Integrand size = 35, antiderivative size = 182 \[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{9/2}} \, dx=\frac {i (c-i c \tan (e+f x))^{3/2}}{9 f (a+i a \tan (e+f x))^{9/2}}+\frac {i (c-i c \tan (e+f x))^{3/2}}{21 a f (a+i a \tan (e+f x))^{7/2}}+\frac {2 i (c-i c \tan (e+f x))^{3/2}}{105 a^2 f (a+i a \tan (e+f x))^{5/2}}+\frac {2 i (c-i c \tan (e+f x))^{3/2}}{315 a^3 f (a+i a \tan (e+f x))^{3/2}} \] Output:
1/9*I*(c-I*c*tan(f*x+e))^(3/2)/f/(a+I*a*tan(f*x+e))^(9/2)+1/21*I*(c-I*c*ta n(f*x+e))^(3/2)/a/f/(a+I*a*tan(f*x+e))^(7/2)+2/105*I*(c-I*c*tan(f*x+e))^(3 /2)/a^2/f/(a+I*a*tan(f*x+e))^(5/2)+2/315*I*(c-I*c*tan(f*x+e))^(3/2)/a^3/f/ (a+I*a*tan(f*x+e))^(3/2)
Time = 2.11 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.57 \[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{9/2}} \, dx=\frac {c (1-i \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \left (58 i-33 \tan (e+f x)-12 i \tan ^2(e+f x)+2 \tan ^3(e+f x)\right )}{315 a^4 f (-i+\tan (e+f x))^4 \sqrt {a+i a \tan (e+f x)}} \] Input:
Integrate[(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(9/2),x]
Output:
(c*(1 - I*Tan[e + f*x])*Sqrt[c - I*c*Tan[e + f*x]]*(58*I - 33*Tan[e + f*x] - (12*I)*Tan[e + f*x]^2 + 2*Tan[e + f*x]^3))/(315*a^4*f*(-I + Tan[e + f*x ])^4*Sqrt[a + I*a*Tan[e + f*x]])
Time = 0.35 (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.
| \(\displaystyle \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{9/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{9/2}}dx\) |
| \(\Big \downarrow \) 4006 |
| \(\displaystyle \frac {a c \int \frac {\sqrt {c-i c \tan (e+f x)}}{(i \tan (e+f x) a+a)^{11/2}}d\tan (e+f x)}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {\int \frac {\sqrt {c-i c \tan (e+f x)}}{(i \tan (e+f x) a+a)^{9/2}}d\tan (e+f x)}{3 a}+\frac {i (c-i c \tan (e+f x))^{3/2}}{9 a c (a+i a \tan (e+f x))^{9/2}}\right )}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {\frac {2 \int \frac {\sqrt {c-i c \tan (e+f x)}}{(i \tan (e+f x) a+a)^{7/2}}d\tan (e+f x)}{7 a}+\frac {i (c-i c \tan (e+f x))^{3/2}}{7 a c (a+i a \tan (e+f x))^{7/2}}}{3 a}+\frac {i (c-i c \tan (e+f x))^{3/2}}{9 a c (a+i a \tan (e+f x))^{9/2}}\right )}{f}\) |
| \(\Big \downarrow \) 55 |
| \(\displaystyle \frac {a c \left (\frac {\frac {2 \left (\frac {\int \frac {\sqrt {c-i c \tan (e+f x)}}{(i \tan (e+f x) a+a)^{5/2}}d\tan (e+f x)}{5 a}+\frac {i (c-i c \tan (e+f x))^{3/2}}{5 a c (a+i a \tan (e+f x))^{5/2}}\right )}{7 a}+\frac {i (c-i c \tan (e+f x))^{3/2}}{7 a c (a+i a \tan (e+f x))^{7/2}}}{3 a}+\frac {i (c-i c \tan (e+f x))^{3/2}}{9 a c (a+i a \tan (e+f x))^{9/2}}\right )}{f}\) |
| \(\Big \downarrow \) 48 |
| \(\displaystyle \frac {a c \left (\frac {\frac {2 \left (\frac {i (c-i c \tan (e+f x))^{3/2}}{15 a^2 c (a+i a \tan (e+f x))^{3/2}}+\frac {i (c-i c \tan (e+f x))^{3/2}}{5 a c (a+i a \tan (e+f x))^{5/2}}\right )}{7 a}+\frac {i (c-i c \tan (e+f x))^{3/2}}{7 a c (a+i a \tan (e+f x))^{7/2}}}{3 a}+\frac {i (c-i c \tan (e+f x))^{3/2}}{9 a c (a+i a \tan (e+f x))^{9/2}}\right )}{f}\) |
Input:
Int[(c - I*c*Tan[e + f*x])^(3/2)/(a + I*a*Tan[e + f*x])^(9/2),x]
Output:
(a*c*(((I/9)*(c - I*c*Tan[e + f*x])^(3/2))/(a*c*(a + I*a*Tan[e + f*x])^(9/ 2)) + (((I/7)*(c - I*c*Tan[e + f*x])^(3/2))/(a*c*(a + I*a*Tan[e + f*x])^(7 /2)) + (2*(((I/5)*(c - I*c*Tan[e + f*x])^(3/2))/(a*c*(a + I*a*Tan[e + f*x] )^(5/2)) + ((I/15)*(c - I*c*Tan[e + f*x])^(3/2))/(a^2*c*(a + I*a*Tan[e + f *x])^(3/2))))/(7*a))/(3*a)))/f
