\(\int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx\) [795]

Optimal result

Integrand size = 45, antiderivative size = 279 \[ \int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=-\frac {a^{3/2} (5 i A-2 B) c^{7/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{4 f}+\frac {a (5 A+2 i B) c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {(5 i A-2 B) c^2 (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac {(5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f} \]

[Out]

-1/4*a^(3/2)*(5*I*A-2*B)*c^(7/2)*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
/8*a*(5*A+2*I*B)*c^3*(a+I*a*tan(f*x+e))^(1/2)*(c-I*c*tan(f*x+e))^(1/2)*tan(f*x+e)/f-1/12*(5*I*A-2*B)*c^2*(a+I*
a*tan(f*x+e))^(3/2)*(c-I*c*tan(f*x+e))^(3/2)/f-1/20*(5*I*A-2*B)*c*(a+I*a*tan(f*x+e))^(3/2)*(c-I*c*tan(f*x+e))^
(5/2)/f+1/5*B*(a+I*a*tan(f*x+e))^(3/2)*(c-I*c*tan(f*x+e))^(7/2)/f

Rubi [A] (verified)

Time = 0.37 (sec) , antiderivative size = 279, 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} \[ \int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=-\frac {a^{3/2} c^{7/2} (-2 B+5 i A) \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{4 f}+\frac {a c^3 (5 A+2 i B) \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {c^2 (-2 B+5 i A) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac {c (-2 B+5 i A) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f} \]

[In]

Int[(a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2),x]

[Out]

-1/4*(a^(3/2)*((5*I)*A - 2*B)*c^(7/2)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e
+ f*x]])])/f + (a*(5*A + (2*I)*B)*c^3*Tan[e + f*x]*Sqrt[a + I*a*Tan[e + f*x]]*Sqrt[c - I*c*Tan[e + f*x]])/(8*f
) - (((5*I)*A - 2*B)*c^2*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(3/2))/(12*f) - (((5*I)*A - 2*B)*
c*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(5/2))/(20*f) + (B*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c
*Tan[e + f*x])^(7/2))/(5*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 \sqrt {a+i a x} (A+B x) (c-i c x)^{5/2} \, dx,x,\tan (e+f x)\right )}{f} \\ & = \frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}+\frac {(a (5 A+2 i B) c) \text {Subst}\left (\int \sqrt {a+i a x} (c-i c x)^{5/2} \, dx,x,\tan (e+f x)\right )}{5 f} \\ & = -\frac {(5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}+\frac {\left (a (5 A+2 i B) c^2\right ) \text {Subst}\left (\int \sqrt {a+i a x} (c-i c x)^{3/2} \, dx,x,\tan (e+f x)\right )}{4 f} \\ & = -\frac {(5 i A-2 B) c^2 (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac {(5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}+\frac {\left (a (5 A+2 i B) c^3\right ) \text {Subst}\left (\int \sqrt {a+i a x} \sqrt {c-i c x} \, dx,x,\tan (e+f x)\right )}{4 f} \\ & = \frac {a (5 A+2 i B) c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {(5 i A-2 B) c^2 (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac {(5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}+\frac {\left (a^2 (5 A+2 i B) c^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+i a x} \sqrt {c-i c x}} \, dx,x,\tan (e+f x)\right )}{8 f} \\ & = \frac {a (5 A+2 i B) c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {(5 i A-2 B) c^2 (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac {(5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}-\frac {\left (a (5 i A-2 B) c^4\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 )}{4 f} \\ & = \frac {a (5 A+2 i B) c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {(5 i A-2 B) c^2 (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac {(5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}-\frac {\left (a (5 i A-2 B) c^4\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 )}{4 f} \\ & = -\frac {a^{3/2} (5 i A-2 B) c^{7/2} \arctan \left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{4 f}+\frac {a (5 A+2 i B) c^3 \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}-\frac {(5 i A-2 B) c^2 (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}-\frac {(5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f} \\ \end{align*}

