\(\int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx\) [140]

Optimal result

Integrand size = 36, antiderivative size = 316 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=\frac {((9+5 i) A-(25-21 i) B) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^2 d}-\frac {((9+5 i) A-(25-21 i) B) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^2 d}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((7+2 i) A+(2+23 i) B) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((7+2 i) A+(2+23 i) B) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{\sqrt {2} a^2 d}+\frac {5 (i A-5 B) \sqrt {\tan (c+d x)}}{8 a^2 d}+\frac {(3 A+7 i B) \tan ^{\frac {3}{2}}(c+d x)}{8 a^2 d (1+i \tan (c+d x))}+\frac {(i A-B) \tan ^{\frac {5}{2}}(c+d x)}{4 d (a+i a \tan (c+d x))^2} \]

[Out]

-1/32*((9+5*I)*A+(-25+21*I)*B)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/a^2/d*2^(1/2)-1/32*((9+5*I)*A+(-25+21*I)*B)
*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))/a^2/d*2^(1/2)+(-1/64+1/64*I)*((7+2*I)*A+(2+23*I)*B)*ln(1-2^(1/2)*tan(d*x+c
)^(1/2)+tan(d*x+c))/a^2/d*2^(1/2)+(1/64-1/64*I)*((7+2*I)*A+(2+23*I)*B)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c
))/a^2/d*2^(1/2)+5/8*(I*A-5*B)*tan(d*x+c)^(1/2)/a^2/d+1/8*(3*A+7*I*B)*tan(d*x+c)^(3/2)/a^2/d/(1+I*tan(d*x+c))+
1/4*(I*A-B)*tan(d*x+c)^(5/2)/d/(a+I*a*tan(d*x+c))^2

Rubi [A] (verified)

Time = 0.67 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3676, 3609, 3615, 1182, 1176, 631, 210, 1179, 642} \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=\frac {((9+5 i) A-(25-21 i) B) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^2 d}-\frac {((9+5 i) A-(25-21 i) B) \arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{16 \sqrt {2} a^2 d}+\frac {(3 A+7 i B) \tan ^{\frac {3}{2}}(c+d x)}{8 a^2 d (1+i \tan (c+d x))}+\frac {5 (-5 B+i A) \sqrt {\tan (c+d x)}}{8 a^2 d}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((7+2 i) A+(2+23 i) B) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((7+2 i) A+(2+23 i) B) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^2 d}+\frac {(-B+i A) \tan ^{\frac {5}{2}}(c+d x)}{4 d (a+i a \tan (c+d x))^2} \]

[In]

Int[(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2,x]

[Out]

(((9 + 5*I)*A - (25 - 21*I)*B)*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^2*d) - (((9 + 5*I)*A - (2
5 - 21*I)*B)*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]])/(16*Sqrt[2]*a^2*d) - ((1/32 - I/32)*((7 + 2*I)*A + (2 + 2
3*I)*B)*Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a^2*d) + ((1/32 - I/32)*((7 + 2*I)*A + (2
+ 23*I)*B)*Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]])/(Sqrt[2]*a^2*d) + (5*(I*A - 5*B)*Sqrt[Tan[c +
d*x]])/(8*a^2*d) + ((3*A + (7*I)*B)*Tan[c + d*x]^(3/2))/(8*a^2*d*(1 + I*Tan[c + d*x])) + ((I*A - B)*Tan[c + d*
x]^(5/2))/(4*d*(a + I*a*Tan[c + d*x])^2)

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 3609

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*
((a + b*Tan[e + f*x])^m/(f*m)), x] + Int[(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]

Rule 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[c^2 + d^2, 0]

Rule 3676

Int[((a_) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*tan[(e_
.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-(A*b - a*B))*(a + b*Tan[e + f*x])^m*((c + d*Tan[e + f*x])^n/(2*a*f
*m)), x] + Dist[1/(2*a^2*m), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^(n - 1)*Simp[A*(a*c*m + b*d
*n) - B*(b*c*m + a*d*n) - d*(b*B*(m - n) - a*A*(m + n))*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, A
, B}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] && GtQ[n, 0]

