3.9.10 \(\int \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x))}{(c-i c \tan (e+f x))^{3/2}} \, dx\) [810]

Optimal. Leaf size=226 \[ -\frac {2 a^{5/2} (i A+4 B) \text {ArcTan}\left (\frac {\sqrt {c} \sqrt {a+i a \tan (e+f x)}}{\sqrt {a} \sqrt {c-i c \tan (e+f x)}}\right )}{c^{3/2} f}-\frac {(i A+B) (a+i a \tan (e+f x))^{5/2}}{3 f (c-i c \tan (e+f x))^{3/2}}+\frac {2 a (i A+4 B) (a+i a \tan (e+f x))^{3/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}+\frac {a^2 (i A+4 B) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c^2 f} \]

[Out]

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

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Rubi [A]
time = 0.20, antiderivative size = 226, 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, 79, 49, 52, 65, 223, 209} \begin {gather*} -\frac {2 a^{5/2} (4 B+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 )}{c^{3/2} f}+\frac {a^2 (4 B+i A) \sqrt {a+i a \tan (e+f x)} \sqrt {c-i c \tan (e+f x)}}{c^2 f}+\frac {2 a (4 B+i A) (a+i a \tan (e+f x))^{3/2}}{3 c f \sqrt {c-i c \tan (e+f x)}}-\frac {(B+i A) (a+i a \tan (e+f x))^{5/2}}{3 f (c-i c \tan (e+f x))^{3/2}} \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]

(-2*a^(5/2)*(I*A + 4*B)*ArcTan[(Sqrt[c]*Sqrt[a + I*a*Tan[e + f*x]])/(Sqrt[a]*Sqrt[c - I*c*Tan[e + f*x]])])/(c^
(3/2)*f) - ((I*A + B)*(a + I*a*Tan[e + f*x])^(5/2))/(3*f*(c - I*c*Tan[e + f*x])^(3/2)) + (2*a*(I*A + 4*B)*(a +
I*a*Tan[e + f*x])^(3/2))/(3*c*f*Sqrt[c - I*c*Tan[e + f*x]]) + (a^2*(I*A + 4*B)*Sqrt[a + I*a*Tan[e + f*x]]*Sqr
t[c - I*c*Tan[e + f*x]])/(c^2*f)

Rule 49

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(a + b*x)^(m + 1)*((c + d*x)^n/(b*(
m + 1))), x] - Dist[d*(n/(b*(m + 1))), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

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 79

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

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

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Mathematica [A]
time = 4.91, size = 227, normalized size = 1.00 \begin {gather*} \frac {(a+i a \tan (e+f x))^{5/2} (A+B \tan (e+f x)) \left (-\frac {6 i (A-4 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)}}}}+\sqrt {\sec (e+f x)} (2 i A+8 B+(2 i A+11 B) \cos (2 (e+f x))+(4 A-13 i B) \sin (2 (e+f x))) \sqrt {c-i c \tan (e+f x)}\right )}{3 c^2 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]))/(c - I*c*Tan[e + f*x])^(3/2),x]

[Out]

((a + I*a*Tan[e + f*x])^(5/2)*(A + B*Tan[e + f*x])*(((-6*I)*(A - (4*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)))]) + Sqrt[Sec[e +
f*x]]*((2*I)*A + 8*B + ((2*I)*A + 11*B)*Cos[2*(e + f*x)] + (4*A - (13*I)*B)*Sin[2*(e + f*x)])*Sqrt[c - I*c*Tan
[e + f*x]]))/(3*c^2*f*Sec[e + f*x]^(7/2)*(A*Cos[e + f*x] + B*Sin[e + f*x]))

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Maple [B] Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (186 ) = 372\).
time = 0.42, size = 667, normalized size = 2.95

