3.9.7 \(\int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx\) [807]

Optimal. Leaf size=222 \[ -\frac {a^{5/2} (4 i A+B) c^{3/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{4 f}+\frac {a^2 (4 A-i B) c \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}+\frac {a (4 i A+B) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}{4 f} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 222, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.156, Rules used = {3669, 81, 51, 38, 65, 223, 209} \begin {gather*} -\frac {a^{5/2} c^{3/2} (B+4 i A) \text {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^2 c (4 A-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 {a (B+4 i A) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}+\frac {B (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}{4 f} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2),x]

[Out]

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

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Mathematica [A]
time = 5.73, size = 145, normalized size = 0.65 \begin {gather*} \frac {a^2 c^2 \sec ^2(e+f x) (1-i \tan (e+f x)) \sqrt {a+i a \tan (e+f x)} \left (16 (i A+B)-12 i (4 A-i B) \text {ArcTan}\left (e^{i (e+f x)}\right ) \cos ^3(e+f x)+3 (4 A+3 i B+(4 A-i B) \cos (2 (e+f x))) \tan (e+f x)\right )}{48 f \sqrt {c-i c \tan (e+f x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*(c - I*c*Tan[e + f*x])^(3/2),x]

[Out]

(a^2*c^2*Sec[e + f*x]^2*(1 - I*Tan[e + f*x])*Sqrt[a + I*a*Tan[e + f*x]]*(16*(I*A + B) - (12*I)*(4*A - I*B)*Arc
Tan[E^(I*(e + f*x))]*Cos[e + f*x]^3 + 3*(4*A + (3*I)*B + (4*A - I*B)*Cos[2*(e + f*x)])*Tan[e + f*x]))/(48*f*Sq
rt[c - I*c*Tan[e + f*x]])

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Maple [A]
time = 0.42, size = 350, normalized size = 1.58

method	result	size
derivativedivides	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} c \left (6 i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{3}\left (f x +e \right )\right )+8 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )-3 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c +3 i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )+8 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+8 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}+12 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c +12 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )+8 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{24 f \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}\)	\(350\)
default	\(\frac {\sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, \sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, a^{2} c \left (6 i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{3}\left (f x +e \right )\right )+8 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )-3 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c +3 i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )+8 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+8 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}+12 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c +12 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )+8 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{24 f \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}\)	\(350\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(3/2),x,method=_RETURNVERBOSE)

[Out]

1/24/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)*a^2*c*(6*I*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^
(1/2)*tan(f*x+e)^3+8*I*A*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^2-3*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+3*I*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)
+8*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^2+8*I*A*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)+12*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+12*A*(a*c*(1+tan(f*x+e)^2))^(1/
2)*(a*c)^(1/2)*tan(f*x+e)+8*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2)/(a*c*(1+tan(f*x+e)^2))^(1/
2)

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Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1451 vs. \(2 (177) = 354\).
time = 1.43, size = 1451, normalized size = 6.54 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

-96*(12*(4*A - I*B)*a^2*c*cos(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 4*(20*A - 53*I*B)*a^2*c*cos(5
/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 44*(4*A - I*B)*a^2*c*cos(3/2*arctan2(sin(2*f*x + 2*e), cos(2
*f*x + 2*e))) - 12*(4*A - I*B)*a^2*c*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 12*(-4*I*A - B)*a^
2*c*sin(7/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 4*(20*I*A + 53*B)*a^2*c*sin(5/2*arctan2(sin(2*f*x +
2*e), cos(2*f*x + 2*e))) - 44*(4*I*A + B)*a^2*c*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 12*(4*
I*A + B)*a^2*c*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 6*((4*A - I*B)*a^2*c*cos(8*f*x + 8*e) +
4*(4*A - I*B)*a^2*c*cos(6*f*x + 6*e) + 6*(4*A - I*B)*a^2*c*cos(4*f*x + 4*e) + 4*(4*A - I*B)*a^2*c*cos(2*f*x +
2*e) - (-4*I*A - B)*a^2*c*sin(8*f*x + 8*e) - 4*(-4*I*A - B)*a^2*c*sin(6*f*x + 6*e) - 6*(-4*I*A - B)*a^2*c*sin(
4*f*x + 4*e) - 4*(-4*I*A - B)*a^2*c*sin(2*f*x + 2*e) + (4*A - I*B)*a^2*c)*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) + 6*((4*A - I*B)*a^2*c*cos
(8*f*x + 8*e) + 4*(4*A - I*B)*a^2*c*cos(6*f*x + 6*e) + 6*(4*A - I*B)*a^2*c*cos(4*f*x + 4*e) + 4*(4*A - I*B)*a^
2*c*cos(2*f*x + 2*e) - (-4*I*A - B)*a^2*c*sin(8*f*x + 8*e) - 4*(-4*I*A - B)*a^2*c*sin(6*f*x + 6*e) - 6*(-4*I*A
- B)*a^2*c*sin(4*f*x + 4*e) - 4*(-4*I*A - B)*a^2*c*sin(2*f*x + 2*e) + (4*A - I*B)*a^2*c)*arctan2(cos(1/2*arct
an2(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) - 3*((-4*
I*A - B)*a^2*c*cos(8*f*x + 8*e) + 4*(-4*I*A - B)*a^2*c*cos(6*f*x + 6*e) + 6*(-4*I*A - B)*a^2*c*cos(4*f*x + 4*e
) + 4*(-4*I*A - B)*a^2*c*cos(2*f*x + 2*e) + (4*A - I*B)*a^2*c*sin(8*f*x + 8*e) + 4*(4*A - I*B)*a^2*c*sin(6*f*x
+ 6*e) + 6*(4*A - I*B)*a^2*c*sin(4*f*x + 4*e) + 4*(4*A - I*B)*a^2*c*sin(2*f*x + 2*e) + (-4*I*A - B)*a^2*c)*lo
g(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) - 3*((4*I*A + B)*a^2*c*cos(8*f*x + 8*e) + 4*
(4*I*A + B)*a^2*c*cos(6*f*x + 6*e) + 6*(4*I*A + B)*a^2*c*cos(4*f*x + 4*e) + 4*(4*I*A + B)*a^2*c*cos(2*f*x + 2*
e) - (4*A - I*B)*a^2*c*sin(8*f*x + 8*e) - 4*(4*A - I*B)*a^2*c*sin(6*f*x + 6*e) - 6*(4*A - I*B)*a^2*c*sin(4*f*x
+ 4*e) - 4*(4*A - I*B)*a^2*c*sin(2*f*x + 2*e) + (4*I*A + B)*a^2*c)*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*(-4608*I*cos(8*f*x + 8*e) - 18432*I*cos(6*f*x + 6*e) - 27648*I*c
os(4*f*x + 4*e) - 18432*I*cos(2*f*x + 2*e) + 4608*sin(8*f*x + 8*e) + 18432*sin(6*f*x + 6*e) + 27648*sin(4*f*x
+ 4*e) + 18432*sin(2*f*x + 2*e) - 4608*I))

