\(\int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{9/2}} \, dx\) [825]

Optimal result

Integrand size = 45, antiderivative size = 102 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{9/2}} \, dx=-\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{9 f (c-i c \tan (e+f x))^{9/2}}-\frac {(i A-8 B) (a+i a \tan (e+f x))^{7/2}}{63 c f (c-i c \tan (e+f x))^{7/2}} \]

[Out]

-1/9*(I*A+B)*(a+I*a*tan(f*x+e))^(7/2)/f/(c-I*c*tan(f*x+e))^(9/2)-1/63*(I*A-8*B)*(a+I*a*tan(f*x+e))^(7/2)/c/f/(
c-I*c*tan(f*x+e))^(7/2)

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 102, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.067, Rules used = {3669, 79, 37} \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{9/2}} \, dx=-\frac {(-8 B+i A) (a+i a \tan (e+f x))^{7/2}}{63 c f (c-i c \tan (e+f x))^{7/2}}-\frac {(B+i A) (a+i a \tan (e+f x))^{7/2}}{9 f (c-i c \tan (e+f x))^{9/2}} \]

[In]

Int[((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(9/2),x]

[Out]

-1/9*((I*A + B)*(a + I*a*Tan[e + f*x])^(7/2))/(f*(c - I*c*Tan[e + f*x])^(9/2)) - ((I*A - 8*B)*(a + I*a*Tan[e +
f*x])^(7/2))/(63*c*f*(c - I*c*Tan[e + f*x])^(7/2))

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 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 {(a+i a x)^{5/2} (A+B x)}{(c-i c x)^{11/2}} \, dx,x,\tan (e+f x)\right )}{f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{9 f (c-i c \tan (e+f x))^{9/2}}+\frac {(a (A+8 i B)) \text {Subst}\left (\int \frac {(a+i a x)^{5/2}}{(c-i c x)^{9/2}} \, dx,x,\tan (e+f x)\right )}{9 f} \\ & = -\frac {(i A+B) (a+i a \tan (e+f x))^{7/2}}{9 f (c-i c \tan (e+f x))^{9/2}}-\frac {(i A-8 B) (a+i a \tan (e+f x))^{7/2}}{63 c f (c-i c \tan (e+f x))^{7/2}} \\ \end{align*}

Mathematica [A] (verified)

Time = 6.96 (sec) , antiderivative size = 111, normalized size of antiderivative = 1.09 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{9/2}} \, dx=-\frac {a^4 \sec ^4(e+f x) (\cos (4 (e+f x))+i \sin (4 (e+f x))) (8 i A-B+(A+8 i B) \tan (e+f x))}{63 c^4 f (i+\tan (e+f x))^4 \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}} \]

[In]

Integrate[((a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(c - I*c*Tan[e + f*x])^(9/2),x]

[Out]

-1/63*(a^4*Sec[e + f*x]^4*(Cos[4*(e + f*x)] + I*Sin[4*(e + f*x)])*((8*I)*A - B + (A + (8*I)*B)*Tan[e + f*x]))/
(c^4*f*(I + Tan[e + f*x])^4*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 = 106, normalized size of antiderivative = 1.04

method	result	size
risch	\(-\frac {a^{3} \sqrt {\frac {a \,{\mathrm e}^{2 i \left (f x +e \right )}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (7 i A \,{\mathrm e}^{8 i \left (f x +e \right )}+7 B \,{\mathrm e}^{8 i \left (f x +e \right )}+9 i A \,{\mathrm e}^{6 i \left (f x +e \right )}-9 B \,{\mathrm e}^{6 i \left (f x +e \right )}\right )}{126 c^{4} \sqrt {\frac {c}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\)	\(106\)
derivativedivides	\(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (8 i B \tan \left (f x +e \right )^{3}+6 i A \tan \left (f x +e \right )^{2}+A \tan \left (f x +e \right )^{3}-6 i \tan \left (f x +e \right ) B +15 B \tan \left (f x +e \right )^{2}-8 i A +15 A \tan \left (f x +e \right )+B \right )}{63 f \,c^{5} \left (i+\tan \left (f x +e \right )\right )^{6}}\)	\(134\)
default	\(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (8 i B \tan \left (f x +e \right )^{3}+6 i A \tan \left (f x +e \right )^{2}+A \tan \left (f x +e \right )^{3}-6 i \tan \left (f x +e \right ) B +15 B \tan \left (f x +e \right )^{2}-8 i A +15 A \tan \left (f x +e \right )+B \right )}{63 f \,c^{5} \left (i+\tan \left (f x +e \right )\right )^{6}}\)	\(134\)
parts	\(-\frac {A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (6 i \tan \left (f x +e \right )^{2}+\tan \left (f x +e \right )^{3}-8 i+15 \tan \left (f x +e \right )\right )}{63 f \,c^{5} \left (i+\tan \left (f x +e \right )\right )^{6}}+\frac {i B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} \left (1+\tan \left (f x +e \right )^{2}\right ) \left (15 i \tan \left (f x +e \right )^{2}-8 \tan \left (f x +e \right )^{3}+i+6 \tan \left (f x +e \right )\right )}{63 f \,c^{5} \left (i+\tan \left (f x +e \right )\right )^{6}}\)	\(193\)

[In]

int((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(9/2),x,method=_RETURNVERBOSE)

[Out]

-1/126*a^3/c^4*(a*exp(2*I*(f*x+e))/(exp(2*I*(f*x+e))+1))^(1/2)/(c/(exp(2*I*(f*x+e))+1))^(1/2)/f*(7*I*A*exp(8*I
*(f*x+e))+7*B*exp(8*I*(f*x+e))+9*I*A*exp(6*I*(f*x+e))-9*B*exp(6*I*(f*x+e)))

