3.144 \(\int \frac {A+B \tan (c+d x)}{\tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2} \, dx\)

Optimal. Leaf size=318 \[ \frac {((25+21 i) A-(9-5 i) B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^2 d}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((2+23 i) A-(7+2 i) B) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^2 d}-\frac {5 (5 A+i B)}{8 a^2 d \sqrt {\tan (c+d x)}}+\frac {7 A+3 i B}{8 a^2 d (1+i \tan (c+d x)) \sqrt {\tan (c+d x)}}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((23+2 i) A+(2+7 i) B) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((23+2 i) A+(2+7 i) B) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^2 d}+\frac {A+i B}{4 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2} \]

[Out]

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

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Rubi [A]  time = 0.58, antiderivative size = 318, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3596, 3529, 3534, 1168, 1162, 617, 204, 1165, 628} \[ \frac {((25+21 i) A-(9-5 i) B) \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{16 \sqrt {2} a^2 d}-\frac {\left (\frac {1}{16}-\frac {i}{16}\right ) ((2+23 i) A-(7+2 i) B) \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^2 d}-\frac {5 (5 A+i B)}{8 a^2 d \sqrt {\tan (c+d x)}}+\frac {7 A+3 i B}{8 a^2 d (1+i \tan (c+d x)) \sqrt {\tan (c+d x)}}-\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((23+2 i) A+(2+7 i) B) \log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^2 d}+\frac {\left (\frac {1}{32}-\frac {i}{32}\right ) ((23+2 i) A+(2+7 i) B) \log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2} a^2 d}+\frac {A+i B}{4 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2),x]

[Out]

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

Rule 204

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

Rule 617

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 628

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 1162

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 1165

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 1168

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 3529

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

Rule 3534

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 3596

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*A + b*B)*(a + b*Tan[e + f*x])^m*(c + d*Tan[e + f*x])^(n + 1))/(2
*f*m*(b*c - a*d)), x] + Dist[1/(2*a*m*(b*c - a*d)), Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Si
mp[A*(b*c*m - a*d*(2*m + n + 1)) + B*(a*c*m - b*d*(n + 1)) + d*(A*b - a*B)*(m + n + 1)*Tan[e + f*x], x], x], x
] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 + b^2, 0] && LtQ[m, 0] &&  !GtQ[n,
0]

Rubi steps

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

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Mathematica [A]  time = 2.63, size = 250, normalized size = 0.79 \[ \frac {\sec (c+d x) (\cos (d x)+i \sin (d x))^2 (A+B \tan (c+d x)) \left ((-2 \cos (2 d x)+2 i \sin (2 d x)) ((-7 B+43 i A) \sin (2 (c+d x))+(41 A+5 i B) \cos (2 (c+d x))-9 A-5 i B)+(-\sin (2 c)+i \cos (2 c)) \sqrt {\sin (2 (c+d x))} \sec (c+d x) \left (((21-25 i) A+(5+9 i) B) \sin ^{-1}(\cos (c+d x)-\sin (c+d x))-(1+i) ((23+2 i) A+(2+7 i) B) \log \left (\sin (c+d x)+\sqrt {\sin (2 (c+d x))}+\cos (c+d x)\right )\right )\right )}{32 d \sqrt {\tan (c+d x)} (a+i a \tan (c+d x))^2 (A \cos (c+d x)+B \sin (c+d x))} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*Tan[c + d*x])/(Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2),x]

[Out]

(Sec[c + d*x]*(Cos[d*x] + I*Sin[d*x])^2*((((21 - 25*I)*A + (5 + 9*I)*B)*ArcSin[Cos[c + d*x] - Sin[c + d*x]] -
(1 + I)*((23 + 2*I)*A + (2 + 7*I)*B)*Log[Cos[c + d*x] + Sin[c + d*x] + Sqrt[Sin[2*(c + d*x)]]])*Sec[c + d*x]*(
I*Cos[2*c] - Sin[2*c])*Sqrt[Sin[2*(c + d*x)]] + (-2*Cos[2*d*x] + (2*I)*Sin[2*d*x])*(-9*A - (5*I)*B + (41*A + (
5*I)*B)*Cos[2*(c + d*x)] + ((43*I)*A - 7*B)*Sin[2*(c + d*x)]))*(A + B*Tan[c + d*x]))/(32*d*(A*Cos[c + d*x] + B
*Sin[c + d*x])*Sqrt[Tan[c + d*x]]*(a + I*a*Tan[c + d*x])^2)

