Integrand size = 41, antiderivative size = 127 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx=-\frac {a^3 (i A+B)}{2 c^8 f (i+\tan (e+f x))^8}+\frac {4 a^3 (A-2 i B)}{7 c^8 f (i+\tan (e+f x))^7}+\frac {a^3 (i A+5 B)}{6 c^8 f (i+\tan (e+f x))^6}+\frac {i a^3 B}{5 c^8 f (i+\tan (e+f x))^5} \] Output:
-1/2*a^3*(I*A+B)/c^8/f/(I+tan(f*x+e))^8+4/7*a^3*(A-2*I*B)/c^8/f/(I+tan(f*x +e))^7+1/6*a^3*(I*A+5*B)/c^8/f/(I+tan(f*x+e))^6+1/5*I*a^3*B/c^8/f/(I+tan(f *x+e))^5
Time = 4.72 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.66 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx=\frac {a^3 \left (2 (-10 i A+B)+2 (25 A-8 i B) \tan (e+f x)+7 (5 i A+7 B) \tan ^2(e+f x)+42 i B \tan ^3(e+f x)\right )}{210 c^8 f (i+\tan (e+f x))^8} \] Input:
Integrate[((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^8,x]
Output:
(a^3*(2*((-10*I)*A + B) + 2*(25*A - (8*I)*B)*Tan[e + f*x] + 7*((5*I)*A + 7 *B)*Tan[e + f*x]^2 + (42*I)*B*Tan[e + f*x]^3))/(210*c^8*f*(I + Tan[e + f*x ])^8)
Time = 0.57 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.80, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, 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.
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8}dx\) |
\(\Big \downarrow \) 4071 |
\(\displaystyle \frac {a c \int \frac {a^2 (i \tan (e+f x)+1)^2 (A+B \tan (e+f x))}{c^9 (1-i \tan (e+f x))^9}d\tan (e+f x)}{f}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {a^3 \int \frac {(i \tan (e+f x)+1)^2 (A+B \tan (e+f x))}{(1-i \tan (e+f x))^9}d\tan (e+f x)}{c^8 f}\) |
\(\Big \downarrow \) 86 |
\(\displaystyle \frac {a^3 \int \left (-\frac {4 (A-2 i B)}{(\tan (e+f x)+i)^8}-\frac {i B}{(\tan (e+f x)+i)^6}-\frac {i (A-5 i B)}{(\tan (e+f x)+i)^7}+\frac {4 (i A+B)}{(\tan (e+f x)+i)^9}\right )d\tan (e+f x)}{c^8 f}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {a^3 \left (\frac {4 (A-2 i B)}{7 (\tan (e+f x)+i)^7}+\frac {5 B+i A}{6 (\tan (e+f x)+i)^6}-\frac {B+i A}{2 (\tan (e+f x)+i)^8}+\frac {i B}{5 (\tan (e+f x)+i)^5}\right )}{c^8 f}\) |
Input:
Int[((a + I*a*Tan[e + f*x])^3*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x]) ^8,x]
Output:
(a^3*(-1/2*(I*A + B)/(I + Tan[e + f*x])^8 + (4*(A - (2*I)*B))/(7*(I + Tan[ e + f*x])^7) + (I*A + 5*B)/(6*(I + Tan[e + f*x])^6) + ((I/5)*B)/(I + Tan[e + f*x])^5))/(c^8*f)
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 = 0.46 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.71
method | result | size |
derivativedivides | \(\frac {a^{3} \left (\frac {i B}{5 \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {-i A -5 B}{6 \left (i+\tan \left (f x +e \right )\right )^{6}}-\frac {4 i A +4 B}{8 \left (i+\tan \left (f x +e \right )\right )^{8}}-\frac {8 i B -4 A}{7 \left (i+\tan \left (f x +e \right )\right )^{7}}\right )}{f \,c^{8}}\) | \(90\) |
