\(\int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx\) [804]

Optimal result

Integrand size = 45, antiderivative size = 261 \[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=-\frac {(i A+B) (a+i a \tan (e+f x))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{231 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac {2 (4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{1155 c^3 f (c-i c \tan (e+f x))^{5/2}}-\frac {2 (4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{3465 c^4 f (c-i c \tan (e+f x))^{3/2}} \]

[Out]

-1/11*(I*A+B)*(a+I*a*tan(f*x+e))^(3/2)/f/(c-I*c*tan(f*x+e))^(11/2)-1/99*(4*I*A-7*B)*(a+I*a*tan(f*x+e))^(3/2)/c
/f/(c-I*c*tan(f*x+e))^(9/2)-1/231*(4*I*A-7*B)*(a+I*a*tan(f*x+e))^(3/2)/c^2/f/(c-I*c*tan(f*x+e))^(7/2)-2/1155*(
4*I*A-7*B)*(a+I*a*tan(f*x+e))^(3/2)/c^3/f/(c-I*c*tan(f*x+e))^(5/2)-2/3465*(4*I*A-7*B)*(a+I*a*tan(f*x+e))^(3/2)
/c^4/f/(c-I*c*tan(f*x+e))^(3/2)

Rubi [A] (verified)

Time = 0.35 (sec) , antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.089, Rules used = {3669, 79, 47, 37} \[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=-\frac {2 (-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{3465 c^4 f (c-i c \tan (e+f x))^{3/2}}-\frac {2 (-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{1155 c^3 f (c-i c \tan (e+f x))^{5/2}}-\frac {(-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{231 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac {(-7 B+4 i A) (a+i a \tan (e+f x))^{3/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}} \]

[In]

Int[((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(11/2),x]

[Out]

-1/11*((I*A + B)*(a + I*a*Tan[e + f*x])^(3/2))/(f*(c - I*c*Tan[e + f*x])^(11/2)) - (((4*I)*A - 7*B)*(a + I*a*T
an[e + f*x])^(3/2))/(99*c*f*(c - I*c*Tan[e + f*x])^(9/2)) - (((4*I)*A - 7*B)*(a + I*a*Tan[e + f*x])^(3/2))/(23
1*c^2*f*(c - I*c*Tan[e + f*x])^(7/2)) - (2*((4*I)*A - 7*B)*(a + I*a*Tan[e + f*x])^(3/2))/(1155*c^3*f*(c - I*c*
Tan[e + f*x])^(5/2)) - (2*((4*I)*A - 7*B)*(a + I*a*Tan[e + f*x])^(3/2))/(3465*c^4*f*(c - I*c*Tan[e + f*x])^(3/
2))

Rule 37

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] && NeQ[b*c - a*d, 0] && EqQ[m + n + 2, 0] && NeQ
[m, -1]

Rule 47

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] - Dist[d*(Simplify[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] && NeQ[b*c - a*d, 0] && 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] ||  !SumSimplerQ[n, 1])

Rule 79

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[(-(b*e - a*f
))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Dist[(a*d*f*(n + p + 2) - b*(d*e*(n + 1
) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)), Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e,
f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || IntegerQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || L
tQ[p, n]))))

Rule 3669

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[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]