Rubi steps \begin{align*} \text {integral}& = \frac {(i A-B) \tan ^{\frac {5}{2}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {\int \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {5}{2} a (i A-B)+\frac {1}{2} a (A+9 i B) \tan (c+d x)\right )}{a+i a \tan (c+d x)} \, dx}{4 a^2} \\ & = \frac {(3 A+7 i B) \tan ^{\frac {3}{2}}(c+d x)}{8 a^2 d (1+i \tan (c+d x))}+\frac {(i A-B) \tan ^{\frac {5}{2}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\int \sqrt {\tan (c+d x)} \left (-\frac {3}{2} a^2 (3 A+7 i B)+\frac {5}{2} a^2 (i A-5 B) \tan (c+d x)\right ) \, dx}{8 a^4} \\ & = \frac {5 (i A-5 B) \sqrt {\tan (c+d x)}}{8 a^2 d}+\frac {(3 A+7 i B) \tan ^{\frac {3}{2}}(c+d x)}{8 a^2 d (1+i \tan (c+d x))}+\frac {(i A-B) \tan ^{\frac {5}{2}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\int \frac {-\frac {5}{2} a^2 (i A-5 B)-\frac {3}{2} a^2 (3 A+7 i B) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx}{8 a^4} \\ & = \frac {5 (i A-5 B) \sqrt {\tan (c+d x)}}{8 a^2 d}+\frac {(3 A+7 i B) \tan ^{\frac {3}{2}}(c+d x)}{8 a^2 d (1+i \tan (c+d x))}+\frac {(i A-B) \tan ^{\frac {5}{2}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}+\frac {\text {Subst}\left (\int \frac {-\frac {5}{2} a^2 (i A-5 B)-\frac {3}{2} a^2 (3 A+7 i B) x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{4 a^4 d} \\ & = \frac {5 (i A-5 B) \sqrt {\tan (c+d x)}}{8 a^2 d}+\frac {(3 A+7 i B) \tan ^{\frac {3}{2}}(c+d x)}{8 a^2 d (1+i \tan (c+d x))}+\frac {(i A-B) \tan ^{\frac {5}{2}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {((9+5 i) A-(25-21 i) B) \text {Subst}\left (\int \frac {1+x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{16 a^2 d}+\frac {((9-5 i) A+(25+21 i) B) \text {Subst}\left (\int \frac {1-x^2}{1+x^4} \, dx,x,\sqrt {\tan (c+d x)}\right )}{16 a^2 d} \\ & = \frac {5 (i A-5 B) \sqrt {\tan (c+d x)}}{8 a^2 d}+\frac {(3 A+7 i B) \tan ^{\frac {3}{2}}(c+d x)}{8 a^2 d (1+i \tan (c+d x))}+\frac {(i A-B) \tan ^{\frac {5}{2}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {((9+5 i) A-(25-21 i) B) \text {Subst}\left (\int \frac {1}{1-\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{32 a^2 d}-\frac {((9+5 i) A-(25-21 i) B) \text {Subst}\left (\int \frac {1}{1+\sqrt {2} x+x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{32 a^2 d}-\frac {((9-5 i) A+(25+21 i) B) \text {Subst}\left (\int \frac {\sqrt {2}+2 x}{-1-\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{32 \sqrt {2} a^2 d}-\frac {((9-5 i) A+(25+21 i) B) \text {Subst}\left (\int \frac {\sqrt {2}-2 x}{-1+\sqrt {2} x-x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{32 \sqrt {2} a^2 d} \\ & = -\frac {((9-5 i) A+(25+21 i) B) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt {2} a^2 d}+\frac {((9-5 i) A+(25+21 i) B) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt {2} a^2 d}+\frac {5 (i A-5 B) \sqrt {\tan (c+d x)}}{8 a^2 d}+\frac {(3 A+7 i B) \tan ^{\frac {3}{2}}(c+d x)}{8 a^2 d (1+i \tan (c+d x))}+\frac {(i A-B) \tan ^{\frac {5}{2}}(c+d x)}{4 d (a+i a \tan (c+d x))^2}-\frac {((9+5 i) A-(25-21 i) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^2 d}+\frac {((9+5 i) A-(25-21 i) B) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^2 d} \\ & = \frac {((9+5 i) A-(25-21 i) B) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^2 d}-\frac {((9+5 i) A-(25-21 i) B) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^2 d}-\frac {((9-5 i) A+(25+21 i) B) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt {2} a^2 d}+\frac {((9-5 i) A+(25+21 i) B) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{32 \sqrt {2} a^2 d}+\frac {5 (i A-5 B) \sqrt {\tan (c+d x)}}{8 a^2 d}+\frac {(3 A+7 i B) \tan ^{\frac {3}{2}}(c+d x)}{8 a^2 d (1+i \tan (c+d x))}+\frac {(i A-B) \tan ^{\frac {5}{2}}(c+d x)}{4 d (a+i a \tan (c+d x))^2} \\ \end{align*}