default | \(\frac {a^{3} \left (\frac {i B}{5 \left (i+\tan \left (f x +e \right )\right )^{5}}-\frac {-i A -5 B}{6 \left (i+\tan \left (f x +e \right )\right )^{6}}-\frac {4 i A +4 B}{8 \left (i+\tan \left (f x +e \right )\right )^{8}}-\frac {8 i B -4 A}{7 \left (i+\tan \left (f x +e \right )\right )^{7}}\right )}{f \,c^{8}}\) | \(90\) |
risch | \(-\frac {a^{3} {\mathrm e}^{16 i \left (f x +e \right )} B}{512 c^{8} f}-\frac {i a^{3} {\mathrm e}^{16 i \left (f x +e \right )} A}{512 c^{8} f}-\frac {3 \,{\mathrm e}^{14 i \left (f x +e \right )} B \,a^{3}}{448 c^{8} f}-\frac {5 i {\mathrm e}^{14 i \left (f x +e \right )} A \,a^{3}}{448 c^{8} f}-\frac {{\mathrm e}^{12 i \left (f x +e \right )} B \,a^{3}}{192 c^{8} f}-\frac {5 i {\mathrm e}^{12 i \left (f x +e \right )} A \,a^{3}}{192 c^{8} f}+\frac {{\mathrm e}^{10 i \left (f x +e \right )} B \,a^{3}}{160 c^{8} f}-\frac {i {\mathrm e}^{10 i \left (f x +e \right )} A \,a^{3}}{32 c^{8} f}+\frac {3 \,{\mathrm e}^{8 i \left (f x +e \right )} B \,a^{3}}{256 c^{8} f}-\frac {5 i {\mathrm e}^{8 i \left (f x +e \right )} A \,a^{3}}{256 c^{8} f}+\frac {a^{3} {\mathrm e}^{6 i \left (f x +e \right )} B}{192 c^{8} f}-\frac {i a^{3} {\mathrm e}^{6 i \left (f x +e \right )} A}{192 c^{8} f}\) | \(260\) |
Input:
int((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^8,x,method=_R ETURNVERBOSE)
Output:
1/f*a^3/c^8*(1/5*I*B/(I+tan(f*x+e))^5-1/6*(-I*A-5*B)/(I+tan(f*x+e))^6-1/8* (4*I*A+4*B)/(I+tan(f*x+e))^8-1/7*(8*I*B-4*A)/(I+tan(f*x+e))^7)
Time = 0.08 (sec) , antiderivative size = 131, normalized size of antiderivative = 1.03 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx=-\frac {105 \, {\left (i \, A + B\right )} a^{3} e^{\left (16 i \, f x + 16 i \, e\right )} + 120 \, {\left (5 i \, A + 3 \, B\right )} a^{3} e^{\left (14 i \, f x + 14 i \, e\right )} + 280 \, {\left (5 i \, A + B\right )} a^{3} e^{\left (12 i \, f x + 12 i \, e\right )} + 336 \, {\left (5 i \, A - B\right )} a^{3} e^{\left (10 i \, f x + 10 i \, e\right )} + 210 \, {\left (5 i \, A - 3 \, B\right )} a^{3} e^{\left (8 i \, f x + 8 i \, e\right )} + 280 \, {\left (i \, A - B\right )} a^{3} e^{\left (6 i \, f x + 6 i \, e\right )}}{53760 \, c^{8} f} \] Input:
integrate((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^8,x, al gorithm="fricas")
Output:
-1/53760*(105*(I*A + B)*a^3*e^(16*I*f*x + 16*I*e) + 120*(5*I*A + 3*B)*a^3* e^(14*I*f*x + 14*I*e) + 280*(5*I*A + B)*a^3*e^(12*I*f*x + 12*I*e) + 336*(5 *I*A - B)*a^3*e^(10*I*f*x + 10*I*e) + 210*(5*I*A - 3*B)*a^3*e^(8*I*f*x + 8 *I*e) + 280*(I*A - B)*a^3*e^(6*I*f*x + 6*I*e))/(c^8*f)
Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 496 vs. \(2 (104) = 208\).
