Integrand size = 43, antiderivative size = 144 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\frac {8 a^3 (i A+B) (c-i c \tan (e+f x))^{5/2}}{5 f}-\frac {8 a^3 (i A+2 B) (c-i c \tan (e+f x))^{7/2}}{7 c f}+\frac {2 a^3 (i A+5 B) (c-i c \tan (e+f x))^{9/2}}{9 c^2 f}-\frac {2 a^3 B (c-i c \tan (e+f x))^{11/2}}{11 c^3 f} \]
8/5*a^3*(I*A+B)*(c-I*c*tan(f*x+e))^(5/2)/f-8/7*a^3*(I*A+2*B)*(c-I*c*tan(f* x+e))^(7/2)/c/f+2/9*a^3*(I*A+5*B)*(c-I*c*tan(f*x+e))^(9/2)/c^2/f-2/11*a^3* B*(c-I*c*tan(f*x+e))^(11/2)/c^3/f
Time = 5.97 (sec) , antiderivative size = 128, normalized size of antiderivative = 0.89 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\frac {a^3 c^3 \sec ^6(e+f x) (143 (11 i A+B) \cos (e+f x)+(781 i A+701 B) \cos (3 (e+f x))-10 (121 A-74 i B+(121 A-137 i B) \cos (2 (e+f x))) \sin (e+f x)) (\cos (3 (e+f x))-i \sin (3 (e+f x)))}{3465 f \sqrt {c-i c \tan (e+f x)}} \]
(a^3*c^3*Sec[e + f*x]^6*(143*((11*I)*A + B)*Cos[e + f*x] + ((781*I)*A + 70 1*B)*Cos[3*(e + f*x)] - 10*(121*A - (74*I)*B + (121*A - (137*I)*B)*Cos[2*( e + f*x)])*Sin[e + f*x])*(Cos[3*(e + f*x)] - I*Sin[3*(e + f*x)]))/(3465*f* Sqrt[c - I*c*Tan[e + f*x]])
Time = 0.38 (sec) , antiderivative size = 131, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.116, Rules used = {3042, 4071, 27, 86, 2009}
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.
| \(\displaystyle \int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a+i a \tan (e+f x))^3 (c-i c \tan (e+f x))^{5/2} (A+B \tan (e+f x))dx\) |
| \(\Big \downarrow \) 4071 |
| \(\displaystyle \frac {a c \int a^2 (i \tan (e+f x)+1)^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {a^3 c \int (i \tan (e+f x)+1)^2 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 86 |
| \(\displaystyle \frac {a^3 c \int \left (\frac {i B (c-i c \tan (e+f x))^{9/2}}{c^3}+\frac {(A-5 i B) (c-i c \tan (e+f x))^{7/2}}{c^2}-\frac {4 (A-2 i B) (c-i c \tan (e+f x))^{5/2}}{c}+4 (A-i B) (c-i c \tan (e+f x))^{3/2}\right )d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {a^3 c \left (\frac {2 (5 B+i A) (c-i c \tan (e+f x))^{9/2}}{9 c^3}-\frac {8 (2 B+i A) (c-i c \tan (e+f x))^{7/2}}{7 c^2}+\frac {8 (B+i A) (c-i c \tan (e+f x))^{5/2}}{5 c}-\frac {2 B (c-i c \tan (e+f x))^{11/2}}{11 c^4}\right )}{f}\) |
(a^3*c*((8*(I*A + B)*(c - I*c*Tan[e + f*x])^(5/2))/(5*c) - (8*(I*A + 2*B)* (c - I*c*Tan[e + f*x])^(7/2))/(7*c^2) + (2*(I*A + 5*B)*(c - I*c*Tan[e + f* x])^(9/2))/(9*c^3) - (2*B*(c - I*c*Tan[e + f*x])^(11/2))/(11*c^4)))/f
3.8.57.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_ .), x_] :> Int[ExpandIntegrand[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && ((ILtQ[n, 0] && ILtQ[p, 0]) || EqQ[p, 1 ] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))
Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Si mp[a*(c/f) Subst[Int[(a + b*x)^(m - 1)*(c + d*x)^(n - 1)*(A + B*x), x], x , Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, B, m, n}, x] && EqQ[b*c + a*d, 0] && EqQ[a^2 + b^2, 0]
Time = 0.60 (sec) , antiderivative size = 121, normalized size of antiderivative = 0.84
| method | result | size |
| derivativedivides | \(\frac {2 i a^{3} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}+\frac {\left (-5 i B c +c A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {\left (-4 \left (-i B c +c A \right ) c +4 i B \,c^{2}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {4 \left (-i B c +c A \right ) c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}\right )}{f \,c^{3}}\) | \(121\) |
| default | \(\frac {2 i a^{3} \left (\frac {i B \left (c -i c \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}+\frac {\left (-5 i B c +c A \right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {\left (-4 \left (-i B c +c A \right ) c +4 i B \,c^{2}\right ) \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {4 \left (-i B c +c A \right ) c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}\right )}{f \,c^{3}}\) | \(121\) |
