}}{2 a}-\frac {1}{9 a (a+i a \tan (c+d x))^{9/2}}\right )}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}\right )}{d}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {i a^3 \left (\frac {11 \left (\frac {\frac {\frac {\frac {\frac {i \arctan \left (\frac {\sqrt {a} \tan (c+d x)}{\sqrt {2}}\right )}{\sqrt {2} a^{3/2}}-\frac {1}{a \sqrt {a+i a \tan (c+d x)}}}{2 a}-\frac {1}{3 a (a+i a \tan (c+d x))^{3/2}}}{2 a}-\frac {1}{5 a (a+i a \tan (c+d x))^{5/2}}}{2 a}-\frac {1}{7 a (a+i a \tan (c+d x))^{7/2}}}{2 a}-\frac {1}{9 a (a+i a \tan (c+d x))^{9/2}}\right )}{4 a}+\frac {1}{2 a (a-i a \tan (c+d x)) (a+i a \tan (c+d x))^{9/2}}\right )}{d}\) |
((-I)*a^3*(1/(2*a*(a - I*a*Tan[c + d*x])*(a + I*a*Tan[c + d*x])^(9/2)) + ( 11*(-1/9*1/(a*(a + I*a*Tan[c + d*x])^(9/2)) + (-1/7*1/(a*(a + I*a*Tan[c + d*x])^(7/2)) + (-1/5*1/(a*(a + I*a*Tan[c + d*x])^(5/2)) + (-1/3*1/(a*(a + I*a*Tan[c + d*x])^(3/2)) + ((I*ArcTan[(Sqrt[a]*Tan[c + d*x])/Sqrt[2]])/(Sq rt[2]*a^(3/2)) - 1/(a*Sqrt[a + I*a*Tan[c + d*x]]))/(2*a))/(2*a))/(2*a))/(2 *a)))/(4*a)))/d
3.4.83.3.1 Defintions of rubi rules used
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && ILtQ[m, -1] && FractionQ[n] && LtQ[n, 0]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(( m + n + 2)/((b*c - a*d)*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && LtQ[m, -1] && !(LtQ[n, -1] && (EqQ[a, 0 ] || (NeQ[c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && IntLinearQ[a, b, c, d , m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
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_)]^(m_)*((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(n_ ), x_Symbol] :> Simp[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]
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 976 vs. \(2 (183 ) = 366\).
Time = 10.36 (sec) , antiderivative size = 977, normalized size of antiderivative = 4.19
-1/40320/d/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)/(cos(d*x+c)+1)/(1+I*tan(d*x+ c))^3/(a*(1+I*tan(d*x+c)))^(1/2)/a^3*(-50424*I*cos(d*x+c)*(-cos(d*x+c)/(co s(d*x+c)+1))^(1/2)-13860*I*arctan(1/2*(cos(d*x+c)+1+I*sin(d*x+c))/(cos(d*x +c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-12320*sin(d*x+c)*cos(d*x+c)^2*( -cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+7840*I*cos(d*x+c)^2*(-cos(d*x+c)/(cos(d* x+c)+1))^(1/2)-3465*I*sec(d*x+c)^3*arctan(1/2*(cos(d*x+c)+1+I*sin(d*x+c))/ (cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))-12320*(-cos(d*x+c)/(cos (d*x+c)+1))^(1/2)*cos(d*x+c)*sin(d*x+c)-50424*I*(-cos(d*x+c)/(cos(d*x+c)+1 ))^(1/2)+25410*I*sec(d*x+c)^2*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+42504*sin (d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+27720*arctan(1/2*(cos(d*x+c)+1+ I*sin(d*x+c))/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))*sin(d*x+c )+27720*I*sec(d*x+c)*arctan(1/2*(cos(d*x+c)+1+I*sin(d*x+c))/(cos(d*x+c)+1) /(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+25410*I*sec(d*x+c)*(-cos(d*x+c)/(cos( d*x+c)+1))^(1/2)+42504*tan(d*x+c)*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)+13860 *tan(d*x+c)*arctan(1/2*(cos(d*x+c)+1+I*sin(d*x+c))/(cos(d*x+c)+1)/(-cos(d* x+c)/(cos(d*x+c)+1))^(1/2))-27720*I*cos(d*x+c)*arctan(1/2*(cos(d*x+c)+1+I* sin(d*x+c))/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2))+7840*I*cos( d*x+c)^3*(-cos(d*x+c)/(cos(d*x+c)+1))^(1/2)-6930*tan(d*x+c)*sec(d*x+c)*(-c os(d*x+c)/(cos(d*x+c)+1))^(1/2)-13860*tan(d*x+c)*sec(d*x+c)*arctan(1/2*(co s(d*x+c)+1+I*sin(d*x+c))/(cos(d*x+c)+1)/(-cos(d*x+c)/(cos(d*x+c)+1))^(1...