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fricas [B]  time = 0.72, size = 762, normalized size = 2.40 \[ \frac {2 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} \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 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {i \, A^{2} + 2 \, A B - i \, B^{2}}{a^{4} d^{2}}} \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 ) + {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {-529 i \, A^{2} + 322 \, A B + 49 i \, B^{2}}{a^{4} d^{2}}} \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 {-529 i \, A^{2} + 322 \, A B + 49 i \, B^{2}}{a^{4} d^{2}}} + 23 \, A + 7 i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) - {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )} \sqrt {\frac {-529 i \, A^{2} + 322 \, A B + 49 i \, B^{2}}{a^{4} d^{2}}} \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 {-529 i \, A^{2} + 322 \, A B + 49 i \, B^{2}}{a^{4} d^{2}}} - 23 \, A - 7 i \, B\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{8 \, a^{2} d}\right ) + 2 \, {\left ({\left (-42 i \, A + 6 \, B\right )} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (-33 i \, A + B\right )} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (10 i \, A - 6 \, 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}}}{32 \, {\left (a^{2} d e^{\left (6 i \, d x + 6 i \, c\right )} - a^{2} d e^{\left (4 i \, d x + 4 i \, c\right )}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/32*(2*(a^2*d*e^(6*I*d*x + 6*I*c) - a^2*d*e^(4*I*d*x + 4*I*c))*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^4*d^2))*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*e^(6*I
*d*x + 6*I*c) - a^2*d*e^(4*I*d*x + 4*I*c))*sqrt((I*A^2 + 2*A*B - I*B^2)/(a^4*d^2))*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*e^(6*I*d*x + 6*I*c) - a^2*d*
e^(4*I*d*x + 4*I*c))*sqrt((-529*I*A^2 + 322*A*B + 49*I*B^2)/(a^4*d^2))*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((-529*I*A^2 + 322*A*B + 49*I*B^2)/(a^4
*d^2)) + 23*A + 7*I*B)*e^(-2*I*d*x - 2*I*c)/(a^2*d)) - (a^2*d*e^(6*I*d*x + 6*I*c) - a^2*d*e^(4*I*d*x + 4*I*c))
*sqrt((-529*I*A^2 + 322*A*B + 49*I*B^2)/(a^4*d^2))*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((-529*I*A^2 + 322*A*B + 49*I*B^2)/(a^4*d^2)) - 23*A - 7*I
*B)*e^(-2*I*d*x - 2*I*c)/(a^2*d)) + 2*((-42*I*A + 6*B)*e^(6*I*d*x + 6*I*c) + (-33*I*A + B)*e^(4*I*d*x + 4*I*c)
+ (10*I*A - 6*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)))
/(a^2*d*e^(6*I*d*x + 6*I*c) - a^2*d*e^(4*I*d*x + 4*I*c))

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {B \tan \left (d x + c\right ) + A}{{\left (i \, a \tan \left (d x + c\right ) + a\right )}^{2} \tan \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate((B*tan(d*x + c) + A)/((I*a*tan(d*x + c) + a)^2*tan(d*x + c)^(3/2)), x)

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maple [A]  time = 0.36, size = 311, normalized size = 0.98 \[ -\frac {\arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right ) A}{2 d \,a^{2} \left (\sqrt {2}+i \sqrt {2}\right )}+\frac {i \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}+i \sqrt {2}}\right ) B}{2 d \,a^{2} \left (\sqrt {2}+i \sqrt {2}\right )}-\frac {2 A}{d \,a^{2} \sqrt {\tan \left (d x +c \right )}}-\frac {9 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) A}{8 d \,a^{2} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {5 i \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right ) B}{8 d \,a^{2} \left (\tan \left (d x +c \right )-i\right )^{2}}+\frac {11 i \left (\sqrt {\tan }\left (d x +c \right )\right ) A}{8 d \,a^{2} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {7 \left (\sqrt {\tan }\left (d x +c \right )\right ) B}{8 d \,a^{2} \left (\tan \left (d x +c \right )-i\right )^{2}}-\frac {7 i \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right ) B}{4 d \,a^{2} \left (\sqrt {2}-i \sqrt {2}\right )}-\frac {23 \arctan \left (\frac {2 \left (\sqrt {\tan }\left (d x +c \right )\right )}{\sqrt {2}-i \sqrt {2}}\right ) A}{4 d \,a^{2} \left (\sqrt {2}-i \sqrt {2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A+B*tan(d*x+c))/tan(d*x+c)^(3/2)/(a+I*a*tan(d*x+c))^2,x)

[Out]