Time = 0.72 (sec) , antiderivative size = 496, normalized size of antiderivative = 3.91 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx=\begin {cases} \frac {\left (- 1803886264320 i A a^{3} c^{40} f^{5} e^{6 i e} + 1803886264320 B a^{3} c^{40} f^{5} e^{6 i e}\right ) e^{6 i f x} + \left (- 6764573491200 i A a^{3} c^{40} f^{5} e^{8 i e} + 4058744094720 B a^{3} c^{40} f^{5} e^{8 i e}\right ) e^{8 i f x} + \left (- 10823317585920 i A a^{3} c^{40} f^{5} e^{10 i e} + 2164663517184 B a^{3} c^{40} f^{5} e^{10 i e}\right ) e^{10 i f x} + \left (- 9019431321600 i A a^{3} c^{40} f^{5} e^{12 i e} - 1803886264320 B a^{3} c^{40} f^{5} e^{12 i e}\right ) e^{12 i f x} + \left (- 3865470566400 i A a^{3} c^{40} f^{5} e^{14 i e} - 2319282339840 B a^{3} c^{40} f^{5} e^{14 i e}\right ) e^{14 i f x} + \left (- 676457349120 i A a^{3} c^{40} f^{5} e^{16 i e} - 676457349120 B a^{3} c^{40} f^{5} e^{16 i e}\right ) e^{16 i f x}}{346346162749440 c^{48} f^{6}} & \text {for}\: c^{48} f^{6} \neq 0 \\\frac {x \left (A a^{3} e^{16 i e} + 5 A a^{3} e^{14 i e} + 10 A a^{3} e^{12 i e} + 10 A a^{3} e^{10 i e} + 5 A a^{3} e^{8 i e} + A a^{3} e^{6 i e} - i B a^{3} e^{16 i e} - 3 i B a^{3} e^{14 i e} - 2 i B a^{3} e^{12 i e} + 2 i B a^{3} e^{10 i e} + 3 i B a^{3} e^{8 i e} + i B a^{3} e^{6 i e}\right )}{32 c^{8}} & \text {otherwise} \end {cases} \] Input:
integrate((a+I*a*tan(f*x+e))**3*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))**8,x)
Output:
Piecewise((((-1803886264320*I*A*a**3*c**40*f**5*exp(6*I*e) + 1803886264320 *B*a**3*c**40*f**5*exp(6*I*e))*exp(6*I*f*x) + (-6764573491200*I*A*a**3*c** 40*f**5*exp(8*I*e) + 4058744094720*B*a**3*c**40*f**5*exp(8*I*e))*exp(8*I*f *x) + (-10823317585920*I*A*a**3*c**40*f**5*exp(10*I*e) + 2164663517184*B*a **3*c**40*f**5*exp(10*I*e))*exp(10*I*f*x) + (-9019431321600*I*A*a**3*c**40 *f**5*exp(12*I*e) - 1803886264320*B*a**3*c**40*f**5*exp(12*I*e))*exp(12*I* f*x) + (-3865470566400*I*A*a**3*c**40*f**5*exp(14*I*e) - 2319282339840*B*a **3*c**40*f**5*exp(14*I*e))*exp(14*I*f*x) + (-676457349120*I*A*a**3*c**40* f**5*exp(16*I*e) - 676457349120*B*a**3*c**40*f**5*exp(16*I*e))*exp(16*I*f* x))/(346346162749440*c**48*f**6), Ne(c**48*f**6, 0)), (x*(A*a**3*exp(16*I* e) + 5*A*a**3*exp(14*I*e) + 10*A*a**3*exp(12*I*e) + 10*A*a**3*exp(10*I*e) + 5*A*a**3*exp(8*I*e) + A*a**3*exp(6*I*e) - I*B*a**3*exp(16*I*e) - 3*I*B*a **3*exp(14*I*e) - 2*I*B*a**3*exp(12*I*e) + 2*I*B*a**3*exp(10*I*e) + 3*I*B* a**3*exp(8*I*e) + I*B*a**3*exp(6*I*e))/(32*c**8), True))
Exception generated. \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx=\text {Exception raised: RuntimeError} \] Input:
integrate((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^8,x, al gorithm="maxima")
Output:
Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negati ve exponent.
Time = 0.71 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.76 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx=-\frac {-42 i \, B a^{3} \tan \left (f x + e\right )^{3} - 35 i \, A a^{3} \tan \left (f x + e\right )^{2} - 49 \, B a^{3} \tan \left (f x + e\right )^{2} - 50 \, A a^{3} \tan \left (f x + e\right ) + 16 i \, B a^{3} \tan \left (f x + e\right ) + 20 i \, A a^{3} - 2 \, B a^{3}}{210 \, c^{8} f {\left (\tan \left (f x + e\right ) + i\right )}^{8}} \] Input:
