\(\int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx\) [818]

Optimal result

Integrand size = 45, antiderivative size = 284 \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=-\frac {a^{7/2} (6 i A+B) c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{8 f}+\frac {a^3 (6 A-i B) c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a^2 (6 A-i B) c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a (6 i A+B) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f} \]

[Out]

-1/8*a^(7/2)*(6*I*A+B)*c^(5/2)*arctan(c^(1/2)*(a+I*a*tan(f*x+e))^(1/2)/a^(1/2)/(c-I*c*tan(f*x+e))^(1/2))/f+1/1
6*a^3*(6*A-I*B)*c^2*(a+I*a*tan(f*x+e))^(1/2)*(c-I*c*tan(f*x+e))^(1/2)*tan(f*x+e)/f+1/24*a^2*(6*A-I*B)*c*tan(f*
x+e)*(a+I*a*tan(f*x+e))^(3/2)*(c-I*c*tan(f*x+e))^(3/2)/f+1/30*a*(6*I*A+B)*(a+I*a*tan(f*x+e))^(5/2)*(c-I*c*tan(
f*x+e))^(5/2)/f+1/6*B*(a+I*a*tan(f*x+e))^(7/2)*(c-I*c*tan(f*x+e))^(5/2)/f

Rubi [A] (verified)

Time = 0.39 (sec) , antiderivative size = 284, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3669, 81, 51, 38, 65, 223, 209} \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=-\frac {a^{7/2} c^{5/2} (B+6 i A) \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{8 f}+\frac {a^3 c^2 (6 A-i B) \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a^2 c (6 A-i B) \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a (B+6 i A) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f} \]

[In]

Int[(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]

[Out]

-1/8*(a^(7/2)*((6*I)*A + B)*c^(5/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e +
f*x]])])/f + (a^3*(6*A - I*B)*c^2*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(16*f) +
(a^2*(6*A - I*B)*c*Tan[e + f*x]*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(24*f) + (a*((6*I)
*A + B)*(a + I*a*Tan[e + f*x])^(5/2)*(c - I*c*Tan[e + f*x])^(5/2))/(30*f) + (B*(a + I*a*Tan[e + f*x])^(7/2)*(c
- I*c*Tan[e + f*x])^(5/2))/(6*f)

Rule 38

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[x*(a + b*x)^m*((c + d*x)^m/(2*m + 1))
, x] + Dist[2*a*c*(m/(2*m + 1)), Int[(a + b*x)^(m - 1)*(c + d*x)^(m - 1), x], x] /; FreeQ[{a, b, c, d}, x] &&
EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0]

Rule 51

Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m
+ n + 1))), x] + Dist[2*c*(n/(m + n + 1)), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x]
&& EqQ[b*c + a*d, 0] && IGtQ[m + 1/2, 0] && IGtQ[n + 1/2, 0] && LtQ[m, n]

Rule 65

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 81

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

Rule 209

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*ArcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /;
FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a, 0] || GtQ[b, 0])

Rule 223

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a,
b}, x] &&  !GtQ[a, 0]

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 (a+i a x)^{5/2} (A+B x) (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {(a (6 A-i B) c) \text {Subst}\left (\int (a+i a x)^{5/2} (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{6 f} \\ & = \frac {a (6 i A+B) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {\left (a^2 (6 A-i B) c\right ) \text {Subst}\left (\int (a+i a x)^{3/2} (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{6 f} \\ & = \frac {a^2 (6 A-i B) c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a (6 i A+B) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {\left (a^3 (6 A-i B) c^2\right ) \text {Subst}\left (\int \sqrt {a+i a x} \sqrt {c-i c x} \, dx,x,\tan (e+f x)\right )}{8 f} \\ & = \frac {a^3 (6 A-i B) c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a^2 (6 A-i B) c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a (6 i A+B) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {\left (a^4 (6 A-i B) c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{16 f} \\ & = \frac {a^3 (6 A-i B) c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a^2 (6 A-i B) c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a (6 i A+B) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}-\frac {\left (a^3 (6 i A+B) c^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {2 c-\frac {c x^2}{a}}} \, dx,x,\sqrt {a+i a \tan (e+f x)}\right )}{8 f} \\ & = \frac {a^3 (6 A-i B) c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a^2 (6 A-i B) c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a (6 i A+B) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}-\frac {\left (a^3 (6 i A+B) c^3\right ) \text {Subst}\left (\int \frac {1}{1+\frac {c x^2}{a}} \, dx,x,\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {c-i c \tan (e+f x)}}\right )}{8 f} \\ & = -\frac {a^{7/2} (6 i A+B) c^{5/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{8 f}+\frac {a^3 (6 A-i B) c^2 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{16 f}+\frac {a^2 (6 A-i B) c \tan (e+f x) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{24 f}+\frac {a (6 i A+B) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{5/2}}{30 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.71 (sec) , antiderivative size = 510, normalized size of antiderivative = 1.80 \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{5/2}}{6 f}+\frac {-\frac {a (6 i A+B) c (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{3/2}}{5 f}+\frac {-\frac {3 a^2 (6 i A+B) c^2 (a+i a \tan (e+f x))^{7/2} \sqrt {c-i c \tan (e+f x)}}{4 f}+\frac {-\frac {a^3 (6 i A+B) c^3 (a+i a \tan (e+f x))^{7/2}}{f \sqrt {c-i c \tan (e+f x)}}+\frac {-\frac {3 a^4 (6 i A+B) c^4 (a+i a \tan (e+f x))^{5/2}}{2 f \sqrt {c-i c \tan (e+f x)}}+\frac {-\frac {15 a^5 (6 i A+B) c^5 (a+i a \tan (e+f x))^{3/2}}{f \sqrt {c-i c \tan (e+f x)}}+\frac {90 i a^7 (6 A-i B) c^5 (1-i \tan (e+f x)) \left (\frac {1+i \tan (e+f x)}{1-i \tan (e+f x)}-\frac {\arcsin \left (\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {1-i \tan (e+f x)}}\right )}{f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}}{2 c}}{3 c}}{4 a}}{5 a}}{6 a} \]