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.02 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{9/2}} \, dx=-\frac {{\left (7 \, {\left (i \, A + B\right )} a^{3} e^{\left (11 i \, f x + 11 i \, e\right )} + 2 \, {\left (8 i \, A - B\right )} a^{3} e^{\left (9 i \, f x + 9 i \, e\right )} + 9 \, {\left (i \, A - B\right )} a^{3} e^{\left (7 i \, f x + 7 i \, e\right )}\right )} \sqrt {\frac {a}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}} \sqrt {\frac {c}{e^{\left (2 i \, f x + 2 i \, e\right )} + 1}}}{126 \, c^{5} f} \]

[In]

integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(9/2),x, algorithm="fricas")

[Out]

-1/126*(7*(I*A + B)*a^3*e^(11*I*f*x + 11*I*e) + 2*(8*I*A - B)*a^3*e^(9*I*f*x + 9*I*e) + 9*(I*A - B)*a^3*e^(7*I
*f*x + 7*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^5*f)

Sympy [F(-1)]

Timed out. \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{9/2}} \, dx=\text {Timed out} \]

[In]

integrate((a+I*a*tan(f*x+e))**(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))**(9/2),x)

[Out]

Timed out

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 166 vs. \(2 (78) = 156\).

Time = 0.44 (sec) , antiderivative size = 166, normalized size of antiderivative = 1.63 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{9/2}} \, dx=-\frac {126 \, {\left (7 \, {\left (A - i \, B\right )} a^{3} \cos \left (11 \, f x + 11 \, e\right ) + 2 \, {\left (8 \, A + i \, B\right )} a^{3} \cos \left (9 \, f x + 9 \, e\right ) + 9 \, {\left (A + i \, B\right )} a^{3} \cos \left (7 \, f x + 7 \, e\right ) - 7 \, {\left (-i \, A - B\right )} a^{3} \sin \left (11 \, f x + 11 \, e\right ) - 2 \, {\left (-8 i \, A + B\right )} a^{3} \sin \left (9 \, f x + 9 \, e\right ) - 9 \, {\left (-i \, A + B\right )} a^{3} \sin \left (7 \, f x + 7 \, e\right )\right )} \sqrt {a} \sqrt {c}}{-15876 \, {\left (i \, c^{5} \cos \left (2 \, f x + 2 \, e\right ) - c^{5} \sin \left (2 \, f x + 2 \, e\right ) + i \, c^{5}\right )} f} \]

[In]

integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(9/2),x, algorithm="maxima")

[Out]

-126*(7*(A - I*B)*a^3*cos(11*f*x + 11*e) + 2*(8*A + I*B)*a^3*cos(9*f*x + 9*e) + 9*(A + I*B)*a^3*cos(7*f*x + 7*
e) - 7*(-I*A - B)*a^3*sin(11*f*x + 11*e) - 2*(-8*I*A + B)*a^3*sin(9*f*x + 9*e) - 9*(-I*A + B)*a^3*sin(7*f*x +
7*e))*sqrt(a)*sqrt(c)/((-15876*I*c^5*cos(2*f*x + 2*e) + 15876*c^5*sin(2*f*x + 2*e) - 15876*I*c^5)*f)

Giac [F]

\[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{9/2}} \, dx=\int { \frac {{\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}}}{{\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {9}{2}}} \,d x } \]

[In]

integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))/(c-I*c*tan(f*x+e))^(9/2),x, algorithm="giac")

[Out]

integrate((B*tan(f*x + e) + A)*(I*a*tan(f*x + e) + a)^(7/2)/(-I*c*tan(f*x + e) + c)^(9/2), x)

Mupad [B] (verification not implemented)

Time = 10.60 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.88 \[ \int \frac {(a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{9/2}} \, dx=-\frac {a^3\,\sqrt {\frac {a\,\left (\cos \left (2\,e+2\,f\,x\right )+1+\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}\,\left (A\,\cos \left (6\,e+6\,f\,x\right )\,9{}\mathrm {i}+A\,\cos \left (8\,e+8\,f\,x\right )\,7{}\mathrm {i}-9\,B\,\cos \left (6\,e+6\,f\,x\right )+7\,B\,\cos \left (8\,e+8\,f\,x\right )-9\,A\,\sin \left (6\,e+6\,f\,x\right )-7\,A\,\sin \left (8\,e+8\,f\,x\right )-B\,\sin \left (6\,e+6\,f\,x\right )\,9{}\mathrm {i}+B\,\sin \left (8\,e+8\,f\,x\right )\,7{}\mathrm {i}\right )}{126\,c^4\,f\,\sqrt {\frac {c\,\left (\cos \left (2\,e+2\,f\,x\right )+1-\sin \left (2\,e+2\,f\,x\right )\,1{}\mathrm {i}\right )}{\cos \left (2\,e+2\,f\,x\right )+1}}} \]

[In]

int(((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(7/2))/(c - c*tan(e + f*x)*1i)^(9/2),x)

[Out]

-(a^3*((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(6*e + 6*f*x)*9i +
A*cos(8*e + 8*f*x)*7i - 9*B*cos(6*e + 6*f*x) + 7*B*cos(8*e + 8*f*x) - 9*A*sin(6*e + 6*f*x) - 7*A*sin(8*e + 8*
f*x) - B*sin(6*e + 6*f*x)*9i + B*sin(8*e + 8*f*x)*7i))/(126*c^4*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))