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} \left (-12 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 \left (\tan ^{3}\left (f x +e \right )\right )+9 i 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 \left (\tan ^{2}\left (f x +e \right )\right )+3 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 \left (\tan ^{3}\left (f x +e \right )\right )+36 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 \tan \left (f x +e \right )+29 i 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 )+36 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 \left (\tan ^{2}\left (f x +e \right )\right )+3 B \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan ^{3}\left (f x +e \right )\right )-3 i 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 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )-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 \tan \left (f x +e \right )-8 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 )-19 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}-12 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 -45 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )+4 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{3 f \,c^{2} \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (i+\tan \left (f x +e \right )\right )^{3}}\)	\(667\)
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} \left (-12 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 \left (\tan ^{3}\left (f x +e \right )\right )+9 i 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 \left (\tan ^{2}\left (f x +e \right )\right )+3 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 \left (\tan ^{3}\left (f x +e \right )\right )+36 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 \tan \left (f x +e \right )+29 i 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 )+36 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 \left (\tan ^{2}\left (f x +e \right )\right )+3 B \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \left (\tan ^{3}\left (f x +e \right )\right )-3 i 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 i A \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \tan \left (f x +e \right )-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 \tan \left (f x +e \right )-8 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 )-19 i B \sqrt {a c}\, \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}-12 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 -45 B \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \tan \left (f x +e \right )+4 A \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\right )}{3 f \,c^{2} \sqrt {a c \left (1+\tan ^{2}\left (f x +e \right )\right )}\, \sqrt {a c}\, \left (i+\tan \left (f x +e \right )\right )^{3}}\)	\(667\)

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]

1/3/f*(a*(1+I*tan(f*x+e)))^(1/2)*(-c*(I*tan(f*x+e)-1))^(1/2)*a^2/c^2*(-12*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*tan(f*x+e)^3+9*I*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*tan(f*x+e)^2+3*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*tan(f*x+e)^3+36*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*tan(f*x+e)+29*I*B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^2+36*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*tan(f*x+e)^2+3*B*(a*c)^(1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*t
an(f*x+e)^3-3*I*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*I*A*(a*c)^(
1/2)*(a*c*(1+tan(f*x+e)^2))^(1/2)*tan(f*x+e)-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*tan(f*x+e)-8*A*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)^2-19*I*B*(a*c*(1+tan(f*x+e
)^2))^(1/2)*(a*c)^(1/2)-12*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-45*
B*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2)*tan(f*x+e)+4*A*(a*c*(1+tan(f*x+e)^2))^(1/2)*(a*c)^(1/2))/(a*c*(1+ta
n(f*x+e)^2))^(1/2)/(a*c)^(1/2)/(I+tan(f*x+e))^3

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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. 841 vs. \(2 (184) = 368\).
time = 0.65, size = 841, normalized size = 3.72 \begin {gather*} -\frac {3 \, {\left (6 \, {\left ({\left (A - 4 i \, B\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right ) - {\left (-i \, A - 4 \, B\right )} a^{2} \sin \left (2 \, f x + 2 \, e\right ) + {\left (A - 4 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 (A - 4 i \, B\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right ) - {\left (-i \, A - 4 \, B\right )} a^{2} \sin \left (2 \, f x + 2 \, e\right ) + {\left (A - 4 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 ) + 4 \, {\left ({\left (A - i \, B\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right ) - {\left (-i \, A - B\right )} a^{2} \sin \left (2 \, f x + 2 \, e\right ) + {\left (A - i \, B\right )} a^{2}\right )} \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 ({\left (A - 3 i \, B\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right ) + {\left (i \, A + 3 \, B\right )} a^{2} \sin \left (2 \, f x + 2 \, e\right ) + {\left (A - 4 i \, B\right )} a^{2}\right )} \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 ) - 3 \, {\left ({\left (-i \, A - 4 \, B\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right ) + {\left (A - 4 i \, B\right )} a^{2} \sin \left (2 \, f x + 2 \, e\right ) + {\left (-i \, A - 4 \, 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 (i \, A + 4 \, B\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right ) - {\left (A - 4 i \, B\right )} a^{2} \sin \left (2 \, f x + 2 \, e\right ) + {\left (i \, A + 4 \, 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 ) - 4 \, {\left ({\left (-i \, A - B\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right ) + {\left (A - i \, B\right )} a^{2} \sin \left (2 \, f x + 2 \, e\right ) + {\left (-i \, A - B\right )} a^{2}\right )} \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 ({\left (i \, A + 3 \, B\right )} a^{2} \cos \left (2 \, f x + 2 \, e\right ) - {\left (A - 3 i \, B\right )} a^{2} \sin \left (2 \, f x + 2 \, e\right ) + {\left (i \, A + 4 \, B\right )} a^{2}\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 )\right )} \sqrt {a} \sqrt {c}}{-18 \, {\left (i \, c^{2} \cos \left (2 \, f x + 2 \, e\right ) - c^{2} \sin \left (2 \, f x + 2 \, e\right ) + i \, c^{2}\right )} f} \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]