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Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 641 vs. \(2 (177) = 354\).
time = 2.41, size = 641, normalized size = 2.89 \begin {gather*} -\frac {3 \, \sqrt {\frac {{\left (16 \, A^{2} - 8 i \, A B - B^{2}\right )} a^{5} c^{3}}{f^{2}}} {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-4 i \, A - B\right )} a^{2} c e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-4 i \, A - B\right )} a^{2} c 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 (16 \, A^{2} - 8 i \, A B - B^{2}\right )} a^{5} c^{3}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (4 i \, A + B\right )} a^{2} c e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (4 i \, A + B\right )} a^{2} c}\right ) - 3 \, \sqrt {\frac {{\left (16 \, A^{2} - 8 i \, A B - B^{2}\right )} a^{5} c^{3}}{f^{2}}} {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-4 i \, A - B\right )} a^{2} c e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-4 i \, A - B\right )} a^{2} c 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 (16 \, A^{2} - 8 i \, A B - B^{2}\right )} a^{5} c^{3}}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (4 i \, A + B\right )} a^{2} c e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (4 i \, A + B\right )} a^{2} c}\right ) + 4 \, {\left (3 \, {\left (4 i \, A + B\right )} a^{2} c e^{\left (7 i \, f x + 7 i \, e\right )} - {\left (20 i \, A + 53 \, B\right )} a^{2} c e^{\left (5 i \, f x + 5 i \, e\right )} + 11 \, {\left (-4 i \, A - B\right )} a^{2} c e^{\left (3 i \, f x + 3 i \, e\right )} + 3 \, {\left (-4 i \, A - B\right )} a^{2} c 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}}}{48 \, {\left (f e^{\left (6 i \, f x + 6 i \, e\right )} + 3 \, f e^{\left (4 i \, f x + 4 i \, e\right )} + 3 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(3/2),x, algorithm="fricas")

[Out]

-1/48*(3*sqrt((16*A^2 - 8*I*A*B - B^2)*a^5*c^3/f^2)*(f*e^(6*I*f*x + 6*I*e) + 3*f*e^(4*I*f*x + 4*I*e) + 3*f*e^(
2*I*f*x + 2*I*e) + f)*log(-4*(2*((-4*I*A - B)*a^2*c*e^(3*I*f*x + 3*I*e) + (-4*I*A - B)*a^2*c*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((16*A^2 - 8*I*A*B - B^2)*a^5*c^3/f^
2)*(f*e^(2*I*f*x + 2*I*e) - f))/((4*I*A + B)*a^2*c*e^(2*I*f*x + 2*I*e) + (4*I*A + B)*a^2*c)) - 3*sqrt((16*A^2
- 8*I*A*B - B^2)*a^5*c^3/f^2)*(f*e^(6*I*f*x + 6*I*e) + 3*f*e^(4*I*f*x + 4*I*e) + 3*f*e^(2*I*f*x + 2*I*e) + f)*
log(-4*(2*((-4*I*A - B)*a^2*c*e^(3*I*f*x + 3*I*e) + (-4*I*A - B)*a^2*c*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((16*A^2 - 8*I*A*B - B^2)*a^5*c^3/f^2)*(f*e^(2*I*f*x + 2*I
*e) - f))/((4*I*A + B)*a^2*c*e^(2*I*f*x + 2*I*e) + (4*I*A + B)*a^2*c)) + 4*(3*(4*I*A + B)*a^2*c*e^(7*I*f*x + 7
*I*e) - (20*I*A + 53*B)*a^2*c*e^(5*I*f*x + 5*I*e) + 11*(-4*I*A - B)*a^2*c*e^(3*I*f*x + 3*I*e) + 3*(-4*I*A - B)
*a^2*c*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^(6*I*f*x + 6
*I*e) + 3*f*e^(4*I*f*x + 4*I*e) + 3*f*e^(2*I*f*x + 2*I*e) + f)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))**(5/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))**(3/2),x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 3062 deep

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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(3/2),x, algorithm="giac")

[Out]

Timed out

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \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 )}^{5/2}\,{\left (c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(5/2)*(c - c*tan(e + f*x)*1i)^(3/2),x)

[Out]

int((A + B*tan(e + f*x))*(a + a*tan(e + f*x)*1i)^(5/2)*(c - c*tan(e + f*x)*1i)^(3/2), x)

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