-1/2/d/a^2/(2^(1/2)+I*2^(1/2))*arctan(2*tan(d*x+c)^(1/2)/(2^(1/2)+I*2^(1/2)))*A+1/2*I/d/a^2/(2^(1/2)+I*2^(1/2)
)*arctan(2*tan(d*x+c)^(1/2)/(2^(1/2)+I*2^(1/2)))*B-2/d/a^2*A/tan(d*x+c)^(1/2)-9/8/d/a^2/(tan(d*x+c)-I)^2*tan(d
*x+c)^(3/2)*A-5/8*I/d/a^2/(tan(d*x+c)-I)^2*tan(d*x+c)^(3/2)*B+11/8*I/d/a^2/(tan(d*x+c)-I)^2*tan(d*x+c)^(1/2)*A
-7/8/d/a^2/(tan(d*x+c)-I)^2*tan(d*x+c)^(1/2)*B-7/4*I/d/a^2/(2^(1/2)-I*2^(1/2))*arctan(2*tan(d*x+c)^(1/2)/(2^(1
/2)-I*2^(1/2)))*B-23/4/d/a^2/(2^(1/2)-I*2^(1/2))*arctan(2*tan(d*x+c)^(1/2)/(2^(1/2)-I*2^(1/2)))*A

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: RuntimeError >> ECL says: Error executing code in Maxima: expt: undefined: 0 to a negative e
xponent.

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mupad [B]  time = 9.83, size = 338, normalized size = 1.06 \[ -\frac {-\frac {5\,B\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}}{8\,a^2\,d}+\frac {B\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,7{}\mathrm {i}}{8\,a^2\,d}}{{\mathrm {tan}\left (c+d\,x\right )}^2\,1{}\mathrm {i}+2\,\mathrm {tan}\left (c+d\,x\right )-\mathrm {i}}+2\,\mathrm {atanh}\left (\frac {8\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {A^2\,1{}\mathrm {i}}{64\,a^4\,d^2}}}{A}\right )\,\sqrt {\frac {A^2\,1{}\mathrm {i}}{64\,a^4\,d^2}}+2\,\mathrm {atanh}\left (\frac {16\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-\frac {A^2\,529{}\mathrm {i}}{256\,a^4\,d^2}}}{23\,A}\right )\,\sqrt {-\frac {A^2\,529{}\mathrm {i}}{256\,a^4\,d^2}}+\mathrm {atan}\left (\frac {8\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {-\frac {B^2\,1{}\mathrm {i}}{64\,a^4\,d^2}}}{B}\right )\,\sqrt {-\frac {B^2\,1{}\mathrm {i}}{64\,a^4\,d^2}}\,2{}\mathrm {i}-\mathrm {atan}\left (\frac {16\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\sqrt {\frac {B^2\,49{}\mathrm {i}}{256\,a^4\,d^2}}}{7\,B}\right )\,\sqrt {\frac {B^2\,49{}\mathrm {i}}{256\,a^4\,d^2}}\,2{}\mathrm {i}-\frac {\frac {43\,A\,\mathrm {tan}\left (c+d\,x\right )}{8\,a^2\,d}-\frac {A\,2{}\mathrm {i}}{a^2\,d}+\frac {A\,{\mathrm {tan}\left (c+d\,x\right )}^2\,25{}\mathrm {i}}{8\,a^2\,d}}{2\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}-\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,1{}\mathrm {i}+{\mathrm {tan}\left (c+d\,x\right )}^{5/2}\,1{}\mathrm {i}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*tan(c + d*x))/(tan(c + d*x)^(3/2)*(a + a*tan(c + d*x)*1i)^2),x)

[Out]

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) - ((B*tan(
c + d*x)^(1/2)*7i)/(8*a^2*d) - (5*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((16*a^2*d*tan(c + d*x)^(1/2)*(-(A^2*529i)/(256*a^4*d^2))^(1/2))/(23*A))*(-(A^2*529i)/(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*49i)/(256*a^4*d^2))^(1/2))/(7*B))*((B^2*49i)/(256*a^4*d^2))^(1/2)*2i
- ((43*A*tan(c + d*x))/(8*a^2*d) - (A*2i)/(a^2*d) + (A*tan(c + d*x)^2*25i)/(8*a^2*d))/(2*tan(c + d*x)^(3/2) -
tan(c + d*x)^(1/2)*1i + tan(c + d*x)^(5/2)*1i)

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sympy [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((A+B*tan(d*x+c))/tan(d*x+c)**(3/2)/(a+I*a*tan(d*x+c))**2,x)

[Out]

Exception raised: TypeError

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