Mathematica [A] (verified)

Time = 3.47 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.60 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=\frac {2 \sqrt [4]{-1} (i A+B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+\sqrt [4]{-1} (-7 i A+23 B) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right ) \sec ^2(c+d x) (\cos (2 (c+d x))+i \sin (2 (c+d x)))+\sqrt {\tan (c+d x)} \left (-5 i A+25 B+(7 A+43 i B) \tan (c+d x)-16 B \tan ^2(c+d x)\right )}{8 a^2 d (-i+\tan (c+d x))^2} \]

[In]

Integrate[(Tan[c + d*x]^(5/2)*(A + B*Tan[c + d*x]))/(a + I*a*Tan[c + d*x])^2,x]

[Out]

(2*(-1)^(1/4)*(I*A + B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sec[c + d*x]^2*(Cos[2*(c + d*x)] + I*Sin[2*(c +
d*x)]) + (-1)^(1/4)*((-7*I)*A + 23*B)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]]*Sec[c + d*x]^2*(Cos[2*(c + d*x)]
+ I*Sin[2*(c + d*x)]) + Sqrt[Tan[c + d*x]]*((-5*I)*A + 25*B + (7*A + (43*I)*B)*Tan[c + d*x] - 16*B*Tan[c + d*x
]^2))/(8*a^2*d*(-I + Tan[c + d*x])^2)

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.50

method	result	size
derivativedivides	\(\frac {-2 B \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {i \left (\frac {\left (-\frac {7 i A}{2}+\frac {11 B}{2}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+\left (-\frac {5 A}{2}-\frac {9 i B}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{\left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {\left (7 i A -23 B \right ) \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}\right )}{4}+\frac {i \left (i A +B \right ) \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right )}{2 \sqrt {2}+2 i \sqrt {2}}}{d \,a^{2}}\)	\(159\)
default	\(\frac {-2 B \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {i \left (\frac {\left (-\frac {7 i A}{2}+\frac {11 B}{2}\right ) \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )+\left (-\frac {5 A}{2}-\frac {9 i B}{2}\right ) \left (\sqrt {\tan }\left (d x +c \right )\right )}{\left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {\left (7 i A -23 B \right ) \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right )}{\sqrt {2}-i \sqrt {2}}\right )}{4}+\frac {i \left (i A +B \right ) \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right )}{2 \sqrt {2}+2 i \sqrt {2}}}{d \,a^{2}}\)	\(159\)

[In]

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

[Out]

1/d/a^2*(-2*B*tan(d*x+c)^(1/2)+1/4*I*(((-7/2*I*A+11/2*B)*tan(d*x+c)^(3/2)+(-5/2*A-9/2*I*B)*tan(d*x+c)^(1/2))/(
tan(d*x+c)-I)^2+(7*I*A-23*B)/(2^(1/2)-I*2^(1/2))*arctan(2*tan(d*x+c)^(1/2)/(2^(1/2)-I*2^(1/2))))+1/2*I*(I*A+B)
/(2^(1/2)+I*2^(1/2))*arctan(2*tan(d*x+c)^(1/2)/(2^(1/2)+I*2^(1/2))))

Fricas [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 664 vs. \(2 (235) = 470\).

Time = 0.27 (sec) , antiderivative size = 664, normalized size of antiderivative = 2.10 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=\frac {{\left (2 \, a^{2} d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac {2 \, {\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} + {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{i \, A + B}\right ) - 2 \, a^{2} d \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {2 \, {\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} - {\left (A - i \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{i \, A + B}\right ) + a^{2} d \sqrt {\frac {-49 i \, A^{2} + 322 \, A B + 529 i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (\frac {{\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-49 i \, A^{2} + 322 \, A B + 529 i \, B^{2}}{a^{4} d^{2}}} + 7 \, A + 23 i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - a^{2} d \sqrt {\frac {-49 i \, A^{2} + 322 \, A B + 529 i \, B^{2}}{a^{4} d^{2}}} e^{\left (4 i \, d x + 4 i \, c\right )} \log \left (-\frac {{\left ({\left (a^{2} d e^{\left (2 i \, d x + 2 i \, c\right )} + a^{2} d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}} \sqrt {\frac {-49 i \, A^{2} + 322 \, A B + 529 i \, B^{2}}{a^{4} d^{2}}} - 7 \, A - 23 i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - 2 \, {\left (6 \, {\left (-i \, A + 7 \, B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} - {\left (5 i \, A - 9 \, B\right )} e^{\left (2 i \, d x + 2 i \, c\right )} + i \, A - B\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-4 i \, d x - 4 i \, c\right )}}{32 \, a^{2} d} \]