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 = 0.35 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.53
| method | result | size |
| derivativedivides | \(\frac {i \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c \left (1+\tan \left (f x +e \right )^{2}\right ) \left (2 i \tan \left (f x +e \right )^{3}-33 i \tan \left (f x +e \right )+12 \tan \left (f x +e \right )^{2}-58\right )}{315 f \,a^{5} \left (-\tan \left (f x +e \right )+i\right )^{6}}\) | \(97\) |
| default | \(\frac {i \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, c \left (1+\tan \left (f x +e \right )^{2}\right ) \left (2 i \tan \left (f x +e \right )^{3}-33 i \tan \left (f x +e \right )+12 \tan \left (f x +e \right )^{2}-58\right )}{315 f \,a^{5} \left (-\tan \left (f x +e \right )+i\right )^{6}}\) | \(97\) |
Input:
int((c-I*c*tan(f*x+e))^(3/2)/(a+I*a*tan(f*x+e))^(9/2),x,method=_RETURNVERB OSE)
Output:
1/315*I/f*(-c*(I*tan(f*x+e)-1))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)/a^5*c*(1+ tan(f*x+e)^2)*(2*I*tan(f*x+e)^3-33*I*tan(f*x+e)+12*tan(f*x+e)^2-58)/(-tan( f*x+e)+I)^6
Time = 0.08 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.57 \[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{9/2}} \, dx=\frac {{\left (105 i \, c e^{\left (8 i \, f x + 8 i \, e\right )} + 294 i \, c e^{\left (6 i \, f x + 6 i \, e\right )} + 324 i \, c e^{\left (4 i \, f x + 4 i \, e\right )} + 170 i \, c e^{\left (2 i \, f x + 2 i \, e\right )} + 35 i \, c\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} e^{\left (-9 i \, f x - 9 i \, e\right )}}{2520 \, a^{5} f} \] Input:
integrate((c-I*c*tan(f*x+e))^(3/2)/(a+I*a*tan(f*x+e))^(9/2),x, algorithm=" fricas")
Output:
1/2520*(105*I*c*e^(8*I*f*x + 8*I*e) + 294*I*c*e^(6*I*f*x + 6*I*e) + 324*I* c*e^(4*I*f*x + 4*I*e) + 170*I*c*e^(2*I*f*x + 2*I*e) + 35*I*c)*sqrt(a/(e^(2 *I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))*e^(-9*I*f*x - 9*I* e)/(a^5*f)
Timed out. \[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{9/2}} \, dx=\text {Timed out} \] Input:
integrate((c-I*c*tan(f*x+e))**(3/2)/(a+I*a*tan(f*x+e))**(9/2),x)
Output:
Timed out
Time = 0.20 (sec) , antiderivative size = 186, normalized size of antiderivative = 1.02 \[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{9/2}} \, dx=\frac {{\left (35 i \, c \cos \left (9 \, f x + 9 \, e\right ) + 135 i \, c \cos \left (\frac {7}{9} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 189 i \, c \cos \left (\frac {5}{9} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 105 i \, c \cos \left (\frac {1}{3} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 35 \, c \sin \left (9 \, f x + 9 \, e\right ) + 135 \, c \sin \left (\frac {7}{9} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 189 \, c \sin \left (\frac {5}{9} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right ) + 105 \, c \sin \left (\frac {1}{3} \, \arctan \left (\sin \left (9 \, f x + 9 \, e\right ), \cos \left (9 \, f x + 9 \, e\right )\right )\right )\right )} \sqrt {c}}{2520 \, a^{\frac {9}{2}} f} \] Input:
integrate((c-I*c*tan(f*x+e))^(3/2)/(a+I*a*tan(f*x+e))^(9/2),x, algorithm=" maxima")
Output:
1/2520*(35*I*c*cos(9*f*x + 9*e) + 135*I*c*cos(7/9*arctan2(sin(9*f*x + 9*e) , cos(9*f*x + 9*e))) + 189*I*c*cos(5/9*arctan2(sin(9*f*x + 9*e), cos(9*f*x + 9*e))) + 105*I*c*cos(1/3*arctan2(sin(9*f*x + 9*e), cos(9*f*x + 9*e))) + 35*c*sin(9*f*x + 9*e) + 135*c*sin(7/9*arctan2(sin(9*f*x + 9*e), cos(9*f*x + 9*e))) + 189*c*sin(5/9*arctan2(sin(9*f*x + 9*e), cos(9*f*x + 9*e))) + 1 05*c*sin(1/3*arctan2(sin(9*f*x + 9*e), cos(9*f*x + 9*e))))*sqrt(c)/(a^(9/2 )*f)