Rubi steps \begin{align*} \text {integral}& = \frac {(a c) \text {Subst}\left (\int \frac {\sqrt {a+i a x} (A+B x)}{(c-i c x)^{13/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}}+\frac {(a (4 A+7 i B)) \text {Subst}\left (\int \frac {\sqrt {a+i a x}}{(c-i c x)^{11/2}} \, dx,x,\tan (e+f x)\right )}{11 f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}+\frac {(a (4 A+7 i B)) \text {Subst}\left (\int \frac {\sqrt {a+i a x}}{(c-i c x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{33 c f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{231 c^2 f (c-i c \tan (e+f x))^{7/2}}+\frac {(2 a (4 A+7 i B)) \text {Subst}\left (\int \frac {\sqrt {a+i a x}}{(c-i c x)^{7/2}} \, dx,x,\tan (e+f x)\right )}{231 c^2 f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{231 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac {2 (4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{1155 c^3 f (c-i c \tan (e+f x))^{5/2}}+\frac {(2 a (4 A+7 i B)) \text {Subst}\left (\int \frac {\sqrt {a+i a x}}{(c-i c x)^{5/2}} \, dx,x,\tan (e+f x)\right )}{1155 c^3 f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{3/2}}{11 f (c-i c \tan (e+f x))^{11/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{99 c f (c-i c \tan (e+f x))^{9/2}}-\frac {(4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{231 c^2 f (c-i c \tan (e+f x))^{7/2}}-\frac {2 (4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{1155 c^3 f (c-i c \tan (e+f x))^{5/2}}-\frac {2 (4 i A-7 B) (a+i a \tan (e+f x))^{3/2}}{3465 c^4 f (c-i c \tan (e+f x))^{3/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 7.70 (sec) , antiderivative size = 153, normalized size of antiderivative = 0.59 \[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=-\frac {a^2 (-i+\tan (e+f x))^2 \left (547 A+91 i B+91 (-4 i A+7 B) \tan (e+f x)-45 (4 A+7 i B) \tan ^2(e+f x)+(56 i A-98 B) \tan ^3(e+f x)+2 (4 A+7 i B) \tan ^4(e+f x)\right )}{3465 c^5 f (i+\tan (e+f x))^5 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]

[In]

Integrate[((a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(11/2),x]

[Out]

-1/3465*(a^2*(-I + Tan[e + f*x])^2*(547*A + (91*I)*B + 91*((-4*I)*A + 7*B)*Tan[e + f*x] - 45*(4*A + (7*I)*B)*T
an[e + f*x]^2 + ((56*I)*A - 98*B)*Tan[e + f*x]^3 + 2*(4*A + (7*I)*B)*Tan[e + f*x]^4))/(c^5*f*(I + Tan[e + f*x]
)^5*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])

Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 158, normalized size of antiderivative = 0.61

method	result	size
derivativedivides	\(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a \left (1+\tan \left (f x +e \right )^{2}\right ) \left (14 i B \tan \left (f x +e \right )^{4}+56 i A \tan \left (f x +e \right )^{3}+8 A \tan \left (f x +e \right )^{4}-315 i B \tan \left (f x +e \right )^{2}-98 B \tan \left (f x +e \right )^{3}-364 i A \tan \left (f x +e \right )-180 A \tan \left (f x +e \right )^{2}+91 i B +637 B \tan \left (f x +e \right )+547 A \right )}{3465 f \,c^{6} \left (i+\tan \left (f x +e \right )\right )^{7}}\)	\(158\)
default	\(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a \left (1+\tan \left (f x +e \right )^{2}\right ) \left (14 i B \tan \left (f x +e \right )^{4}+56 i A \tan \left (f x +e \right )^{3}+8 A \tan \left (f x +e \right )^{4}-315 i B \tan \left (f x +e \right )^{2}-98 B \tan \left (f x +e \right )^{3}-364 i A \tan \left (f x +e \right )-180 A \tan \left (f x +e \right )^{2}+91 i B +637 B \tan \left (f x +e \right )+547 A \right )}{3465 f \,c^{6} \left (i+\tan \left (f x +e \right )\right )^{7}}\)	\(158\)
risch	\(-\frac {a \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (315 i A \,{\mathrm e}^{10 i \left (f x +e \right )}+315 B \,{\mathrm e}^{10 i \left (f x +e \right )}+1540 i A \,{\mathrm e}^{8 i \left (f x +e \right )}+770 B \,{\mathrm e}^{8 i \left (f x +e \right )}+2970 i A \,{\mathrm e}^{6 i \left (f x +e \right )}+2772 i A \,{\mathrm e}^{4 i \left (f x +e \right )}-1386 B \,{\mathrm e}^{4 i \left (f x +e \right )}+1155 i A \,{\mathrm e}^{2 i \left (f x +e \right )}-1155 B \,{\mathrm e}^{2 i \left (f x +e \right )}\right )}{55440 c^{5} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\)	\(167\)
parts	\(-\frac {A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a \left (1+\tan \left (f x +e \right )^{2}\right ) \left (56 i \tan \left (f x +e \right )^{3}+8 \tan \left (f x +e \right )^{4}-364 i \tan \left (f x +e \right )-180 \tan \left (f x +e \right )^{2}+547\right )}{3465 f \,c^{6} \left (i+\tan \left (f x +e \right )\right )^{7}}-\frac {i B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a \left (1+\tan \left (f x +e \right )^{2}\right ) \left (14 i \tan \left (f x +e \right )^{3}+2 \tan \left (f x +e \right )^{4}-91 i \tan \left (f x +e \right )-45 \tan \left (f x +e \right )^{2}+13\right )}{495 f \,c^{6} \left (i+\tan \left (f x +e \right )\right )^{7}}\)	\(211\)