Mathematica [A] (verified)

Time = 11.54 (sec) , antiderivative size = 456, normalized size of antiderivative = 1.63 \[ \int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=\frac {B (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{7/2}}{5 f}+\frac {-\frac {a (5 i A-2 B) c (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{5/2}}{4 f}+\frac {-\frac {5 a^2 (5 i A-2 B) c^2 (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{3 f}+\frac {-\frac {15 a^3 (5 i A-2 B) c^3 (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {-\frac {15 a^4 (5 i A-2 B) c^4 (a+i a \tan (e+f x))^{3/2}}{f \sqrt {c-i c \tan (e+f x)}}+\frac {30 a^6 (5 i A-2 B) c^4 (1-i \tan (e+f x)) \left (\frac {1+i \tan (e+f x)}{1-i \tan (e+f x)}-\frac {\arcsin \left (\frac {\sqrt {a+i a \tan (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {1-i \tan (e+f x)}}\right )}{f \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}}{2 a}}{3 a}}{4 a}}{5 a} \]

[In]

Integrate[(a + I*a*Tan[e + f*x])^(3/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(7/2),x]

[Out]

(B*(a + I*a*Tan[e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(7/2))/(5*f) + (-1/4*(a*((5*I)*A - 2*B)*c*(a + I*a*Tan[
e + f*x])^(3/2)*(c - I*c*Tan[e + f*x])^(5/2))/f + ((-5*a^2*((5*I)*A - 2*B)*c^2*(a + I*a*Tan[e + f*x])^(3/2)*(c
- I*c*Tan[e + f*x])^(3/2))/(3*f) + ((-15*a^3*((5*I)*A - 2*B)*c^3*(a + I*a*Tan[e + f*x])^(3/2)*Sqrt[c - I*c*Ta
n[e + f*x]])/(2*f) + ((-15*a^4*((5*I)*A - 2*B)*c^4*(a + I*a*Tan[e + f*x])^(3/2))/(f*Sqrt[c - I*c*Tan[e + f*x]]
) + (30*a^6*((5*I)*A - 2*B)*c^4*(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*a))/(3*a))/(4*a))/(5*a)

Maple [A] (verified)

Time = 0.35 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.48

method	result	size
derivativedivides	\(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, c^{3} a \left (60 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}+24 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{4}+80 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+30 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-30 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +30 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-32 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+80 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-75 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -45 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-56 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{120 f \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}\)	\(412\)
default	\(-\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, c^{3} a \left (60 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}+24 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{4}+80 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+30 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{3}-30 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c +30 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-32 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )^{2}+80 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-75 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -45 A \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )-56 B \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{120 f \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}\)	\(412\)
parts	\(-\frac {A \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, c^{3} a \left (16 i \tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+6 \tan \left (f x +e \right )^{3} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+16 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-15 a c \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right )-9 \tan \left (f x +e \right ) \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}\right )}{24 f \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}}+\frac {B \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, c^{3} a \left (-30 i \tan \left (f x +e \right )^{3} \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}-12 \tan \left (f x +e \right )^{4} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+15 i \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}}{\sqrt {a c}}\right ) a c -15 i \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \tan \left (f x +e \right )+16 \tan \left (f x +e \right )^{2} \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}+28 \sqrt {a c}\, \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\right )}{60 f \sqrt {a c \left (1+\tan \left (f x +e \right )^{2}\right )}\, \sqrt {a c}}\)	\(465\)

[In]

int((a+I*a*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2),x,method=_RETURNVERBOSE)

[Out]

-1/120/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)*c^3*a*(60*I*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2
))^(1/2)*tan(f*x+e)^3+24*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^4+80*I*A*(a*c)^(1/2)*(a*c*(1+ta
n(f*x+e)^2))^(1/2)*tan(f*x+e)^2+30*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^3-30*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+30*I*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1
/2)*tan(f*x+e)-32*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)^2+80*I*A*(a*c)^(1/2)*(a*c*(1+tan(f*x+e
)^2))^(1/2)-75*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-45*A*(a*c)^(1/2
)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)-56*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2))/(a*c)^(1/2)/(a*c*(1+t
an(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. 682 vs. \(2 (213) = 426\).