[In]

Integrate[(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(5/2),x]

[Out]

(B*(a + I*a*Tan[e + f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(5/2))/(6*f) + (-1/5*(a*((6*I)*A + B)*c*(a + I*a*Tan[e
+ f*x])^(7/2)*(c - I*c*Tan[e + f*x])^(3/2))/f + ((-3*a^2*((6*I)*A + B)*c^2*(a + I*a*Tan[e + f*x])^(7/2)*Sqrt[c
- I*c*Tan[e + f*x]])/(4*f) + (-((a^3*((6*I)*A + B)*c^3*(a + I*a*Tan[e + f*x])^(7/2))/(f*Sqrt[c - I*c*Tan[e +
f*x]])) + ((-3*a^4*((6*I)*A + B)*c^4*(a + I*a*Tan[e + f*x])^(5/2))/(2*f*Sqrt[c - I*c*Tan[e + f*x]]) + ((-15*a^
5*((6*I)*A + B)*c^5*(a + I*a*Tan[e + f*x])^(3/2))/(f*Sqrt[c - I*c*Tan[e + f*x]]) + ((90*I)*a^7*(6*A - I*B)*c^5
*(1 - I*Tan[e + f*x])*((1 + I*Tan[e + f*x])/(1 - I*Tan[e + f*x]) - (ArcSin[Sqrt[a + I*a*Tan[e + f*x]]/(Sqrt[2]
*Sqrt[a])]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[1 - I*Tan[e + f*x]])))/(f*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt
[c - I*c*Tan[e + f*x]]))/(2*c))/(3*c))/(4*a))/(5*a))/(6*a)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (232 ) = 464\).

Time = 0.49 (sec) , antiderivative size = 478, normalized size of antiderivative = 1.68

method	result	size
derivativedivides	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} c^{2} \left (40 i B \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{5}+48 i A \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{4}+70 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}+48 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{4}+96 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+60 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-15 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +15 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+96 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+48 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+90 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +150 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+48 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{240 f \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}\)	\(478\)
default	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} c^{2} \left (40 i B \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{5}+48 i A \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )^{4}+70 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}+48 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{4}+96 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+60 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-15 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +15 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+96 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+48 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+90 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +150 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+48 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{240 f \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}\)	\(478\)
parts	\(\frac {A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} c^{2} \left (8 i \tan \left (f x +e \right )^{4} \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+16 i \tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+10 \tan \left (f x +e \right )^{3} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+8 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}+15 a c \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right )+25 \tan \left (f x +e \right ) \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\right )}{40 f \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}}-\frac {B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{3} c^{2} \left (-40 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{5}-70 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-48 \tan \left (f x +e \right )^{4} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+15 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -15 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-96 \tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}-48 \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{240 f \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}}\)	\(531\)

[In]

int((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x,method=_RETURNVERBOSE)

[Out]

1/240/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)*a^3*c^2*(40*I*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a
*c)^(1/2)*tan(f*x+e)^5+48*I*A*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^4+70*I*B*(a*c)^(1/2)*(a*c*(1
+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^3+48*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^4+96*I*A*(a*c)^(1/
2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^2+60*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^3-15*I*B
*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c+15*I*B*(a*c)^(1/2)*(a*c*(1+tan(
f*x+e)^2))^(1/2)*tan(f*x+e)+96*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^2+48*I*A*(a*c)^(1/2)*(a*c
*(1+tan(f*x+e)^2))^(1/2)+90*A*ln((a*c*tan(f*x+e)+(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2))*a*c+15
0*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)+48*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(
1/2)/(a*c*(1+tan(f*x+e)^2))^(1/2)

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 756 vs. \(2 (218) = 436\).