-3*(6*((A - 4*I*B)*a^2*cos(2*f*x + 2*e) - (-I*A - 4*B)*a^2*sin(2*f*x + 2*e) + (A - 4*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*(
(A - 4*I*B)*a^2*cos(2*f*x + 2*e) - (-I*A - 4*B)*a^2*sin(2*f*x + 2*e) + (A - 4*I*B)*a^2)*arctan2(cos(1/2*arctan
2(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) + 4*((A - I
*B)*a^2*cos(2*f*x + 2*e) - (-I*A - B)*a^2*sin(2*f*x + 2*e) + (A - I*B)*a^2)*cos(3/2*arctan2(sin(2*f*x + 2*e),
cos(2*f*x + 2*e))) - 12*((A - 3*I*B)*a^2*cos(2*f*x + 2*e) + (I*A + 3*B)*a^2*sin(2*f*x + 2*e) + (A - 4*I*B)*a^2
)*cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 3*((-I*A - 4*B)*a^2*cos(2*f*x + 2*e) + (A - 4*I*B)*a^
2*sin(2*f*x + 2*e) + (-I*A - 4*B)*a^2)*log(cos(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e)))^2 + sin(1/2*ar
ctan2(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
*((I*A + 4*B)*a^2*cos(2*f*x + 2*e) - (A - 4*I*B)*a^2*sin(2*f*x + 2*e) + (I*A + 4*B)*a^2)*log(cos(1/2*arctan2(s
in(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*arc
tan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) + 1) - 4*((-I*A - B)*a^2*cos(2*f*x + 2*e) + (A - I*B)*a^2*sin(2*f*x
+ 2*e) + (-I*A - B)*a^2)*sin(3/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e))) - 12*((I*A + 3*B)*a^2*cos(2*f*x
+ 2*e) - (A - 3*I*B)*a^2*sin(2*f*x + 2*e) + (I*A + 4*B)*a^2)*sin(1/2*arctan2(sin(2*f*x + 2*e), cos(2*f*x + 2*e
))))*sqrt(a)*sqrt(c)/((-18*I*c^2*cos(2*f*x + 2*e) + 18*c^2*sin(2*f*x + 2*e) - 18*I*c^2)*f)

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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. 510 vs. \(2 (184) = 368\).
time = 4.76, size = 510, normalized size = 2.26 \begin {gather*} -\frac {3 \, c^{2} \sqrt {\frac {{\left (A^{2} - 8 i \, A B - 16 \, B^{2}\right )} a^{5}}{c^{3} f^{2}}} f \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-i \, A - 4 \, B\right )} a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-i \, A - 4 \, 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}} + {\left (c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - c^{2} f\right )} \sqrt {\frac {{\left (A^{2} - 8 i \, A B - 16 \, B^{2}\right )} a^{5}}{c^{3} f^{2}}}\right )}}{{\left (i \, A + 4 \, B\right )} a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, A + 4 \, B\right )} a^{2}}\right ) - 3 \, c^{2} \sqrt {\frac {{\left (A^{2} - 8 i \, A B - 16 \, B^{2}\right )} a^{5}}{c^{3} f^{2}}} f \log \left (-\frac {4 \, {\left (2 \, {\left ({\left (-i \, A - 4 \, B\right )} a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + {\left (-i \, A - 4 \, 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}} - {\left (c^{2} f e^{\left (2 i \, f x + 2 i \, e\right )} - c^{2} f\right )} \sqrt {\frac {{\left (A^{2} - 8 i \, A B - 16 \, B^{2}\right )} a^{5}}{c^{3} f^{2}}}\right )}}{{\left (i \, A + 4 \, B\right )} a^{2} e^{\left (2 i \, f x + 2 i \, e\right )} + {\left (i \, A + 4 \, B\right )} a^{2}}\right ) + 4 \, {\left ({\left (i \, A + B\right )} a^{2} e^{\left (5 i \, f x + 5 i \, e\right )} + 2 \, {\left (-i \, A - 4 \, B\right )} a^{2} e^{\left (3 i \, f x + 3 i \, e\right )} + 3 \, {\left (-i \, A - 4 \, 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}}}{6 \, c^{2} f} \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]

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

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (i a \left (\tan {\left (e + f x \right )} - i\right )\right )^{\frac {5}{2}} \left (A + B \tan {\left (e + f x \right )}\right )}{\left (- i c \left (\tan {\left (e + f x \right )} + i\right )\right )^{\frac {3}{2}}}\, dx \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]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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]

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

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\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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