Time = 0.28 (sec) , antiderivative size = 682, normalized size of antiderivative = 2.44 \[ \int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=\frac {15 \, \sqrt {\frac {{\left (25 \, A^{2} + 20 i \, A B - 4 \, B^{2}\right )} a^{3} c^{7}}{f^{2}}} {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (5 i \, A - 2 \, B\right )} a c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (5 i \, A - 2 \, B\right )} a c^{3} 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 (25 \, A^{2} + 20 i \, A B - 4 \, B^{2}\right )} a^{3} c^{7}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (-5 i \, A + 2 \, B\right )} a c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-5 i \, A + 2 \, B\right )} a c^{3}}\right ) - 15 \, \sqrt {\frac {{\left (25 \, A^{2} + 20 i \, A B - 4 \, B^{2}\right )} a^{3} c^{7}}{f^{2}}} {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (5 i \, A - 2 \, B\right )} a c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (5 i \, A - 2 \, B\right )} a c^{3} 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 (25 \, A^{2} + 20 i \, A B - 4 \, B^{2}\right )} a^{3} c^{7}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (-5 i \, A + 2 \, B\right )} a c^{3} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (-5 i \, A + 2 \, B\right )} a c^{3}}\right ) - 4 \, {\left (15 \, {\left (5 i \, A - 2 \, B\right )} a c^{3} e^{\left (9 i \, f x + 9 i \, e\right )} + 70 \, {\left (5 i \, A - 2 \, B\right )} a c^{3} e^{\left (7 i \, f x + 7 i \, e\right )} + 128 \, {\left (5 i \, A - 2 \, B\right )} a c^{3} e^{\left (5 i \, f x + 5 i \, e\right )} + 10 \, {\left (29 i \, A - 50 \, B\right )} a c^{3} e^{\left (3 i \, f x + 3 i \, e\right )} + 15 \, {\left (-5 i \, A + 2 \, B\right )} a c^{3} 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}}}{240 \, {\left (f e^{\left (8 i \, f x + 8 i \, e\right )} + 4 \, f e^{\left (6 i \, f x + 6 i \, e\right )} + 6 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 4 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \]

[In]

integrate((a+I*a*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2),x, algorithm="fricas")

[Out]

1/240*(15*sqrt((25*A^2 + 20*I*A*B - 4*B^2)*a^3*c^7/f^2)*(f*e^(8*I*f*x + 8*I*e) + 4*f*e^(6*I*f*x + 6*I*e) + 6*f
*e^(4*I*f*x + 4*I*e) + 4*f*e^(2*I*f*x + 2*I*e) + f)*log(-4*(2*((5*I*A - 2*B)*a*c^3*e^(3*I*f*x + 3*I*e) + (5*I*
A - 2*B)*a*c^3*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((25
*A^2 + 20*I*A*B - 4*B^2)*a^3*c^7/f^2)*(f*e^(2*I*f*x + 2*I*e) - f))/((-5*I*A + 2*B)*a*c^3*e^(2*I*f*x + 2*I*e) +
(-5*I*A + 2*B)*a*c^3)) - 15*sqrt((25*A^2 + 20*I*A*B - 4*B^2)*a^3*c^7/f^2)*(f*e^(8*I*f*x + 8*I*e) + 4*f*e^(6*I
*f*x + 6*I*e) + 6*f*e^(4*I*f*x + 4*I*e) + 4*f*e^(2*I*f*x + 2*I*e) + f)*log(-4*(2*((5*I*A - 2*B)*a*c^3*e^(3*I*f
*x + 3*I*e) + (5*I*A - 2*B)*a*c^3*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((25*A^2 + 20*I*A*B - 4*B^2)*a^3*c^7/f^2)*(f*e^(2*I*f*x + 2*I*e) - f))/((-5*I*A + 2*B)*a*c^3*e^
(2*I*f*x + 2*I*e) + (-5*I*A + 2*B)*a*c^3)) - 4*(15*(5*I*A - 2*B)*a*c^3*e^(9*I*f*x + 9*I*e) + 70*(5*I*A - 2*B)*
a*c^3*e^(7*I*f*x + 7*I*e) + 128*(5*I*A - 2*B)*a*c^3*e^(5*I*f*x + 5*I*e) + 10*(29*I*A - 50*B)*a*c^3*e^(3*I*f*x
+ 3*I*e) + 15*(-5*I*A + 2*B)*a*c^3*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^(8*I*f*x + 8*I*e) + 4*f*e^(6*I*f*x + 6*I*e) + 6*f*e^(4*I*f*x + 4*I*e) + 4*f*e^(2*I*f*x + 2*I*e
) + f)

