3.9.19 \(\int (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x)) (c-i c \tan (e+f x))^{3/2} \, dx\) [819]

Optimal. Leaf size=279 \[ -\frac {a^{7/2} (5 i A+2 B) c^{3/2} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{4 f}+\frac {a^3 (5 A-2 i B) c \tan (e+f x) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{8 f}+\frac {a^2 (5 i A+2 B) (a+i a \tan (e+f x))^{3/2} (c-i c \tan (e+f x))^{3/2}}{12 f}+\frac {a (5 i A+2 B) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{3/2}}{5 f} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.23, antiderivative size = 279, normalized size of antiderivative = 1.00, number of steps used = 8, 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^{7/2} c^{3/2} (2 B+5 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^3 c (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 {a^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 {a (2 B+5 i A) (a+i a \tan (e+f x))^{5/2} (c-i c \tan (e+f x))^{3/2}}{20 f}+\frac {B (a+i a \tan (e+f x))^{7/2} (c-i c \tan (e+f x))^{3/2}}{5 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

Mathematica [A]
time = 8.72, size = 507, normalized size = 1.82 \begin {gather*} -\frac {i (5 A-2 i B) c^2 e^{-i (4 e+f x)} \sqrt {e^{i f x}} \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} \text {ArcTan}\left (e^{i (e+f x)}\right ) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{4 \sqrt {\frac {c}{1+e^{2 i (e+f x)}}} f \sec ^{\frac {9}{2}}(e+f x) (\cos (f x)+i \sin (f x))^{7/2} (A \cos (e+f x)+B \sin (e+f x))}+\frac {\cos ^4(e+f x) \sqrt {\sec (e+f x) (c \cos (e+f x)-i c \sin (e+f x))} \left (\sec (e) \sec ^2(e+f x) (8 i A \cos (e)+8 B \cos (e)-3 A \sin (e)+6 i B \sin (e)) \left (\frac {1}{12} c \cos (3 e)-\frac {1}{12} i c \sin (3 e)\right )+\sec ^4(e+f x) \left (-\frac {1}{5} B c \cos (3 e)+\frac {1}{5} i B c \sin (3 e)\right )+\sec (e) \sec (e+f x) \left (\frac {1}{8} \cos (3 e)-\frac {1}{8} i \sin (3 e)\right ) (5 A c \sin (f x)-2 i B c \sin (f x))+\sec (e) \sec ^3(e+f x) \left (\frac {1}{4} \cos (3 e)-\frac {1}{4} i \sin (3 e)\right ) (-A c \sin (f x)+2 i B c \sin (f x))+(5 A-2 i B) \left (\frac {1}{8} c \cos (3 e)-\frac {1}{8} i c \sin (3 e)\right ) \tan (e)\right ) (a+i a \tan (e+f x))^{7/2} (A+B \tan (e+f x))}{f (\cos (f x)+i \sin (f x))^3 (A \cos (e+f x)+B \sin (e+f x))} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

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

[Out]

((-1/4*I)*(5*A - (2*I)*B)*c^2*Sqrt[E^(I*f*x)]*Sqrt[E^(I*(e + f*x))/(1 + E^((2*I)*(e + f*x)))]*ArcTan[E^(I*(e +
f*x))]*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(E^(I*(4*e + f*x))*Sqrt[c/(1 + E^((2*I)*(e + f*x)))
]*f*Sec[e + f*x]^(9/2)*(Cos[f*x] + I*Sin[f*x])^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x])) + (Cos[e + f*x]^4*Sqrt
[Sec[e + f*x]*(c*Cos[e + f*x] - I*c*Sin[e + f*x])]*(Sec[e]*Sec[e + f*x]^2*((8*I)*A*Cos[e] + 8*B*Cos[e] - 3*A*S
in[e] + (6*I)*B*Sin[e])*((c*Cos[3*e])/12 - (I/12)*c*Sin[3*e]) + Sec[e + f*x]^4*(-1/5*(B*c*Cos[3*e]) + (I/5)*B*
c*Sin[3*e]) + Sec[e]*Sec[e + f*x]*(Cos[3*e]/8 - (I/8)*Sin[3*e])*(5*A*c*Sin[f*x] - (2*I)*B*c*Sin[f*x]) + Sec[e]
*Sec[e + f*x]^3*(Cos[3*e]/4 - (I/4)*Sin[3*e])*(-(A*c*Sin[f*x]) + (2*I)*B*c*Sin[f*x]) + (5*A - (2*I)*B)*((c*Cos
[3*e])/8 - (I/8)*c*Sin[3*e])*Tan[e])*(a + I*a*Tan[e + f*x])^(7/2)*(A + B*Tan[e + f*x]))/(f*(Cos[f*x] + I*Sin[f
*x])^3*(A*Cos[e + f*x] + B*Sin[e + f*x]))

