Integrand size = 35, antiderivative size = 200 \[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}} \, dx=\frac {i}{7 f (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}+\frac {6 \tan (e+f x)}{35 a f (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}+\frac {8 \tan (e+f x)}{35 a^2 c f (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}+\frac {16 \tan (e+f x)}{35 a^3 c^2 f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]
16/35*tan(f*x+e)/a^3/c^2/f/(a+I*a*tan(f*x+e))^(1/2)/(c-I*c*tan(f*x+e))^(1/ 2)+1/7*I/f/(a+I*a*tan(f*x+e))^(7/2)/(c-I*c*tan(f*x+e))^(5/2)+6/35*tan(f*x+ e)/a/f/(a+I*a*tan(f*x+e))^(5/2)/(c-I*c*tan(f*x+e))^(5/2)+8/35*tan(f*x+e)/a ^2/c/f/(a+I*a*tan(f*x+e))^(3/2)/(c-I*c*tan(f*x+e))^(3/2)
Time = 6.67 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.68 \[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}} \, dx=\frac {5-30 i \tan (e+f x)+30 \tan ^2(e+f x)-40 i \tan ^3(e+f x)+40 \tan ^4(e+f x)-16 i \tan ^5(e+f x)+16 \tan ^6(e+f x)}{35 a^3 c^2 f (-i+\tan (e+f x))^3 (i+\tan (e+f x))^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]
(5 - (30*I)*Tan[e + f*x] + 30*Tan[e + f*x]^2 - (40*I)*Tan[e + f*x]^3 + 40* Tan[e + f*x]^4 - (16*I)*Tan[e + f*x]^5 + 16*Tan[e + f*x]^6)/(35*a^3*c^2*f* (-I + Tan[e + f*x])^3*(I + Tan[e + f*x])^2*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt [c - I*c*Tan[e + f*x]])
Time = 0.33 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.11, 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, 42, 42, 41}
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.
| \(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}dx\) |
| \(\Big \downarrow \) 4006 |
| \(\displaystyle \frac {a c \int \frac {1}{(i \tan (e+f x) a+a)^{9/2} (c-i c \tan (e+f x))^{7/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {6 \int \frac {1}{(i \tan (e+f x) a+a)^{7/2} (c-i c \tan (e+f x))^{7/2}}d\tan (e+f x)}{7 a}+\frac {i}{7 a c (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}\right )}{f}\) |
| \(\Big \downarrow \) 42 |
| \(\displaystyle \frac {a c \left (\frac {6 \left (\frac {4 \int \frac {1}{(i \tan (e+f x) a+a)^{5/2} (c-i c \tan (e+f x))^{5/2}}d\tan (e+f x)}{5 a c}+\frac {\tan (e+f x)}{5 a c (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}\right )}{7 a}+\frac {i}{7 a c (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}\right )}{f}\) |
| \(\Big \downarrow \) 42 |
| \(\displaystyle \frac {a c \left (\frac {6 \left (\frac {4 \left (\frac {2 \int \frac {1}{(i \tan (e+f x) a+a)^{3/2} (c-i c \tan (e+f x))^{3/2}}d\tan (e+f x)}{3 a c}+\frac {\tan (e+f x)}{3 a c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}\right )}{5 a c}+\frac {\tan (e+f x)}{5 a c (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}\right )}{7 a}+\frac {i}{7 a c (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}\right )}{f}\) |
| \(\Big \downarrow \) 41 |
| \(\displaystyle \frac {a c \left (\frac {6 \left (\frac {4 \left (\frac {2 \tan (e+f x)}{3 a^2 c^2 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}+\frac {\tan (e+f x)}{3 a c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}\right )}{5 a c}+\frac {\tan (e+f x)}{5 a c (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}\right )}{7 a}+\frac {i}{7 a c (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}\right )}{f}\) |
(a*c*((I/7)/(a*c*(a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(5/2) ) + (6*(Tan[e + f*x]/(5*a*c*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2)) + (4*(Tan[e + f*x]/(3*a*c*(a + I*a*Tan[e + f*x])^(3/2)*(c - I *c*Tan[e + f*x])^(3/2)) + (2*Tan[e + f*x])/(3*a^2*c^2*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])))/(5*a*c)))/(7*a)))/f
3.11.42.3.1 Defintions of rubi rules used
Int[1/(((a_) + (b_.)*(x_))^(3/2)*((c_) + (d_.)*(x_))^(3/2)), x_Symbol] :> S imp[x/(a*c*Sqrt[a + b*x]*Sqrt[c + d*x]), x] /; FreeQ[{a, b, c, d}, x] && Eq Q[b*c + a*d, 0]
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[(- x)*(a + b*x)^(m + 1)*((c + d*x)^(m + 1)/(2*a*c*(m + 1))), x] + Simp[(2*m + 3)/(2*a*c*(m + 1)) Int[(a + b*x)^(m + 1)*(c + d*x)^(m + 1), x], x] /; Fre eQ[{a, b, c, d}, x] && EqQ[b*c + a*d, 0] && ILtQ[m + 3/2, 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*(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 = 0.84 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.76