Sympy [F(-1)]

Timed out. \[ \int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=\text {Timed out} \]

[In]

integrate((a+I*a*tan(f*x+e))**(3/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**(7/2),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. 1655 vs. \(2 (213) = 426\).

Time = 2.24 (sec) , antiderivative size = 1655, normalized size of antiderivative = 5.93 \[ \int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=\text {Too large to display} \]

[In]

integrate((a+I*a*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2),x, algorithm="maxima")

[Out]

-480*(60*(5*A + 2*I*B)*a*c^3*cos(9/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 280*(5*A + 2*I*B)*a*c^3*co
s(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 512*(5*A + 2*I*B)*a*c^3*cos(5/2*arctan2(sin(2*f*x + 2*e),
cos(2*f*x + 2*e))) + 40*(29*A + 50*I*B)*a*c^3*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 60*(5*A
+ 2*I*B)*a*c^3*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 60*(5*I*A - 2*B)*a*c^3*sin(9/2*arctan2(s
in(2*f*x + 2*e), cos(2*f*x + 2*e))) + 280*(5*I*A - 2*B)*a*c^3*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*
e))) + 512*(5*I*A - 2*B)*a*c^3*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 40*(29*I*A - 50*B)*a*c^3
*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 60*(-5*I*A + 2*B)*a*c^3*sin(1/2*arctan2(sin(2*f*x + 2*
e), cos(2*f*x + 2*e))) + 30*((5*A + 2*I*B)*a*c^3*cos(10*f*x + 10*e) + 5*(5*A + 2*I*B)*a*c^3*cos(8*f*x + 8*e) +
10*(5*A + 2*I*B)*a*c^3*cos(6*f*x + 6*e) + 10*(5*A + 2*I*B)*a*c^3*cos(4*f*x + 4*e) + 5*(5*A + 2*I*B)*a*c^3*cos
(2*f*x + 2*e) + (5*I*A - 2*B)*a*c^3*sin(10*f*x + 10*e) + 5*(5*I*A - 2*B)*a*c^3*sin(8*f*x + 8*e) + 10*(5*I*A -
2*B)*a*c^3*sin(6*f*x + 6*e) + 10*(5*I*A - 2*B)*a*c^3*sin(4*f*x + 4*e) + 5*(5*I*A - 2*B)*a*c^3*sin(2*f*x + 2*e)
+ (5*A + 2*I*B)*a*c^3)*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) + 30*((5*A + 2*I*B)*a*c^3*cos(10*f*x + 10*e) + 5*(5*A + 2*I*B)*a*c^3*cos(8*f
*x + 8*e) + 10*(5*A + 2*I*B)*a*c^3*cos(6*f*x + 6*e) + 10*(5*A + 2*I*B)*a*c^3*cos(4*f*x + 4*e) + 5*(5*A + 2*I*B
)*a*c^3*cos(2*f*x + 2*e) + (5*I*A - 2*B)*a*c^3*sin(10*f*x + 10*e) + 5*(5*I*A - 2*B)*a*c^3*sin(8*f*x + 8*e) + 1
0*(5*I*A - 2*B)*a*c^3*sin(6*f*x + 6*e) + 10*(5*I*A - 2*B)*a*c^3*sin(4*f*x + 4*e) + 5*(5*I*A - 2*B)*a*c^3*sin(2
