3.9.8 \(\int (a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \sqrt {c-i c \tan (e+f x)} \, dx\) [808]

Optimal. Leaf size=217 \[ -\frac {a^{5/2} (3 i A+2 B) \sqrt {c} \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{f}+\frac {a^2 (3 i A+2 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {a (3 i A+2 B) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 f}+\frac {B (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 f} \]

[Out]

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

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Rubi [A]
time = 0.19, antiderivative size = 217, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 45, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3669, 81, 52, 65, 223, 209} \begin {gather*} -\frac {a^{5/2} \sqrt {c} (2 B+3 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 )}{f}+\frac {a^2 (2 B+3 i A) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{2 f}+\frac {a (2 B+3 i A) (a+i a \tan (e+f x))^{3/2} \sqrt {c-i c \tan (e+f x)}}{6 f}+\frac {B (a+i a \tan (e+f x))^{5/2} \sqrt {c-i c \tan (e+f x)}}{3 f} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 52

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[n*((b*c - a*d)/(b*(m + n + 1))), Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a
, b, c, d}, x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !IntegerQ[n] || (G
tQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinearQ[a, b, c, d, m, n, x]

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

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Mathematica [A]
time = 3.95, size = 253, normalized size = 1.17 \begin {gather*} \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \left (-\frac {i (3 A-2 i B) c e^{-3 i (e+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 )}{\sqrt {\frac {c}{1+e^{2 i (e+f x)}}}}+\frac {\sec ^{\frac {5}{2}}(e+f x) (i \cos (2 e)+\sin (2 e)) (12 A-8 i B+12 (A-i B) \cos (2 (e+f x))+(3 i A+6 B) \sin (2 (e+f x))) \sqrt {c-i c \tan (e+f x)}}{12 (\cos (f x)+i \sin (f x))^2}\right )}{f \sec ^{\frac {7}{2}}(e+f x) (A \cos (e+f x)+B \sin (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])*Sqrt[c - I*c*Tan[e + f*x]],x]

[Out]

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

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Maple [A]
time = 0.41, size = 285, normalized size = 1.31

method	result	size
derivativedivides	\(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, a^{2} \left (-6 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c +6 i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )-2 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+12 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}+9 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c -3 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )+10 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{6 f \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}\)	\(285\)
default	\(\frac {\sqrt {-c \left (i \tan \left (f x +e \right )-1\right )}\, \sqrt {a \left (1+i \tan \left (f x +e \right )\right )}\, a^{2} \left (-6 i B \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c +6 i B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )-2 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (\tan ^{2}\left (f x +e \right )\right )+12 i A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}+9 A \ln \left (\frac {a c \tan \left (f x +e \right )+\sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}}{\sqrt {a c}}\right ) a c -3 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )+10 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{6 f \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}}\)	\(285\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c-I*c*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)),x,method=_RETURNVERBOSE)

[Out]

