Integrand size = 35, antiderivative size = 182 \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{11/2}} \, dx=\frac {i (c-i c \tan (e+f x))^{5/2}}{11 f (a+i a \tan (e+f x))^{11/2}}+\frac {i (c-i c \tan (e+f x))^{5/2}}{33 a f (a+i a \tan (e+f x))^{9/2}}+\frac {2 i (c-i c \tan (e+f x))^{5/2}}{231 a^2 f (a+i a \tan (e+f x))^{7/2}}+\frac {2 i (c-i c \tan (e+f x))^{5/2}}{1155 a^3 f (a+i a \tan (e+f x))^{5/2}} \]
1/11*I*(c-I*c*tan(f*x+e))^(5/2)/f/(a+I*a*tan(f*x+e))^(11/2)+1/33*I*(c-I*c* tan(f*x+e))^(5/2)/a/f/(a+I*a*tan(f*x+e))^(9/2)+2/231*I*(c-I*c*tan(f*x+e))^ (5/2)/a^2/f/(a+I*a*tan(f*x+e))^(7/2)+2/1155*I*(c-I*c*tan(f*x+e))^(5/2)/a^3 /f/(a+I*a*tan(f*x+e))^(5/2)
Time = 5.89 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.58 \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{11/2}} \, dx=\frac {c^2 (i+\tan (e+f x))^2 \sqrt {c-i c \tan (e+f x)} \left (-152-61 i \tan (e+f x)+16 \tan ^2(e+f x)+2 i \tan ^3(e+f x)\right )}{1155 a^5 f (-i+\tan (e+f x))^5 \sqrt {a+i a \tan (e+f x)}} \]
(c^2*(I + Tan[e + f*x])^2*Sqrt[c - I*c*Tan[e + f*x]]*(-152 - (61*I)*Tan[e + f*x] + 16*Tan[e + f*x]^2 + (2*I)*Tan[e + f*x]^3))/(1155*a^5*f*(-I + Tan[ e + f*x])^5*Sqrt[a + I*a*Tan[e + f*x]])
Time = 0.32 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.14, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.171, Rules used = {3042, 4006, 55, 55, 55, 48}
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.
\(\displaystyle \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{11/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{11/2}}dx\) |
\(\Big \downarrow \) 4006 |
\(\displaystyle \frac {a c \int \frac {(c-i c \tan (e+f x))^{3/2}}{(i \tan (e+f x) a+a)^{13/2}}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {a c \left (\frac {3 \int \frac {(c-i c \tan (e+f x))^{3/2}}{(i \tan (e+f x) a+a)^{11/2}}d\tan (e+f x)}{11 a}+\frac {i (c-i c \tan (e+f x))^{5/2}}{11 a c (a+i a \tan (e+f x))^{11/2}}\right )}{f}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {a c \left (\frac {3 \left (\frac {2 \int \frac {(c-i c \tan (e+f x))^{3/2}}{(i \tan (e+f x) a+a)^{9/2}}d\tan (e+f x)}{9 a}+\frac {i (c-i c \tan (e+f x))^{5/2}}{9 a c (a+i a \tan (e+f x))^{9/2}}\right )}{11 a}+\frac {i (c-i c \tan (e+f x))^{5/2}}{11 a c (a+i a \tan (e+f x))^{11/2}}\right )}{f}\) |
\(\Big \downarrow \) 55 |
\(\displaystyle \frac {a c \left (\frac {3 \left (\frac {2 \left (\frac {\int \frac {(c-i c \tan (e+f x))^{3/2}}{(i \tan (e+f x) a+a)^{7/2}}d\tan (e+f x)}{7 a}+\frac {i (c-i c \tan (e+f x))^{5/2}}{7 a c (a+i a \tan (e+f x))^{7/2}}\right )}{9 a}+\frac {i (c-i c \tan (e+f x))^{5/2}}{9 a c (a+i a \tan (e+f x))^{9/2}}\right )}{11 a}+\frac {i (c-i c \tan (e+f x))^{5/2}}{11 a c (a+i a \tan (e+f x))^{11/2}}\right )}{f}\) |
\(\Big \downarrow \) 48 |
\(\displaystyle \frac {a c \left (\frac {3 \left (\frac {2 \left (\frac {i (c-i c \tan (e+f x))^{5/2}}{35 a^2 c (a+i a \tan (e+f x))^{5/2}}+\frac {i (c-i c \tan (e+f x))^{5/2}}{7 a c (a+i a \tan (e+f x))^{7/2}}\right )}{9 a}+\frac {i (c-i c \tan (e+f x))^{5/2}}{9 a c (a+i a \tan (e+f x))^{9/2}}\right )}{11 a}+\frac {i (c-i c \tan (e+f x))^{5/2}}{11 a c (a+i a \tan (e+f x))^{11/2}}\right )}{f}\) |