*f*x + 2*e) + (5*A + 2*I*B)*a*c^3)*arctan2(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))), -sin(1/2*arct
an2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) + 15*((5*I*A - 2*B)*a*c^3*cos(10*f*x + 10*e) + 5*(5*I*A - 2*B)*a
*c^3*cos(8*f*x + 8*e) + 10*(5*I*A - 2*B)*a*c^3*cos(6*f*x + 6*e) + 10*(5*I*A - 2*B)*a*c^3*cos(4*f*x + 4*e) + 5*
(5*I*A - 2*B)*a*c^3*cos(2*f*x + 2*e) - (5*A + 2*I*B)*a*c^3*sin(10*f*x + 10*e) - 5*(5*A + 2*I*B)*a*c^3*sin(8*f*
x + 8*e) - 10*(5*A + 2*I*B)*a*c^3*sin(6*f*x + 6*e) - 10*(5*A + 2*I*B)*a*c^3*sin(4*f*x + 4*e) - 5*(5*A + 2*I*B)
*a*c^3*sin(2*f*x + 2*e) + (5*I*A - 2*B)*a*c^3)*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + si
n(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*((-5*I*A + 2*B)*a*c^3*cos(10*f*x + 10*e) + 5*(-5*I*A + 2*B)*a*c^3*cos(8*f*x + 8*e) + 10*(-5*I*A + 2*
B)*a*c^3*cos(6*f*x + 6*e) + 10*(-5*I*A + 2*B)*a*c^3*cos(4*f*x + 4*e) + 5*(-5*I*A + 2*B)*a*c^3*cos(2*f*x + 2*e)
+ (5*A + 2*I*B)*a*c^3*sin(10*f*x + 10*e) + 5*(5*A + 2*I*B)*a*c^3*sin(8*f*x + 8*e) + 10*(5*A + 2*I*B)*a*c^3*si
n(6*f*x + 6*e) + 10*(5*A + 2*I*B)*a*c^3*sin(4*f*x + 4*e) + 5*(5*A + 2*I*B)*a*c^3*sin(2*f*x + 2*e) + (-5*I*A +
2*B)*a*c^3)*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))*sqrt(a)*sqrt(c)/(f*(-115200*I
*cos(10*f*x + 10*e) - 576000*I*cos(8*f*x + 8*e) - 1152000*I*cos(6*f*x + 6*e) - 1152000*I*cos(4*f*x + 4*e) - 57
6000*I*cos(2*f*x + 2*e) + 115200*sin(10*f*x + 10*e) + 576000*sin(8*f*x + 8*e) + 1152000*sin(6*f*x + 6*e) + 115
2000*sin(4*f*x + 4*e) + 576000*sin(2*f*x + 2*e) - 115200*I))

Giac [F(-1)]

Timed out. \[ \int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=\text {Timed out} \]

[In]

integrate((a+I*a*tan(f*x+e))^(3/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(7/2),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int (a+i a \tan (e+f x))^{3/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{7/2} \, dx=\int \left (A+B\,\mathrm {tan}\left (e+f\,x\right )\right )\,{\left (a+a\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2}\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{7/2} \,d x \]

[In]

int((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(3/2)*(c - c*tan(e + f*x)*1i)^(7/2),x)

[Out]

int((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(3/2)*(c - c*tan(e + f*x)*1i)^(7/2), x)