1/6/f*(-c*(I*tan(f*x+e)-1))^(1/2)*(a*(1+I*tan(f*x+e)))^(1/2)*a^2*(-6*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+6*I*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)-2*B*(a*c*(1
+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^2+12*I*A*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)+9*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-3*A*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2
)*tan(f*x+e)+10*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c*(1+tan(f*x+e)^2))^(1/2)/(a*c)^(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. 1149 vs. \(2 (173) = 346\).
time = 0.87, size = 1149, normalized size = 5.29 \begin {gather*} \frac {6 \, {\left (12 \, {\left (5 \, A - 6 i \, B\right )} a^{2} \cos \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 32 \, {\left (3 \, A - 2 i \, B\right )} a^{2} \cos \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 12 \, {\left (3 \, A - 2 i \, B\right )} a^{2} \cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 12 \, {\left (5 i \, A + 6 \, B\right )} a^{2} \sin \left (\frac {5}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 32 \, {\left (3 i \, A + 2 \, B\right )} a^{2} \sin \left (\frac {3}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 12 \, {\left (3 i \, A + 2 \, B\right )} a^{2} \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) - 6 \, {\left ({\left (3 \, A - 2 i \, B\right )} a^{2} \cos \left (6 \, f x + 6 \, e\right ) + 3 \, {\left (3 \, A - 2 i \, B\right )} a^{2} \cos \left (4 \, f x + 4 \, e\right ) + 3 \, {\left (3 \, A - 2 i \, B\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right ) - {\left (-3 i \, A - 2 \, B\right )} a^{2} \sin \left (6 \, f x + 6 \, e\right ) - 3 \, {\left (-3 i \, A - 2 \, B\right )} a^{2} \sin \left (4 \, f x + 4 \, e\right ) - 3 \, {\left (-3 i \, A - 2 \, B\right )} a^{2} \sin \left (2 \, f x + 2 \, e\right ) + {\left (3 \, A - 2 i \, B\right )} a^{2}\right )} \arctan \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) - 6 \, {\left ({\left (3 \, A - 2 i \, B\right )} a^{2} \cos \left (6 \, f x + 6 \, e\right ) + 3 \, {\left (3 \, A - 2 i \, B\right )} a^{2} \cos \left (4 \, f x + 4 \, e\right ) + 3 \, {\left (3 \, A - 2 i \, B\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right ) - {\left (-3 i \, A - 2 \, B\right )} a^{2} \sin \left (6 \, f x + 6 \, e\right ) - 3 \, {\left (-3 i \, A - 2 \, B\right )} a^{2} \sin \left (4 \, f x + 4 \, e\right ) - 3 \, {\left (-3 i \, A - 2 \, B\right )} a^{2} \sin \left (2 \, f x + 2 \, e\right ) + {\left (3 \, A - 2 i \, B\right )} a^{2}\right )} \arctan \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ), -\sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 3 \, {\left ({\left (-3 i \, A - 2 \, B\right )} a^{2} \cos \left (6 \, f x + 6 \, e\right ) + 3 \, {\left (-3 i \, A - 2 \, B\right )} a^{2} \cos \left (4 \, f x + 4 \, e\right ) + 3 \, {\left (-3 i \, A - 2 \, B\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right ) + {\left (3 \, A - 2 i \, B\right )} a^{2} \sin \left (6 \, f x + 6 \, e\right ) + 3 \, {\left (3 \, A - 2 i \, B\right )} a^{2} \sin \left (4 \, f x + 4 \, e\right ) + 3 \, {\left (3 \, A - 2 i \, B\right )} a^{2} \sin \left (2 \, f x + 2 \, e\right ) + {\left (-3 i \, A - 2 \, B\right )} a^{2}\right )} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right ) + 3 \, {\left ({\left (3 i \, A + 2 \, B\right )} a^{2} \cos \left (6 \, f x + 6 \, e\right ) + 3 \, {\left (3 i \, A + 2 \, B\right )} a^{2} \cos \left (4 \, f x + 4 \, e\right ) + 3 \, {\left (3 i \, A + 2 \, B\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right ) - {\left (3 \, A - 2 i \, B\right )} a^{2} \sin \left (6 \, f x + 6 \, e\right ) - 3 \, {\left (3 \, A - 2 i \, B\right )} a^{2} \sin \left (4 \, f x + 4 \, e\right ) - 3 \, {\left (3 \, A - 2 i \, B\right )} a^{2} \sin \left (2 \, f x + 2 \, e\right ) + {\left (3 i \, A + 2 \, B\right )} a^{2}\right )} \log \left (\cos \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} + \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right )^{2} - 2 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (2 \, f x + 2 \, e\right ), \cos \left (2 \, f x + 2 \, e\right )\right )\right ) + 1\right )\right )} \sqrt {a} \sqrt {c}}{-72 \, f {\left (i \, \cos \left (6 \, f x + 6 \, e\right ) + 3 i \, \cos \left (4 \, f x + 4 \, e\right ) + 3 i \, \cos \left (2 \, f x + 2 \, e\right ) - \sin \left (6 \, f x + 6 \, e\right ) - 3 \, \sin \left (4 \, f x + 4 \, e\right ) - 3 \, \sin \left (2 \, f x + 2 \, e\right ) + i\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-I*c*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)),x, algorithm="maxima")