Time = 0.29 (sec) , antiderivative size = 756, normalized size of antiderivative = 2.66 \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=-\frac {15 \, \sqrt {\frac {{\left (36 \, A^{2} - 12 i \, A B - B^{2}\right )} a^{7} c^{5}}{f^{2}}} {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-6 i \, A - B\right )} a^{3} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-6 i \, A - B\right )} a^{3} c^{2} e^{\left (i \, f x + 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}} + \sqrt {\frac {{\left (36 \, A^{2} - 12 i \, A B - B^{2}\right )} a^{7} c^{5}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (6 i \, A + B\right )} a^{3} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (6 i \, A + B\right )} a^{3} c^{2}}\right ) - 15 \, \sqrt {\frac {{\left (36 \, A^{2} - 12 i \, A B - B^{2}\right )} a^{7} c^{5}}{f^{2}}} {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-6 i \, A - B\right )} a^{3} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-6 i \, A - B\right )} a^{3} c^{2} e^{\left (i \, f x + 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}} - \sqrt {\frac {{\left (36 \, A^{2} - 12 i \, A B - B^{2}\right )} a^{7} c^{5}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (6 i \, A + B\right )} a^{3} c^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (6 i \, A + B\right )} a^{3} c^{2}}\right ) + 4 \, {\left (15 \, {\left (6 i \, A + B\right )} a^{3} c^{2} e^{\left (11 i \, f x + 11 i \, e\right )} + 85 \, {\left (6 i \, A + B\right )} a^{3} c^{2} e^{\left (9 i \, f x + 9 i \, e\right )} + 6 \, {\left (-58 i \, A - 223 \, B\right )} a^{3} c^{2} e^{\left (7 i \, f x + 7 i \, e\right )} + 198 \, {\left (-6 i \, A - B\right )} a^{3} c^{2} e^{\left (5 i \, f x + 5 i \, e\right )} + 85 \, {\left (-6 i \, A - B\right )} a^{3} c^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + 15 \, {\left (-6 i \, A - B\right )} a^{3} c^{2} e^{\left (i \, f x + 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}}}{480 \, {\left (f e^{\left (10 i \, f x + 10 i \, e\right )} + 5 \, f e^{\left (8 i \, f x + 8 i \, e\right )} + 10 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 10 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 5 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]

[In]

integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x, algorithm="fricas")

[Out]

-1/480*(15*sqrt((36*A^2 - 12*I*A*B - B^2)*a^7*c^5/f^2)*(f*e^(10*I*f*x + 10*I*e) + 5*f*e^(8*I*f*x + 8*I*e) + 10
*f*e^(6*I*f*x + 6*I*e) + 10*f*e^(4*I*f*x + 4*I*e) + 5*f*e^(2*I*f*x + 2*I*e) + f)*log(-4*(2*((-6*I*A - B)*a^3*c
^2*e^(3*I*f*x + 3*I*e) + (-6*I*A - B)*a^3*c^2*e^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*
I*f*x + 2*I*e) + 1)) + sqrt((36*A^2 - 12*I*A*B - B^2)*a^7*c^5/f^2)*(f*e^(2*I*f*x + 2*I*e) - f))/((6*I*A + B)*a
^3*c^2*e^(2*I*f*x + 2*I*e) + (6*I*A + B)*a^3*c^2)) - 15*sqrt((36*A^2 - 12*I*A*B - B^2)*a^7*c^5/f^2)*(f*e^(10*I
*f*x + 10*I*e) + 5*f*e^(8*I*f*x + 8*I*e) + 10*f*e^(6*I*f*x + 6*I*e) + 10*f*e^(4*I*f*x + 4*I*e) + 5*f*e^(2*I*f*
x + 2*I*e) + f)*log(-4*(2*((-6*I*A - B)*a^3*c^2*e^(3*I*f*x + 3*I*e) + (-6*I*A - B)*a^3*c^2*e^(I*f*x + I*e))*sq
rt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) + 1)) - sqrt((36*A^2 - 12*I*A*B - B^2)*a^7*c^5/f^2
)*(f*e^(2*I*f*x + 2*I*e) - f))/((6*I*A + B)*a^3*c^2*e^(2*I*f*x + 2*I*e) + (6*I*A + B)*a^3*c^2)) + 4*(15*(6*I*A
+ B)*a^3*c^2*e^(11*I*f*x + 11*I*e) + 85*(6*I*A + B)*a^3*c^2*e^(9*I*f*x + 9*I*e) + 6*(-58*I*A - 223*B)*a^3*c^2
*e^(7*I*f*x + 7*I*e) + 198*(-6*I*A - B)*a^3*c^2*e^(5*I*f*x + 5*I*e) + 85*(-6*I*A - B)*a^3*c^2*e^(3*I*f*x + 3*I
*e) + 15*(-6*I*A - B)*a^3*c^2*e^(I*f*x + I*e))*sqrt(a/(e^(2*I*f*x + 2*I*e) + 1))*sqrt(c/(e^(2*I*f*x + 2*I*e) +
1)))/(f*e^(10*I*f*x + 10*I*e) + 5*f*e^(8*I*f*x + 8*I*e) + 10*f*e^(6*I*f*x + 6*I*e) + 10*f*e^(4*I*f*x + 4*I*e)
+ 5*f*e^(2*I*f*x + 2*I*e) + f)