\[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{9/2}} \, dx=\int { \frac {{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {3}{2}}}{{\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {9}{2}}} \,d x } \] Input:
integrate((c-I*c*tan(f*x+e))^(3/2)/(a+I*a*tan(f*x+e))^(9/2),x, algorithm=" giac")
Output:
integrate((-I*c*tan(f*x + e) + c)^(3/2)/(I*a*tan(f*x + e) + a)^(9/2), x)
Time = 4.38 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.13 \[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{9/2}} \, dx=\frac {c\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (105\,\sin \left (2\,e+2\,f\,x\right )+294\,\sin \left (4\,e+4\,f\,x\right )+324\,\sin \left (6\,e+6\,f\,x\right )+170\,\sin \left (8\,e+8\,f\,x\right )+35\,\sin \left (10\,e+10\,f\,x\right )+\cos \left (2\,e+2\,f\,x\right )\,105{}\mathrm {i}+\cos \left (4\,e+4\,f\,x\right )\,294{}\mathrm {i}+\cos \left (6\,e+6\,f\,x\right )\,324{}\mathrm {i}+\cos \left (8\,e+8\,f\,x\right )\,170{}\mathrm {i}+\cos \left (10\,e+10\,f\,x\right )\,35{}\mathrm {i}\right )}{5040\,a^5\,f} \] Input:
int((c - c*tan(e + f*x)*1i)^(3/2)/(a + a*tan(e + f*x)*1i)^(9/2),x)
Output:
(c*((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(2*e + 2*f*x)*105i + cos(4*e + 4*f*x)*294i + cos(6*e + 6*f*x)*324i + cos(8*e + 8*f*x)*170i + cos(10*e + 10*f*x)*35i + 105*sin(2*e + 2*f*x) + 294*sin(4*e + 4*f*x) + 324*sin(6*e + 6*f*x) + 170*sin(8*e + 8* f*x) + 35*sin(10*e + 10*f*x)))/(5040*a^5*f)
\[ \int \frac {(c-i c \tan (e+f x))^{3/2}}{(a+i a \tan (e+f x))^{9/2}} \, dx=\frac {\sqrt {c}\, \sqrt {a}\, c \left (2 \left (\int -\frac {\sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {-\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )}{\tan \left (f x +e \right )^{6}-4 \tan \left (f x +e \right )^{5} i -5 \tan \left (f x +e \right )^{4}-5 \tan \left (f x +e \right )^{2}+4 \tan \left (f x +e \right ) i +1}d x \right ) i -\left (\int -\frac {\sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {-\tan \left (f x +e \right ) i +1}}{\tan \left (f x +e \right )^{6}-4 \tan \left (f x +e \right )^{5} i -5 \tan \left (f x +e \right )^{4}-5 \tan \left (f x +e \right )^{2}+4 \tan \left (f x +e \right ) i +1}d x \right )-\left (\int \frac {\sqrt {\tan \left (f x +e \right ) i +1}\, \sqrt {-\tan \left (f x +e \right ) i +1}\, \tan \left (f x +e \right )^{2}}{\tan \left (f x +e \right )^{6}-4 \tan \left (f x +e \right )^{5} i -5 \tan \left (f x +e \right )^{4}-5 \tan \left (f x +e \right )^{2}+4 \tan \left (f x +e \right ) i +1}d x \right )\right )}{a^{5}} \] Input:
int((c-I*c*tan(f*x+e))^(3/2)/(a+I*a*tan(f*x+e))^(9/2),x)
Output:
(sqrt(c)*sqrt(a)*c*(2*int(( - sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x )*i + 1)*tan(e + f*x))/(tan(e + f*x)**6 - 4*tan(e + f*x)**5*i - 5*tan(e + f*x)**4 - 5*tan(e + f*x)**2 + 4*tan(e + f*x)*i + 1),x)*i - int(( - sqrt(ta n(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1))/(tan(e + f*x)**6 - 4*tan(e + f*x)**5*i - 5*tan(e + f*x)**4 - 5*tan(e + f*x)**2 + 4*tan(e + f*x)*i + 1 ),x) - int((sqrt(tan(e + f*x)*i + 1)*sqrt( - tan(e + f*x)*i + 1)*tan(e + f *x)**2)/(tan(e + f*x)**6 - 4*tan(e + f*x)**5*i - 5*tan(e + f*x)**4 - 5*tan (e + f*x)**2 + 4*tan(e + f*x)*i + 1),x)))/a**5