[Out]

6*(12*(5*A - 6*I*B)*a^2*cos(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 32*(3*A - 2*I*B)*a^2*cos(3/2*ar
ctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 12*(3*A - 2*I*B)*a^2*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x
+ 2*e))) + 12*(5*I*A + 6*B)*a^2*sin(5/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 32*(3*I*A + 2*B)*a^2*si
n(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 12*(3*I*A + 2*B)*a^2*sin(1/2*arctan2(sin(2*f*x + 2*e), co
s(2*f*x + 2*e))) - 6*((3*A - 2*I*B)*a^2*cos(6*f*x + 6*e) + 3*(3*A - 2*I*B)*a^2*cos(4*f*x + 4*e) + 3*(3*A - 2*I
*B)*a^2*cos(2*f*x + 2*e) - (-3*I*A - 2*B)*a^2*sin(6*f*x + 6*e) - 3*(-3*I*A - 2*B)*a^2*sin(4*f*x + 4*e) - 3*(-3
*I*A - 2*B)*a^2*sin(2*f*x + 2*e) + (3*A - 2*I*B)*a^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) - 6*((3*A - 2*I*B)*a^2*cos(6*f*x + 6*e) + 3*(3
*A - 2*I*B)*a^2*cos(4*f*x + 4*e) + 3*(3*A - 2*I*B)*a^2*cos(2*f*x + 2*e) - (-3*I*A - 2*B)*a^2*sin(6*f*x + 6*e)
- 3*(-3*I*A - 2*B)*a^2*sin(4*f*x + 4*e) - 3*(-3*I*A - 2*B)*a^2*sin(2*f*x + 2*e) + (3*A - 2*I*B)*a^2)*arctan2(c
os(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
) + 3*((-3*I*A - 2*B)*a^2*cos(6*f*x + 6*e) + 3*(-3*I*A - 2*B)*a^2*cos(4*f*x + 4*e) + 3*(-3*I*A - 2*B)*a^2*cos(
2*f*x + 2*e) + (3*A - 2*I*B)*a^2*sin(6*f*x + 6*e) + 3*(3*A - 2*I*B)*a^2*sin(4*f*x + 4*e) + 3*(3*A - 2*I*B)*a^2
*sin(2*f*x + 2*e) + (-3*I*A - 2*B)*a^2)*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*a
rctan2(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*((3*I*A + 2*B)*a^2*cos(6*f*x + 6*e) + 3*(3*I*A + 2*B)*a^2*cos(4*f*x + 4*e) + 3*(3*I*A + 2*B)*a^2*cos(2*f*x +
2*e) - (3*A - 2*I*B)*a^2*sin(6*f*x + 6*e) - 3*(3*A - 2*I*B)*a^2*sin(4*f*x + 4*e) - 3*(3*A - 2*I*B)*a^2*sin(2*
f*x + 2*e) + (3*I*A + 2*B)*a^2)*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*arctan2(s
in(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)*s
qrt(c)/(f*(-72*I*cos(6*f*x + 6*e) - 216*I*cos(4*f*x + 4*e) - 216*I*cos(2*f*x + 2*e) + 72*sin(6*f*x + 6*e) + 21
6*sin(4*f*x + 4*e) + 216*sin(2*f*x + 2*e) - 72*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. 570 vs. \(2 (173) = 346\).
time = 2.33, size = 570, normalized size = 2.63 \begin {gather*} -\frac {3 \, \sqrt {\frac {{\left (9 \, A^{2} - 12 i \, A B - 4 \, B^{2}\right )} a^{5} c}{f^{2}}} {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-3 i \, A - 2 \, B\right )} a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-3 i \, A - 2 \, B\right )} a^{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 (9 \, A^{2} - 12 i \, A B - 4 \, B^{2}\right )} a^{5} c}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (3 i \, A + 2 \, B\right )} a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (3 i \, A + 2 \, B\right )} a^{2}}\right ) - 3 \, \sqrt {\frac {{\left (9 \, A^{2} - 12 i \, A B - 4 \, B^{2}\right )} a^{5} c}{f^{2}}} {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, f e^{\left (2 i \, f x + 2 i \, e\right )} + f\right )} \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-3 i \, A - 2 \, B\right )} a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-3 i \, A - 2 \, B\right )} a^{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 (9 \, A^{2} - 12 i \, A B - 4 \, B^{2}\right )} a^{5} c}{f^{2}}} {\left (f e^{\left (2 i \, f x + 2 i \, e\right )} - f\right )}\right )}}{{\left (3 i \, A + 2 \, B\right )} a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (3 i \, A + 2 \, B\right )} a^{2}}\right ) + 4 \, {\left (3 \, {\left (-5 i \, A - 6 \, B\right )} a^{2} e^{\left (5 i \, f x + 5 i \, e\right )} + 8 \, {\left (-3 i \, A - 2 \, B\right )} a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + 3 \, {\left (-3 i \, A - 2 \, B\right )} a^{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}}}{12 \, {\left (f e^{\left (4 i \, f x + 4 i \, e\right )} + 2 \, 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((c-I*c*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)),x, algorithm="fricas")