Sympy [F(-1)]

Timed out. \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\text {Timed out} \]

[In]

integrate((a+I*a*tan(f*x+e))**(7/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**(5/2),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. 2033 vs. \(2 (218) = 436\).

Time = 5.96 (sec) , antiderivative size = 2033, normalized size of antiderivative = 7.16 \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\text {Too large to display} \]

[In]

integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x, algorithm="maxima")

[Out]

-3840*(60*(6*A - I*B)*a^3*c^2*cos(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 340*(6*A - I*B)*a^3*c^2*
cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 24*(58*A - 223*I*B)*a^3*c^2*cos(7/2*arctan2(sin(2*f*x +
2*e), cos(2*f*x + 2*e))) - 792*(6*A - I*B)*a^3*c^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 340
*(6*A - I*B)*a^3*c^2*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 60*(6*A - I*B)*a^3*c^2*cos(1/2*arc
tan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 60*(-6*I*A - B)*a^3*c^2*sin(11/2*arctan2(sin(2*f*x + 2*e), cos(2*f
*x + 2*e))) - 340*(-6*I*A - B)*a^3*c^2*sin(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 24*(58*I*A + 223
*B)*a^3*c^2*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 792*(6*I*A + B)*a^3*c^2*sin(5/2*arctan2(sin
(2*f*x + 2*e), cos(2*f*x + 2*e))) - 340*(6*I*A + B)*a^3*c^2*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)
)) - 60*(6*I*A + B)*a^3*c^2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 30*((6*A - I*B)*a^3*c^2*cos
(12*f*x + 12*e) + 6*(6*A - I*B)*a^3*c^2*cos(10*f*x + 10*e) + 15*(6*A - I*B)*a^3*c^2*cos(8*f*x + 8*e) + 20*(6*A
- I*B)*a^3*c^2*cos(6*f*x + 6*e) + 15*(6*A - I*B)*a^3*c^2*cos(4*f*x + 4*e) + 6*(6*A - I*B)*a^3*c^2*cos(2*f*x +
2*e) - (-6*I*A - B)*a^3*c^2*sin(12*f*x + 12*e) - 6*(-6*I*A - B)*a^3*c^2*sin(10*f*x + 10*e) - 15*(-6*I*A - B)*
a^3*c^2*sin(8*f*x + 8*e) - 20*(-6*I*A - B)*a^3*c^2*sin(6*f*x + 6*e) - 15*(-6*I*A - B)*a^3*c^2*sin(4*f*x + 4*e)
- 6*(-6*I*A - B)*a^3*c^2*sin(2*f*x + 2*e) + (6*A - I*B)*a^3*c^2)*arctan2(cos(1/2*arctan2(sin(2*f*x + 2*e), co
s(2*f*x + 2*e))), sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 30*((6*A - I*B)*a^3*c^2*cos(12*f
*x + 12*e) + 6*(6*A - I*B)*a^3*c^2*cos(10*f*x + 10*e) + 15*(6*A - I*B)*a^3*c^2*cos(8*f*x + 8*e) + 20*(6*A - I*
B)*a^3*c^2*cos(6*f*x + 6*e) + 15*(6*A - I*B)*a^3*c^2*cos(4*f*x + 4*e) + 6*(6*A - I*B)*a^3*c^2*cos(2*f*x + 2*e)
- (-6*I*A - B)*a^3*c^2*sin(12*f*x + 12*e) - 6*(-6*I*A - B)*a^3*c^2*sin(10*f*x + 10*e) - 15*(-6*I*A - B)*a^3*c
^2*sin(8*f*x + 8*e) - 20*(-6*I*A - B)*a^3*c^2*sin(6*f*x + 6*e) - 15*(-6*I*A - B)*a^3*c^2*sin(4*f*x + 4*e) - 6*
(-6*I*A - B)*a^3*c^2*sin(2*f*x + 2*e) + (6*A - I*B)*a^3*c^2)*arctan2(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f