[In]

int((a+I*a*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(11/2),x,method=_RETURNVERBOSE)

[Out]

-1/3465/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)*a/c^6*(1+tan(f*x+e)^2)*(14*I*B*tan(f*x+e)^4+5
6*I*A*tan(f*x+e)^3+8*A*tan(f*x+e)^4-315*I*B*tan(f*x+e)^2-98*B*tan(f*x+e)^3-364*I*A*tan(f*x+e)-180*A*tan(f*x+e)
^2+91*I*B+637*B*tan(f*x+e)+547*A)/(I+tan(f*x+e))^7

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.59 \[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=-\frac {{\left (315 \, {\left (i \, A + B\right )} a e^{\left (13 i \, f x + 13 i \, e\right )} + 35 \, {\left (53 i \, A + 31 \, B\right )} a e^{\left (11 i \, f x + 11 i \, e\right )} + 110 \, {\left (41 i \, A + 7 \, B\right )} a e^{\left (9 i \, f x + 9 i \, e\right )} + 198 \, {\left (29 i \, A - 7 \, B\right )} a e^{\left (7 i \, f x + 7 i \, e\right )} + 231 \, {\left (17 i \, A - 11 \, B\right )} a e^{\left (5 i \, f x + 5 i \, e\right )} + 1155 \, {\left (i \, A - B\right )} a e^{\left (3 i \, f x + 3 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{55440 \, c^{6} f} \]

[In]

integrate((a+I*a*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(11/2),x, algorithm="fricas")

[Out]

-1/55440*(315*(I*A + B)*a*e^(13*I*f*x + 13*I*e) + 35*(53*I*A + 31*B)*a*e^(11*I*f*x + 11*I*e) + 110*(41*I*A + 7
*B)*a*e^(9*I*f*x + 9*I*e) + 198*(29*I*A - 7*B)*a*e^(7*I*f*x + 7*I*e) + 231*(17*I*A - 11*B)*a*e^(5*I*f*x + 5*I*
e) + 1155*(I*A - B)*a*e^(3*I*f*x + 3*I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1))
/(c^6*f)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+I*a*tan(f*x+e))**(3/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))**(11/2),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.43 (sec) , antiderivative size = 312, normalized size of antiderivative = 1.20 \[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=\frac {{\left (315 \, {\left (-i \, A - B\right )} a \cos \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 770 \, {\left (-2 i \, A - B\right )} a \cos \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 2970 i \, A a \cos \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1386 \, {\left (-2 i \, A + B\right )} a \cos \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1155 \, {\left (-i \, A + B\right )} a \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 315 \, {\left (A - i \, B\right )} a \sin \left (\frac {11}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 770 \, {\left (2 \, A - i \, B\right )} a \sin \left (\frac {9}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 2970 \, A a \sin \left (\frac {7}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1386 \, {\left (2 \, A + i \, B\right )} a \sin \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1155 \, {\left (A + i \, B\right )} a \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )\right )} \sqrt {a}}{55440 \, c^{\frac {11}{2}} f} \]