| parts | \(\frac {2 i a^{3} A c \left (-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-2 c \sqrt {c -i c \tan \left (f x +e \right )}+2 c^{\frac {3}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f}+\frac {a^{3} \left (3 i A +B \right ) \left (\frac {2 \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+\frac {2 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}+4 \sqrt {c -i c \tan \left (f x +e \right )}\, c^{2}-4 c^{\frac {5}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f}+\frac {2 B \,a^{3} \left (-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {11}{2}}}{11}+\frac {2 c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}-\frac {2 c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}} c^{4}}{3}-2 \sqrt {c -i c \tan \left (f x +e \right )}\, c^{5}+2 c^{\frac {11}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f \,c^{3}}-\frac {6 i a^{3} \left (-i B +A \right ) \left (\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}} c^{2}}{3}+2 \sqrt {c -i c \tan \left (f x +e \right )}\, c^{3}-2 c^{\frac {7}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f c}-\frac {2 a^{3} \left (i A +3 B \right ) \left (-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {c \left (c -i c \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {c^{2} \left (c -i c \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {\left (c -i c \tan \left (f x +e \right )\right )^{\frac {3}{2}} c^{3}}{3}-2 \sqrt {c -i c \tan \left (f x +e \right )}\, c^{4}+2 c^{\frac {9}{2}} \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c -i c \tan \left (f x +e \right )}\, \sqrt {2}}{2 \sqrt {c}}\right )\right )}{f \,c^{2}}\) | \(554\) |
2*I/f*a^3/c^3*(1/11*I*B*(c-I*c*tan(f*x+e))^(11/2)+1/9*(-5*I*B*c+c*A)*(c-I* c*tan(f*x+e))^(9/2)+1/7*(-4*(-I*B*c+c*A)*c+4*I*B*c^2)*(c-I*c*tan(f*x+e))^( 7/2)+4/5*(-I*B*c+c*A)*c^2*(c-I*c*tan(f*x+e))^(5/2))
Time = 0.38 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.20 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=-\frac {32 \, \sqrt {2} {\left (693 \, {\left (-i \, A - B\right )} a^{3} c^{2} e^{\left (6 i \, f x + 6 i \, e\right )} + 99 \, {\left (-11 i \, A - B\right )} a^{3} c^{2} e^{\left (4 i \, f x + 4 i \, e\right )} + 44 \, {\left (-11 i \, A - B\right )} a^{3} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + 8 \, {\left (-11 i \, A - B\right )} a^{3} c^{2}\right )} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{3465 \, {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]
integrate((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x , algorithm="fricas")
-32/3465*sqrt(2)*(693*(-I*A - B)*a^3*c^2*e^(6*I*f*x + 6*I*e) + 99*(-11*I*A - B)*a^3*c^2*e^(4*I*f*x + 4*I*e) + 44*(-11*I*A - B)*a^3*c^2*e^(2*I*f*x + 2*I*e) + 8*(-11*I*A - B)*a^3*c^2)*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))/(f*e^( 10*I*f*x + 10*I*e) + 5*f*e^(8*I*f*x + 8*I*e) + 10*f*e^(6*I*f*x + 6*I*e) + 10*f*e^(4*I*f*x + 4*I*e) + 5*f*e^(2*I*f*x + 2*I*e) + f)
\[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=- i a^{3} \left (\int i A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c}\, dx + \int \left (- A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\right )\, dx + \int \left (- 2 A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\right )\, dx + \int \left (- A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )}\right )\, dx + \int \left (- B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\right )\, dx + \int \left (- 2 B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\right )\, dx + \int \left (- B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{6}{\left (e + f x \right )}\right )\, dx + \int 2 i A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{2}{\left (e + f x \right )}\, dx + \int i A c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{4}{\left (e + f x \right )}\, dx + \int i B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan {\left (e + f x \right )}\, dx + \int 2 i B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{3}{\left (e + f x \right )}\, dx + \int i B c^{2} \sqrt {- i c \tan {\left (e + f x \right )} + c} \tan ^{5}{\left (e + f x \right )}\, dx\right ) \]