| method | result | size |
| derivativedivides | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (16 i \left (\tan ^{7}\left (f x +e \right )\right )-16 \left (\tan ^{8}\left (f x +e \right )\right )+56 i \left (\tan ^{5}\left (f x +e \right )\right )-56 \left (\tan ^{6}\left (f x +e \right )\right )+70 i \left (\tan ^{3}\left (f x +e \right )\right )-70 \left (\tan ^{4}\left (f x +e \right )\right )+30 i \tan \left (f x +e \right )-35 \left (\tan ^{2}\left (f x +e \right )\right )-5\right )}{35 f \,a^{4} c^{3} \left (-\tan \left (f x +e \right )+i\right )^{5} \left (\tan \left (f x +e \right )+i\right )^{4}}\) | \(151\) |
| default | \(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \left (16 i \left (\tan ^{7}\left (f x +e \right )\right )-16 \left (\tan ^{8}\left (f x +e \right )\right )+56 i \left (\tan ^{5}\left (f x +e \right )\right )-56 \left (\tan ^{6}\left (f x +e \right )\right )+70 i \left (\tan ^{3}\left (f x +e \right )\right )-70 \left (\tan ^{4}\left (f x +e \right )\right )+30 i \tan \left (f x +e \right )-35 \left (\tan ^{2}\left (f x +e \right )\right )-5\right )}{35 f \,a^{4} c^{3} \left (-\tan \left (f x +e \right )+i\right )^{5} \left (\tan \left (f x +e \right )+i\right )^{4}}\) | \(151\) |
1/35/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)/a^4/c^3*(16* I*tan(f*x+e)^7-16*tan(f*x+e)^8+56*I*tan(f*x+e)^5-56*tan(f*x+e)^6+70*I*tan( f*x+e)^3-70*tan(f*x+e)^4+30*I*tan(f*x+e)-35*tan(f*x+e)^2-5)/(-tan(f*x+e)+I )^5/(tan(f*x+e)+I)^4
Time = 0.26 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.78 \[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}} \, dx=\frac {\sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} {\left (-7 i \, e^{\left (14 i \, f x + 14 i \, e\right )} - 77 i \, e^{\left (12 i \, f x + 12 i \, e\right )} - 595 i \, e^{\left (10 i \, f x + 10 i \, e\right )} - 320 i \, e^{\left (9 i \, f x + 9 i \, e\right )} + 175 i \, e^{\left (8 i \, f x + 8 i \, e\right )} - 320 i \, e^{\left (7 i \, f x + 7 i \, e\right )} + 875 i \, e^{\left (6 i \, f x + 6 i \, e\right )} + 217 i \, e^{\left (4 i \, f x + 4 i \, e\right )} + 47 i \, e^{\left (2 i \, f x + 2 i \, e\right )} + 5 i\right )} e^{\left (-7 i \, f x - 7 i \, e\right )}}{2240 \, a^{4} c^{3} f} \]
1/2240*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)) *(-7*I*e^(14*I*f*x + 14*I*e) - 77*I*e^(12*I*f*x + 12*I*e) - 595*I*e^(10*I* f*x + 10*I*e) - 320*I*e^(9*I*f*x + 9*I*e) + 175*I*e^(8*I*f*x + 8*I*e) - 32 0*I*e^(7*I*f*x + 7*I*e) + 875*I*e^(6*I*f*x + 6*I*e) + 217*I*e^(4*I*f*x + 4 *I*e) + 47*I*e^(2*I*f*x + 2*I*e) + 5*I)*e^(-7*I*f*x - 7*I*e)/(a^4*c^3*f)
Timed out. \[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}} \, dx=\text {Timed out} \]
Exception generated. \[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}} \, dx=\text {Exception raised: RuntimeError} \]
\[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}} \, dx=\int { \frac {1}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}}} \,d x } \]
Time = 7.41 (sec) , antiderivative size = 186, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}} \, dx=\frac {\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (\cos \left (2\,e+2\,f\,x\right )\,630{}\mathrm {i}+\cos \left (4\,e+4\,f\,x\right )\,168{}\mathrm {i}+\cos \left (6\,e+6\,f\,x\right )\,42{}\mathrm {i}+\cos \left (8\,e+8\,f\,x\right )\,5{}\mathrm {i}+770\,\sin \left (2\,e+2\,f\,x\right )+182\,\sin \left (4\,e+4\,f\,x\right )+42\,\sin \left (6\,e+6\,f\,x\right )+5\,\sin \left (8\,e+8\,f\,x\right )-525{}\mathrm {i}\right )}{2240\,a^4\,c^2\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]
(((a*(cos(2*e + 2*f*x) + sin(2*e + 2*f*x)*1i + 1))/(cos(2*e + 2*f*x) + 1)) ^(1/2)*(cos(2*e + 2*f*x)*630i + cos(4*e + 4*f*x)*168i + cos(6*e + 6*f*x)*4 2i + cos(8*e + 8*f*x)*5i + 770*sin(2*e + 2*f*x) + 182*sin(4*e + 4*f*x) + 4 2*sin(6*e + 6*f*x) + 5*sin(8*e + 8*f*x) - 525i))/(2240*a^4*c^2*f*((c*(cos( 2*e + 2*f*x) - sin(2*e + 2*f*x)*1i + 1))/(cos(2*e + 2*f*x) + 1))^(1/2))