[In]

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

[Out]

1/32*(2*a^2*d*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^4*d^2))*e^(4*I*d*x + 4*I*c)*log(2*((a^2*d*e^(2*I*d*x + 2*I*c) +
a^2*d)*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^4*d^2)) +
(A - I*B)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c)/(I*A + B)) - 2*a^2*d*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^4*d^2
))*e^(4*I*d*x + 4*I*c)*log(-2*((a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d
*x + 2*I*c) + 1))*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^4*d^2)) - (A - I*B)*e^(2*I*d*x + 2*I*c))*e^(-2*I*d*x - 2*I*c
)/(I*A + B)) + a^2*d*sqrt((-49*I*A^2 + 322*A*B + 529*I*B^2)/(a^4*d^2))*e^(4*I*d*x + 4*I*c)*log(1/8*((a^2*d*e^(
2*I*d*x + 2*I*c) + a^2*d)*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-49*I*A^2 + 322*A
*B + 529*I*B^2)/(a^4*d^2)) + 7*A + 23*I*B)*e^(-2*I*d*x - 2*I*c)/(a^2*d)) - a^2*d*sqrt((-49*I*A^2 + 322*A*B + 5
29*I*B^2)/(a^4*d^2))*e^(4*I*d*x + 4*I*c)*log(-1/8*((a^2*d*e^(2*I*d*x + 2*I*c) + a^2*d)*sqrt((-I*e^(2*I*d*x + 2
*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1))*sqrt((-49*I*A^2 + 322*A*B + 529*I*B^2)/(a^4*d^2)) - 7*A - 23*I*B)*e^(-2*
I*d*x - 2*I*c)/(a^2*d)) - 2*(6*(-I*A + 7*B)*e^(4*I*d*x + 4*I*c) - (5*I*A - 9*B)*e^(2*I*d*x + 2*I*c) + I*A - B)
*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-4*I*d*x - 4*I*c)/(a^2*d)

Sympy [F]

\[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=- \frac {\int \frac {A \tan ^{\frac {5}{2}}{\left (c + d x \right )}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx + \int \frac {B \tan ^{\frac {7}{2}}{\left (c + d x \right )}}{\tan ^{2}{\left (c + d x \right )} - 2 i \tan {\left (c + d x \right )} - 1}\, dx}{a^{2}} \]

[In]

integrate(tan(d*x+c)**(5/2)*(A+B*tan(d*x+c))/(a+I*a*tan(d*x+c))**2,x)

[Out]

-(Integral(A*tan(c + d*x)**(5/2)/(tan(c + d*x)**2 - 2*I*tan(c + d*x) - 1), x) + Integral(B*tan(c + d*x)**(7/2)
/(tan(c + d*x)**2 - 2*I*tan(c + d*x) - 1), x))/a**2

Maxima [F(-2)]

Exception generated. \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=\text {Exception raised: RuntimeError} \]

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: expt: undefined: 0 to a negative exponent.

Giac [A] (verification not implemented)

none

Time = 0.77 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.45 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=-\frac {\left (i + 1\right ) \, \sqrt {2} {\left (7 \, A + 23 i \, B\right )} \arctan \left (\left (\frac {1}{2} i + \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{16 \, a^{2} d} + \frac {\left (i - 1\right ) \, \sqrt {2} {\left (A - i \, B\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{8 \, a^{2} d} - \frac {2 \, B \sqrt {\tan \left (d x + c\right )}}{a^{2} d} + \frac {7 \, A \tan \left (d x + c\right )^{\frac {3}{2}} + 11 i \, B \tan \left (d x + c\right )^{\frac {3}{2}} - 5 i \, A \sqrt {\tan \left (d x + c\right )} + 9 \, B \sqrt {\tan \left (d x + c\right )}}{8 \, a^{2} d {\left (\tan \left (d x + c\right ) - i\right )}^{2}} \]