-I*a**3*(Integral(I*A*c**2*sqrt(-I*c*tan(e + f*x) + c), x) + Integral(-A*c **2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x), x) + Integral(-2*A*c**2*sqrt (-I*c*tan(e + f*x) + c)*tan(e + f*x)**3, x) + Integral(-A*c**2*sqrt(-I*c*t an(e + f*x) + c)*tan(e + f*x)**5, x) + Integral(-B*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**2, x) + Integral(-2*B*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**4, x) + Integral(-B*c**2*sqrt(-I*c*tan(e + f*x) + c)*ta n(e + f*x)**6, x) + Integral(2*I*A*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**2, x) + Integral(I*A*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x) **4, x) + Integral(I*B*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x), x) + Integral(2*I*B*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**3, x) + Int egral(I*B*c**2*sqrt(-I*c*tan(e + f*x) + c)*tan(e + f*x)**5, x))
Time = 0.21 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.72 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\frac {2 i \, {\left (315 i \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {11}{2}} B a^{3} + 385 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {9}{2}} {\left (A - 5 i \, B\right )} a^{3} c - 1980 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {7}{2}} {\left (A - 2 i \, B\right )} a^{3} c^{2} + 2772 \, {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} {\left (A - i \, B\right )} a^{3} c^{3}\right )}}{3465 \, c^{3} f} \]
integrate((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x , algorithm="maxima")
2/3465*I*(315*I*(-I*c*tan(f*x + e) + c)^(11/2)*B*a^3 + 385*(-I*c*tan(f*x + e) + c)^(9/2)*(A - 5*I*B)*a^3*c - 1980*(-I*c*tan(f*x + e) + c)^(7/2)*(A - 2*I*B)*a^3*c^2 + 2772*(-I*c*tan(f*x + e) + c)^(5/2)*(A - I*B)*a^3*c^3)/(c ^3*f)
\[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\int { {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{3} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]
integrate((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x , algorithm="giac")
Time = 12.11 (sec) , antiderivative size = 349, normalized size of antiderivative = 2.42 \[ \int (a+i a \tan (e+f x))^3 (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=-\frac {\left (\frac {a^3\,c^2\,\left (A-B\,1{}\mathrm {i}\right )\,32{}\mathrm {i}}{7\,f}+\frac {a^3\,c^2\,\left (A-B\,3{}\mathrm {i}\right )\,32{}\mathrm {i}}{7\,f}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^3}-\frac {\left (\frac {a^3\,c^2\,\left (A-B\,1{}\mathrm {i}\right )\,32{}\mathrm {i}}{11\,f}-\frac {a^3\,c^2\,\left (A+B\,1{}\mathrm {i}\right )\,32{}\mathrm {i}}{11\,f}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^5}+\frac {\left (\frac {128\,B\,a^3\,c^2}{9\,f}+\frac {a^3\,c^2\,\left (A-B\,1{}\mathrm {i}\right )\,32{}\mathrm {i}}{9\,f}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}}{{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^4}+\frac {a^3\,c^2\,\left (A-B\,1{}\mathrm {i}\right )\,\sqrt {c+\frac {c\,\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}\,1{}\mathrm {i}-\mathrm {i}\right )\,1{}\mathrm {i}}{{\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1}}\,32{}\mathrm {i}}{5\,f\,{\left ({\mathrm {e}}^{e\,2{}\mathrm {i}+f\,x\,2{}\mathrm {i}}+1\right )}^2} \]
(((a^3*c^2*(A - B*1i)*32i)/(9*f) + (128*B*a^3*c^2)/(9*f))*(c + (c*(exp(e*2 i + f*x*2i)*1i - 1i)*1i)/(exp(e*2i + f*x*2i) + 1))^(1/2))/(exp(e*2i + f*x* 2i) + 1)^4 - (((a^3*c^2*(A - B*1i)*32i)/(11*f) - (a^3*c^2*(A + B*1i)*32i)/ (11*f))*(c + (c*(exp(e*2i + f*x*2i)*1i - 1i)*1i)/(exp(e*2i + f*x*2i) + 1)) ^(1/2))/(exp(e*2i + f*x*2i) + 1)^5 - (((a^3*c^2*(A - B*1i)*32i)/(7*f) + (a ^3*c^2*(A - B*3i)*32i)/(7*f))*(c + (c*(exp(e*2i + f*x*2i)*1i - 1i)*1i)/(ex p(e*2i + f*x*2i) + 1))^(1/2))/(exp(e*2i + f*x*2i) + 1)^3 + (a^3*c^2*(A - B *1i)*(c + (c*(exp(e*2i + f*x*2i)*1i - 1i)*1i)/(exp(e*2i + f*x*2i) + 1))^(1 /2)*32i)/(5*f*(exp(e*2i + f*x*2i) + 1)^2)