[Out]

-1/12*(3*sqrt((9*A^2 - 12*I*A*B - 4*B^2)*a^5*c/f^2)*(f*e^(4*I*f*x + 4*I*e) + 2*f*e^(2*I*f*x + 2*I*e) + f)*log(
-4*(2*((-3*I*A - 2*B)*a^2*e^(3*I*f*x + 3*I*e) + (-3*I*A - 2*B)*a^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((9*A^2 - 12*I*A*B - 4*B^2)*a^5*c/f^2)*(f*e^(2*I*f*x + 2*I*e)
- f))/((3*I*A + 2*B)*a^2*e^(2*I*f*x + 2*I*e) + (3*I*A + 2*B)*a^2)) - 3*sqrt((9*A^2 - 12*I*A*B - 4*B^2)*a^5*c/f
^2)*(f*e^(4*I*f*x + 4*I*e) + 2*f*e^(2*I*f*x + 2*I*e) + f)*log(-4*(2*((-3*I*A - 2*B)*a^2*e^(3*I*f*x + 3*I*e) +
(-3*I*A - 2*B)*a^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
((9*A^2 - 12*I*A*B - 4*B^2)*a^5*c/f^2)*(f*e^(2*I*f*x + 2*I*e) - f))/((3*I*A + 2*B)*a^2*e^(2*I*f*x + 2*I*e) + (
3*I*A + 2*B)*a^2)) + 4*(3*(-5*I*A - 6*B)*a^2*e^(5*I*f*x + 5*I*e) + 8*(-3*I*A - 2*B)*a^2*e^(3*I*f*x + 3*I*e) +
3*(-3*I*A - 2*B)*a^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^(4*I*f*x + 4*I*e) + 2*f*e^(2*I*f*x + 2*I*e) + f)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}} \sqrt {- i c \left (\tan {\left (e + f x \right )} + i\right )} \left (A + B \tan {\left (e + f x \right )}\right )\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c-I*c*tan(f*x+e))**(1/2)*(a+I*a*tan(f*x+e))**(5/2)*(A+B*tan(f*x+e)),x)

[Out]

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

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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((c-I*c*tan(f*x+e))^(1/2)*(a+I*a*tan(f*x+e))^(5/2)*(A+B*tan(f*x+e)),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}\,\sqrt {c-c\,\mathrm {tan}\left (e+f\,x\right )\,1{}\mathrm {i}} \,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)^(1/2),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)^(1/2), x)

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