integrate((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^8,x, al gorithm="giac")
Output:
-1/210*(-42*I*B*a^3*tan(f*x + e)^3 - 35*I*A*a^3*tan(f*x + e)^2 - 49*B*a^3* tan(f*x + e)^2 - 50*A*a^3*tan(f*x + e) + 16*I*B*a^3*tan(f*x + e) + 20*I*A* a^3 - 2*B*a^3)/(c^8*f*(tan(f*x + e) + I)^8)
Time = 6.00 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.26 \[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx=\frac {-\frac {a^3\,\left (-2\,B+A\,20{}\mathrm {i}\right )}{210}+\frac {a^3\,\mathrm {tan}\left (e+f\,x\right )\,\left (50\,A-B\,16{}\mathrm {i}\right )}{210}+\frac {B\,a^3\,{\mathrm {tan}\left (e+f\,x\right )}^3\,1{}\mathrm {i}}{5}+\frac {a^3\,{\mathrm {tan}\left (e+f\,x\right )}^2\,\left (49\,B+A\,35{}\mathrm {i}\right )}{210}}{c^8\,f\,\left ({\mathrm {tan}\left (e+f\,x\right )}^8+{\mathrm {tan}\left (e+f\,x\right )}^7\,8{}\mathrm {i}-28\,{\mathrm {tan}\left (e+f\,x\right )}^6-{\mathrm {tan}\left (e+f\,x\right )}^5\,56{}\mathrm {i}+70\,{\mathrm {tan}\left (e+f\,x\right )}^4+{\mathrm {tan}\left (e+f\,x\right )}^3\,56{}\mathrm {i}-28\,{\mathrm {tan}\left (e+f\,x\right )}^2-\mathrm {tan}\left (e+f\,x\right )\,8{}\mathrm {i}+1\right )} \] Input:
int(((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^3)/(c - c*tan(e + f*x)*1 i)^8,x)
Output:
((a^3*tan(e + f*x)*(50*A - B*16i))/210 - (a^3*(A*20i - 2*B))/210 + (B*a^3* tan(e + f*x)^3*1i)/5 + (a^3*tan(e + f*x)^2*(A*35i + 49*B))/210)/(c^8*f*(ta n(e + f*x)^3*56i - 28*tan(e + f*x)^2 - tan(e + f*x)*8i + 70*tan(e + f*x)^4 - tan(e + f*x)^5*56i - 28*tan(e + f*x)^6 + tan(e + f*x)^7*8i + tan(e + f* x)^8 + 1))
\[ \int \frac {(a+i a \tan (e+f x))^3 (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^8} \, dx =\text {Too large to display} \] Input:
int((a+I*a*tan(f*x+e))^3*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^8,x)
Output:
(a**3*(int(tan(e + f*x)**4/(tan(e + f*x)**8*i - 8*tan(e + f*x)**7 - 28*tan (e + f*x)**6*i + 56*tan(e + f*x)**5 + 70*tan(e + f*x)**4*i - 56*tan(e + f* x)**3 - 28*tan(e + f*x)**2*i + 8*tan(e + f*x) + i),x)*b + int(tan(e + f*x) **3/(tan(e + f*x)**8*i - 8*tan(e + f*x)**7 - 28*tan(e + f*x)**6*i + 56*tan (e + f*x)**5 + 70*tan(e + f*x)**4*i - 56*tan(e + f*x)**3 - 28*tan(e + f*x) **2*i + 8*tan(e + f*x) + i),x)*a - 3*int(tan(e + f*x)**3/(tan(e + f*x)**8* i - 8*tan(e + f*x)**7 - 28*tan(e + f*x)**6*i + 56*tan(e + f*x)**5 + 70*tan (e + f*x)**4*i - 56*tan(e + f*x)**3 - 28*tan(e + f*x)**2*i + 8*tan(e + f*x ) + i),x)*b*i - 3*int(tan(e + f*x)**2/(tan(e + f*x)**8*i - 8*tan(e + f*x)* *7 - 28*tan(e + f*x)**6*i + 56*tan(e + f*x)**5 + 70*tan(e + f*x)**4*i - 56 *tan(e + f*x)**3 - 28*tan(e + f*x)**2*i + 8*tan(e + f*x) + i),x)*a*i - 3*i nt(tan(e + f*x)**2/(tan(e + f*x)**8*i - 8*tan(e + f*x)**7 - 28*tan(e + f*x )**6*i + 56*tan(e + f*x)**5 + 70*tan(e + f*x)**4*i - 56*tan(e + f*x)**3 - 28*tan(e + f*x)**2*i + 8*tan(e + f*x) + i),x)*b - 3*int(tan(e + f*x)/(tan( e + f*x)**8*i - 8*tan(e + f*x)**7 - 28*tan(e + f*x)**6*i + 56*tan(e + f*x) **5 + 70*tan(e + f*x)**4*i - 56*tan(e + f*x)**3 - 28*tan(e + f*x)**2*i + 8 *tan(e + f*x) + i),x)*a + int(tan(e + f*x)/(tan(e + f*x)**8*i - 8*tan(e + f*x)**7 - 28*tan(e + f*x)**6*i + 56*tan(e + f*x)**5 + 70*tan(e + f*x)**4*i - 56*tan(e + f*x)**3 - 28*tan(e + f*x)**2*i + 8*tan(e + f*x) + i),x)*b*i + int(1/(tan(e + f*x)**8*i - 8*tan(e + f*x)**7 - 28*tan(e + f*x)**6*i +...