*x + 2*e))), -sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) - 15*((-6*I*A - B)*a^3*c^2*cos(12*f*x
+ 12*e) + 6*(-6*I*A - B)*a^3*c^2*cos(10*f*x + 10*e) + 15*(-6*I*A - B)*a^3*c^2*cos(8*f*x + 8*e) + 20*(-6*I*A -
B)*a^3*c^2*cos(6*f*x + 6*e) + 15*(-6*I*A - B)*a^3*c^2*cos(4*f*x + 4*e) + 6*(-6*I*A - B)*a^3*c^2*cos(2*f*x + 2*
e) + (6*A - I*B)*a^3*c^2*sin(12*f*x + 12*e) + 6*(6*A - I*B)*a^3*c^2*sin(10*f*x + 10*e) + 15*(6*A - I*B)*a^3*c^
2*sin(8*f*x + 8*e) + 20*(6*A - I*B)*a^3*c^2*sin(6*f*x + 6*e) + 15*(6*A - I*B)*a^3*c^2*sin(4*f*x + 4*e) + 6*(6*
A - I*B)*a^3*c^2*sin(2*f*x + 2*e) + (-6*I*A - B)*a^3*c^2)*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*
e)))^2 + sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + 2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*
x + 2*e))) + 1) - 15*((6*I*A + B)*a^3*c^2*cos(12*f*x + 12*e) + 6*(6*I*A + B)*a^3*c^2*cos(10*f*x + 10*e) + 15*(
6*I*A + B)*a^3*c^2*cos(8*f*x + 8*e) + 20*(6*I*A + B)*a^3*c^2*cos(6*f*x + 6*e) + 15*(6*I*A + B)*a^3*c^2*cos(4*f
*x + 4*e) + 6*(6*I*A + B)*a^3*c^2*cos(2*f*x + 2*e) - (6*A - I*B)*a^3*c^2*sin(12*f*x + 12*e) - 6*(6*A - I*B)*a^
3*c^2*sin(10*f*x + 10*e) - 15*(6*A - I*B)*a^3*c^2*sin(8*f*x + 8*e) - 20*(6*A - I*B)*a^3*c^2*sin(6*f*x + 6*e) -
15*(6*A - I*B)*a^3*c^2*sin(4*f*x + 4*e) - 6*(6*A - I*B)*a^3*c^2*sin(2*f*x + 2*e) + (6*I*A + B)*a^3*c^2)*log(c
os(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2
- 2*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1))*sqrt(a)*sqrt(c)/(f*(-1843200*I*cos(12*f*x + 12
*e) - 11059200*I*cos(10*f*x + 10*e) - 27648000*I*cos(8*f*x + 8*e) - 36864000*I*cos(6*f*x + 6*e) - 27648000*I*c
os(4*f*x + 4*e) - 11059200*I*cos(2*f*x + 2*e) + 1843200*sin(12*f*x + 12*e) + 11059200*sin(10*f*x + 10*e) + 276
48000*sin(8*f*x + 8*e) + 36864000*sin(6*f*x + 6*e) + 27648000*sin(4*f*x + 4*e) + 11059200*sin(2*f*x + 2*e) - 1
843200*I))

Giac [F]

\[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\int { {\left (B \tan \left (f x + e\right ) + A\right )} {\left (i \, a \tan \left (f x + e\right ) + a\right )}^{\frac {7}{2}} {\left (-i \, c \tan \left (f x + e\right ) + c\right )}^{\frac {5}{2}} \,d x } \]

[In]

integrate((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(5/2),x, algorithm="giac")

[Out]

sage0*x

Mupad [F(-1)]

Timed out. \[ \int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{5/2} \, dx=\int \left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2}\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{5/2} \,d x \]

[In]

int((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(7/2)*(c - c*tan(e + f*x)*1i)^(5/2),x)

[Out]

int((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(7/2)*(c - c*tan(e + f*x)*1i)^(5/2), x)