(a*c*(((I/11)*(c - I*c*Tan[e + f*x])^(5/2))/(a*c*(a + I*a*Tan[e + f*x])^(1 1/2)) + (3*(((I/9)*(c - I*c*Tan[e + f*x])^(5/2))/(a*c*(a + I*a*Tan[e + f*x ])^(9/2)) + (2*(((I/7)*(c - I*c*Tan[e + f*x])^(5/2))/(a*c*(a + I*a*Tan[e + f*x])^(7/2)) + ((I/35)*(c - I*c*Tan[e + f*x])^(5/2))/(a^2*c*(a + I*a*Tan[ e + f*x])^(5/2))))/(9*a)))/(11*a)))/f
3.11.15.3.1 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] /; FreeQ[{ a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
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*(S implify[m + n + 2]/((b*c - a*d)*(m + 1))) Int[(a + b*x)^Simplify[m + 1]*( c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 2], 0] && NeQ[m, -1] && !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] || !SumSimp lerQ[n, 1])
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_) + (d_.)*tan[(e_.) + ( f_.)*(x_)])^(n_), x_Symbol] :> Simp[a*(c/f) Subst[Int[(a + b*x)^(m - 1)*( c + d*x)^(n - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, m, n }, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
Time = 0.78 (sec) , antiderivative size = 110, normalized size of antiderivative = 0.60
method | result | size |
derivativedivides | \(\frac {i \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{2} \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (2 i \left (\tan ^{4}\left (f x +e \right )\right )-45 i \left (\tan ^{2}\left (f x +e \right )\right )+14 \left (\tan ^{3}\left (f x +e \right )\right )-152 i-91 \tan \left (f x +e \right )\right )}{1155 f \,a^{6} \left (-\tan \left (f x +e \right )+i\right )^{7}}\) | \(110\) |
default | \(\frac {i \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c^{2} \left (1+\tan ^{2}\left (f x +e \right )\right ) \left (2 i \left (\tan ^{4}\left (f x +e \right )\right )-45 i \left (\tan ^{2}\left (f x +e \right )\right )+14 \left (\tan ^{3}\left (f x +e \right )\right )-152 i-91 \tan \left (f x +e \right )\right )}{1155 f \,a^{6} \left (-\tan \left (f x +e \right )+i\right )^{7}}\) | \(110\) |
1/1155*I/f*(-c*(I*tan(f*x+e)-1))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)*c^2/a^6* (1+tan(f*x+e)^2)*(2*I*tan(f*x+e)^4-45*I*tan(f*x+e)^2+14*tan(f*x+e)^3-152*I -91*tan(f*x+e))/(-tan(f*x+e)+I)^7
Time = 0.23 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.62 \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{11/2}} \, dx=\frac {{\left (231 i \, c^{2} e^{\left (8 i \, f x + 8 i \, e\right )} + 726 i \, c^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 880 i \, c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 490 i \, c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 105 i \, c^{2}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-11 i \, f x - 11 i \, e\right )}}{9240 \, a^{6} f} \]
1/9240*(231*I*c^2*e^(8*I*f*x + 8*I*e) + 726*I*c^2*e^(6*I*f*x + 6*I*e) + 88 0*I*c^2*e^(4*I*f*x + 4*I*e) + 490*I*c^2*e^(2*I*f*x + 2*I*e) + 105*I*c^2)*s qrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))*e^(-11* I*f*x - 11*I*e)/(a^6*f)
Timed out. \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{11/2}} \, dx=\text {Timed out} \]