________________________________________________________________________________________

Maple [A]
time = 0.40, size = 412, normalized size = 1.48

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+I*a*tan(f*x+e))^(7/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(3/2),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)*a^3*c*(60*I*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c
)^(1/2)*tan(f*x+e)^3-24*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^4+80*I*A*(a*c*(1+tan(f*x+e)^2))^
(1/2)*(a*c)^(1/2)*tan(f*x+e)^2-30*A*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(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+tan(f*x+e)^2))^(1/2)*(a*c)^(1/
2)*tan(f*x+e)+32*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^2+80*I*A*(a*c*(1+tan(f*x+e)^2))^(1/2)*(
a*c)^(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+tan
(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)+56*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c)^(1/2)/(a*c*(1+ta
n(f*x+e)^2))^(1/2)

________________________________________________________________________________________

Maxima [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 1747 vs. \(2 (224) = 448\).
time = 2.93, size = 1747, normalized size = 6.26 \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))^(7/2)*(A+B*tan(f*x+e))*(c-I*c*tan(f*x+e))^(3/2),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

Fricas [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 713 vs. \(2 (224) = 448\).
time = 4.76, size = 713, normalized size = 2.56 \begin {gather*} -\frac {15 \, \sqrt {\frac {{\left (25 \, A^{2} - 20 i \, A B - 4 \, B^{2}\right )} a^{7} c^{3}}{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^{3} c e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-5 i \, A - 2 \, B\right )} a^{3} 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 (25 \, A^{2} - 20 i \, A B - 4 \, B^{2}\right )} a^{7} c^{3}}{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^{3} c e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (5 i \, A + 2 \, B\right )} a^{3} c}\right ) - 15 \, \sqrt {\frac {{\left (25 \, A^{2} - 20 i \, A B - 4 \, B^{2}\right )} a^{7} c^{3}}{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^{3} c e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-5 i \, A - 2 \, B\right )} a^{3} 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 (25 \, A^{2} - 20 i \, A B - 4 \, B^{2}\right )} a^{7} c^{3}}{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^{3} c e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (5 i \, A + 2 \, B\right )} a^{3} c}\right ) + 4 \, {\left (15 \, {\left (5 i \, A + 2 \, B\right )} a^{3} c e^{\left (9 i \, f x + 9 i \, e\right )} + 10 \, {\left (-29 i \, A - 50 \, B\right )} a^{3} c e^{\left (7 i \, f x + 7 i \, e\right )} + 128 \, {\left (-5 i \, A - 2 \, B\right )} a^{3} c e^{\left (5 i \, f x + 5 i \, e\right )} + 70 \, {\left (-5 i \, A - 2 \, B\right )} a^{3} c e^{\left (3 i \, f x + 3 i \, e\right )} + 15 \, {\left (-5 i \, A - 2 \, B\right )} a^{3} 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}}}{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 )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/240*(15*sqrt((25*A^2 - 20*I*A*B - 4*B^2)*a^7*c^3/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^3*c*e^(3*I*f*x + 3*I*e) + (-5
*I*A - 2*B)*a^3*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(
(25*A^2 - 20*I*A*B - 4*B^2)*a^7*c^3/f^2)*(f*e^(2*I*f*x + 2*I*e) - f))/((5*I*A + 2*B)*a^3*c*e^(2*I*f*x + 2*I*e)
+ (5*I*A + 2*B)*a^3*c)) - 15*sqrt((25*A^2 - 20*I*A*B - 4*B^2)*a^7*c^3/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^3*c*e^(3*I
*f*x + 3*I*e) + (-5*I*A - 2*B)*a^3*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((25*A^2 - 20*I*A*B - 4*B^2)*a^7*c^3/f^2)*(f*e^(2*I*f*x + 2*I*e) - f))/((5*I*A + 2*B)*a^3*c*
e^(2*I*f*x + 2*I*e) + (5*I*A + 2*B)*a^3*c)) + 4*(15*(5*I*A + 2*B)*a^3*c*e^(9*I*f*x + 9*I*e) + 10*(-29*I*A - 50
*B)*a^3*c*e^(7*I*f*x + 7*I*e) + 128*(-5*I*A - 2*B)*a^3*c*e^(5*I*f*x + 5*I*e) + 70*(-5*I*A - 2*B)*a^3*c*e^(3*I*
f*x + 3*I*e) + 15*(-5*I*A - 2*B)*a^3*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^(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(-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))**(7/2)*(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 7317 deep

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

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 )}^{7/2}\,{\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)^(7/2)*(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)^(7/2)*(c - c*tan(e + f*x)*1i)^(3/2), x)

________________________________________________________________________________________