[In]

integrate((a+I*a*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(11/2),x, algorithm="maxima")

[Out]

1/55440*(315*(-I*A - B)*a*cos(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 770*(-2*I*A - B)*a*cos(9/2*a
rctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 2970*I*A*a*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))
+ 1386*(-2*I*A + B)*a*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1155*(-I*A + B)*a*cos(3/2*arctan2
(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 315*(A - I*B)*a*sin(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))
+ 770*(2*A - I*B)*a*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 2970*A*a*sin(7/2*arctan2(sin(2*f*x
+ 2*e), cos(2*f*x + 2*e))) + 1386*(2*A + I*B)*a*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1155*(A
+ I*B)*a*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))))*sqrt(a)/(c^(11/2)*f)

Giac [F]

\[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {11}{2}}} \,d x } \]

[In]

integrate((a+I*a*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(11/2),x, algorithm="giac")

[Out]

integrate((B*tan(f*x + e) + A)*(I*a*tan(f*x + e) + a)^(3/2)/(-I*c*tan(f*x + e) + c)^(11/2), x)

Mupad [B] (verification not implemented)

Time = 12.51 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.21 \[ \int \frac {(a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{11/2}} \, dx=-\frac {a\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,\cos \left (2\,e+2\,f\,x\right )\,1155{}\mathrm {i}+A\,\cos \left (4\,e+4\,f\,x\right )\,2772{}\mathrm {i}+A\,\cos \left (6\,e+6\,f\,x\right )\,2970{}\mathrm {i}+A\,\cos \left (8\,e+8\,f\,x\right )\,1540{}\mathrm {i}+A\,\cos \left (10\,e+10\,f\,x\right )\,315{}\mathrm {i}-1155\,B\,\cos \left (2\,e+2\,f\,x\right )-1386\,B\,\cos \left (4\,e+4\,f\,x\right )+770\,B\,\cos \left (8\,e+8\,f\,x\right )+315\,B\,\cos \left (10\,e+10\,f\,x\right )-1155\,A\,\sin \left (2\,e+2\,f\,x\right )-2772\,A\,\sin \left (4\,e+4\,f\,x\right )-2970\,A\,\sin \left (6\,e+6\,f\,x\right )-1540\,A\,\sin \left (8\,e+8\,f\,x\right )-315\,A\,\sin \left (10\,e+10\,f\,x\right )-B\,\sin \left (2\,e+2\,f\,x\right )\,1155{}\mathrm {i}-B\,\sin \left (4\,e+4\,f\,x\right )\,1386{}\mathrm {i}+B\,\sin \left (8\,e+8\,f\,x\right )\,770{}\mathrm {i}+B\,\sin \left (10\,e+10\,f\,x\right )\,315{}\mathrm {i}\right )}{55440\,c^5\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]

[In]

int(((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(3/2))/(c - c*tan(e + f*x)*1i)^(11/2),x)

[Out]

-(a*((a*(cos(2*e + 2*f*x) + sin(2*e + 2*f*x)*1i + 1))/(cos(2*e + 2*f*x) + 1))^(1/2)*(A*cos(2*e + 2*f*x)*1155i
+ A*cos(4*e + 4*f*x)*2772i + A*cos(6*e + 6*f*x)*2970i + A*cos(8*e + 8*f*x)*1540i + A*cos(10*e + 10*f*x)*315i -
1155*B*cos(2*e + 2*f*x) - 1386*B*cos(4*e + 4*f*x) + 770*B*cos(8*e + 8*f*x) + 315*B*cos(10*e + 10*f*x) - 1155*
A*sin(2*e + 2*f*x) - 2772*A*sin(4*e + 4*f*x) - 2970*A*sin(6*e + 6*f*x) - 1540*A*sin(8*e + 8*f*x) - 315*A*sin(1
0*e + 10*f*x) - B*sin(2*e + 2*f*x)*1155i - B*sin(4*e + 4*f*x)*1386i + B*sin(8*e + 8*f*x)*770i + B*sin(10*e + 1
0*f*x)*315i))/(55440*c^5*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))