Time = 0.44 (sec) , antiderivative size = 202, normalized size of antiderivative = 1.11 \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{11/2}} \, dx=\frac {{\left (105 i \, c^{2} \cos \left (11 \, f x + 11 \, e\right ) + 385 i \, c^{2} \cos \left (\frac {9}{11} \, \arctan \left (\sin \left (11 \, f x + 11 \, e\right ), \cos \left (11 \, f x + 11 \, e\right )\right )\right ) + 495 i \, c^{2} \cos \left (\frac {7}{11} \, \arctan \left (\sin \left (11 \, f x + 11 \, e\right ), \cos \left (11 \, f x + 11 \, e\right )\right )\right ) + 231 i \, c^{2} \cos \left (\frac {5}{11} \, \arctan \left (\sin \left (11 \, f x + 11 \, e\right ), \cos \left (11 \, f x + 11 \, e\right )\right )\right ) + 105 \, c^{2} \sin \left (11 \, f x + 11 \, e\right ) + 385 \, c^{2} \sin \left (\frac {9}{11} \, \arctan \left (\sin \left (11 \, f x + 11 \, e\right ), \cos \left (11 \, f x + 11 \, e\right )\right )\right ) + 495 \, c^{2} \sin \left (\frac {7}{11} \, \arctan \left (\sin \left (11 \, f x + 11 \, e\right ), \cos \left (11 \, f x + 11 \, e\right )\right )\right ) + 231 \, c^{2} \sin \left (\frac {5}{11} \, \arctan \left (\sin \left (11 \, f x + 11 \, e\right ), \cos \left (11 \, f x + 11 \, e\right )\right )\right )\right )} \sqrt {c}}{9240 \, a^{\frac {11}{2}} f} \]
1/9240*(105*I*c^2*cos(11*f*x + 11*e) + 385*I*c^2*cos(9/11*arctan2(sin(11*f *x + 11*e), cos(11*f*x + 11*e))) + 495*I*c^2*cos(7/11*arctan2(sin(11*f*x + 11*e), cos(11*f*x + 11*e))) + 231*I*c^2*cos(5/11*arctan2(sin(11*f*x + 11* e), cos(11*f*x + 11*e))) + 105*c^2*sin(11*f*x + 11*e) + 385*c^2*sin(9/11*a rctan2(sin(11*f*x + 11*e), cos(11*f*x + 11*e))) + 495*c^2*sin(7/11*arctan2 (sin(11*f*x + 11*e), cos(11*f*x + 11*e))) + 231*c^2*sin(5/11*arctan2(sin(1 1*f*x + 11*e), cos(11*f*x + 11*e))))*sqrt(c)/(a^(11/2)*f)
\[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{11/2}} \, dx=\int { \frac {{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {11}{2}}} \,d x } \]
Time = 9.80 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.14 \[ \int \frac {(c-i c \tan (e+f x))^{5/2}}{(a+i a \tan (e+f x))^{11/2}} \, dx=\frac {c^2\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (231\,\sin \left (4\,e+4\,f\,x\right )+726\,\sin \left (6\,e+6\,f\,x\right )+880\,\sin \left (8\,e+8\,f\,x\right )+490\,\sin \left (10\,e+10\,f\,x\right )+105\,\sin \left (12\,e+12\,f\,x\right )+\cos \left (4\,e+4\,f\,x\right )\,231{}\mathrm {i}+\cos \left (6\,e+6\,f\,x\right )\,726{}\mathrm {i}+\cos \left (8\,e+8\,f\,x\right )\,880{}\mathrm {i}+\cos \left (10\,e+10\,f\,x\right )\,490{}\mathrm {i}+\cos \left (12\,e+12\,f\,x\right )\,105{}\mathrm {i}\right )}{18480\,a^6\,f} \]
(c^2*((a*(cos(2*e + 2*f*x) + sin(2*e + 2*f*x)*1i + 1))/(cos(2*e + 2*f*x) + 1))^(1/2)*((c*(cos(2*e + 2*f*x) - sin(2*e + 2*f*x)*1i + 1))/(cos(2*e + 2* f*x) + 1))^(1/2)*(cos(4*e + 4*f*x)*231i + cos(6*e + 6*f*x)*726i + cos(8*e + 8*f*x)*880i + cos(10*e + 10*f*x)*490i + cos(12*e + 12*f*x)*105i + 231*si n(4*e + 4*f*x) + 726*sin(6*e + 6*f*x) + 880*sin(8*e + 8*f*x) + 490*sin(10* e + 10*f*x) + 105*sin(12*e + 12*f*x)))/(18480*a^6*f)