[In]

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

[Out]

-(1/16*I + 1/16)*sqrt(2)*(7*A + 23*I*B)*arctan((1/2*I + 1/2)*sqrt(2)*sqrt(tan(d*x + c)))/(a^2*d) + (1/8*I - 1/
8)*sqrt(2)*(A - I*B)*arctan(-(1/2*I - 1/2)*sqrt(2)*sqrt(tan(d*x + c)))/(a^2*d) - 2*B*sqrt(tan(d*x + c))/(a^2*d
) + 1/8*(7*A*tan(d*x + c)^(3/2) + 11*I*B*tan(d*x + c)^(3/2) - 5*I*A*sqrt(tan(d*x + c)) + 9*B*sqrt(tan(d*x + c)
))/(a^2*d*(tan(d*x + c) - I)^2)

Mupad [B] (verification not implemented)

Time = 12.04 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.06 \[ \int \frac {\tan ^{\frac {5}{2}}(c+d x) (A+B \tan (c+d x))}{(a+i a \tan (c+d x))^2} \, dx=\frac {\frac {5\,A\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}}{8\,a^2\,d}+\frac {A\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}\,7{}\mathrm {i}}{8\,a^2\,d}}{{\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+2\,\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}}+\frac {-\frac {11\,B\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}}{8\,a^2\,d}+\frac {B\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,9{}\mathrm {i}}{8\,a^2\,d}}{{\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+2\,\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}}+2\,\mathrm {atanh}\left (\frac {8\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {A^2\,1{}\mathrm {i}}{64\,a^4\,d^2}}}{A}\right )\,\sqrt {\frac {A^2\,1{}\mathrm {i}}{64\,a^4\,d^2}}+2\,\mathrm {atanh}\left (\frac {16\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-\frac {A^2\,49{}\mathrm {i}}{256\,a^4\,d^2}}}{7\,A}\right )\,\sqrt {-\frac {A^2\,49{}\mathrm {i}}{256\,a^4\,d^2}}-\frac {2\,B\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}}{a^2\,d}+\mathrm {atan}\left (\frac {8\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-\frac {B^2\,1{}\mathrm {i}}{64\,a^4\,d^2}}}{B}\right )\,\sqrt {-\frac {B^2\,1{}\mathrm {i}}{64\,a^4\,d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {16\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {B^2\,529{}\mathrm {i}}{256\,a^4\,d^2}}}{23\,B}\right )\,\sqrt {\frac {B^2\,529{}\mathrm {i}}{256\,a^4\,d^2}}\,2{}\mathrm {i} \]

[In]

int((tan(c + d*x)^(5/2)*(A + B*tan(c + d*x)))/(a + a*tan(c + d*x)*1i)^2,x)

[Out]

((5*A*tan(c + d*x)^(1/2))/(8*a^2*d) + (A*tan(c + d*x)^(3/2)*7i)/(8*a^2*d))/(2*tan(c + d*x) + tan(c + d*x)^2*1i
- 1i) + ((B*tan(c + d*x)^(1/2)*9i)/(8*a^2*d) - (11*B*tan(c + d*x)^(3/2))/(8*a^2*d))/(2*tan(c + d*x) + tan(c +
d*x)^2*1i - 1i) + 2*atanh((8*a^2*d*tan(c + d*x)^(1/2)*((A^2*1i)/(64*a^4*d^2))^(1/2))/A)*((A^2*1i)/(64*a^4*d^2
))^(1/2) + 2*atanh((16*a^2*d*tan(c + d*x)^(1/2)*(-(A^2*49i)/(256*a^4*d^2))^(1/2))/(7*A))*(-(A^2*49i)/(256*a^4*
d^2))^(1/2) + atan((8*a^2*d*tan(c + d*x)^(1/2)*(-(B^2*1i)/(64*a^4*d^2))^(1/2))/B)*(-(B^2*1i)/(64*a^4*d^2))^(1/
2)*2i - atan((16*a^2*d*tan(c + d*x)^(1/2)*((B^2*529i)/(256*a^4*d^2))^(1/2))/(23*B))*((B^2*529i)/(256*a^4*d^2))
^(1/2)*2i - (2*B*tan(c + d*x)^(1/2))/(a^2*d)