Time = 0.25 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.36 \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {{\left (-3465 i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (9 i \, d x + 9 i \, c\right )} \log \left (4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} + a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + 3465 i \, \sqrt {\frac {1}{2}} a^{4} d \sqrt {\frac {1}{a^{7} d^{2}}} e^{\left (9 i \, d x + 9 i \, c\right )} \log \left (-4 \, {\left (\sqrt {2} \sqrt {\frac {1}{2}} {\left (a^{4} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{4} d\right )} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {1}{a^{7} d^{2}}} - a e^{\left (i \, d x + i \, c\right )}\right )} e^{\left (-i \, d x - i \, c\right )}\right ) + \sqrt {2} \sqrt {\frac {a}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} {\left (-315 i \, e^{\left (12 i \, d x + 12 i \, c\right )} + 4303 i \, e^{\left (10 i \, d x + 10 i \, c\right )} + 7034 i \, e^{\left (8 i \, d x + 8 i \, c\right )} + 3754 i \, e^{\left (6 i \, d x + 6 i \, c\right )} + 1798 i \, e^{\left (4 i \, d x + 4 i \, c\right )} + 530 i \, e^{\left (2 i \, d x + 2 i \, c\right )} + 70 i\right )}\right )} e^{\left (-9 i \, d x - 9 i \, c\right )}}{40320 \, a^{4} d} \]
1/40320*(-3465*I*sqrt(1/2)*a^4*d*sqrt(1/(a^7*d^2))*e^(9*I*d*x + 9*I*c)*log (4*(sqrt(2)*sqrt(1/2)*(a^4*d*e^(2*I*d*x + 2*I*c) + a^4*d)*sqrt(a/(e^(2*I*d *x + 2*I*c) + 1))*sqrt(1/(a^7*d^2)) + a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) + 3465*I*sqrt(1/2)*a^4*d*sqrt(1/(a^7*d^2))*e^(9*I*d*x + 9*I*c)*log(-4*(sq rt(2)*sqrt(1/2)*(a^4*d*e^(2*I*d*x + 2*I*c) + a^4*d)*sqrt(a/(e^(2*I*d*x + 2 *I*c) + 1))*sqrt(1/(a^7*d^2)) - a*e^(I*d*x + I*c))*e^(-I*d*x - I*c)) + sqr t(2)*sqrt(a/(e^(2*I*d*x + 2*I*c) + 1))*(-315*I*e^(12*I*d*x + 12*I*c) + 430 3*I*e^(10*I*d*x + 10*I*c) + 7034*I*e^(8*I*d*x + 8*I*c) + 3754*I*e^(6*I*d*x + 6*I*c) + 1798*I*e^(4*I*d*x + 4*I*c) + 530*I*e^(2*I*d*x + 2*I*c) + 70*I) )*e^(-9*I*d*x - 9*I*c)/(a^4*d)
Timed out. \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\text {Timed out} \]
Time = 0.31 (sec) , antiderivative size = 195, normalized size of antiderivative = 0.84 \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\frac {i \, {\left (\frac {4 \, {\left (3465 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{5} - 4620 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{4} a - 1848 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{3} a^{2} - 1584 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} a^{3} - 1760 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )} a^{4} - 2240 \, a^{5}\right )}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {11}{2}} a^{2} - 2 \, {\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a^{3}} + \frac {3465 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {i \, a \tan \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {i \, a \tan \left (d x + c\right ) + a}}\right )}{a^{\frac {5}{2}}}\right )}}{80640 \, a d} \]
1/80640*I*(4*(3465*(I*a*tan(d*x + c) + a)^5 - 4620*(I*a*tan(d*x + c) + a)^ 4*a - 1848*(I*a*tan(d*x + c) + a)^3*a^2 - 1584*(I*a*tan(d*x + c) + a)^2*a^ 3 - 1760*(I*a*tan(d*x + c) + a)*a^4 - 2240*a^5)/((I*a*tan(d*x + c) + a)^(1 1/2)*a^2 - 2*(I*a*tan(d*x + c) + a)^(9/2)*a^3) + 3465*sqrt(2)*log(-(sqrt(2 )*sqrt(a) - sqrt(I*a*tan(d*x + c) + a))/(sqrt(2)*sqrt(a) + sqrt(I*a*tan(d* x + c) + a)))/a^(5/2))/(a*d)
\[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\int { \frac {\cos \left (d x + c\right )^{2}}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]
Timed out. \[ \int \frac {\cos ^2(c+d x)}{(a+i a \tan (c+d x))^{7/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^2}{{\left (a+a\,\mathrm {tan}\left (c+d\,x\right )\,1{}\mathrm {i}\right )}^{7/2}} \,d x \]