\(\int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx\) [122]

Optimal result

Integrand size = 36, antiderivative size = 104 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=-\frac {4 \sqrt [4]{-1} a^2 (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}-\frac {2 a^2 (3 A-5 i B) \sqrt {\tan (c+d x)}}{3 d}+\frac {2 i B \sqrt {\tan (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.32 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {3675, 3673, 3614, 211} \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=-\frac {4 \sqrt [4]{-1} a^2 (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}-\frac {2 a^2 (3 A-5 i B) \sqrt {\tan (c+d x)}}{3 d}+\frac {2 i B \sqrt {\tan (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d} \]

[In]

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

[Out]

(-4*(-1)^(1/4)*a^2*(A - I*B)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]])/d - (2*a^2*(3*A - (5*I)*B)*Sqrt[Tan[c + d*
x]])/(3*d) + (((2*I)/3)*B*Sqrt[Tan[c + d*x]]*(a^2 + I*a^2*Tan[c + d*x]))/d

Rule 211

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

Rule 3614

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2*(c^2/f), S
ubst[Int[1/(b*c - d*x^2), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 + d^2, 0]

Rule 3673

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

Rule 3675

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

Rubi steps \begin{align*} \text {integral}& = \frac {2 i B \sqrt {\tan (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d}+\frac {2}{3} \int \frac {(a+i a \tan (c+d x)) \left (\frac {1}{2} a (3 A-i B)+\frac {1}{2} a (3 i A+5 B) \tan (c+d x)\right )}{\sqrt {\tan (c+d x)}} \, dx \\ & = -\frac {2 a^2 (3 A-5 i B) \sqrt {\tan (c+d x)}}{3 d}+\frac {2 i B \sqrt {\tan (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d}+\frac {2}{3} \int \frac {3 a^2 (A-i B)+3 a^2 (i A+B) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx \\ & = -\frac {2 a^2 (3 A-5 i B) \sqrt {\tan (c+d x)}}{3 d}+\frac {2 i B \sqrt {\tan (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d}+\frac {\left (12 a^4 (A-i B)^2\right ) \text {Subst}\left (\int \frac {1}{3 a^2 (A-i B)-3 a^2 (i A+B) x^2} \, dx,x,\sqrt {\tan (c+d x)}\right )}{d} \\ & = -\frac {4 \sqrt [4]{-1} a^2 (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}-\frac {2 a^2 (3 A-5 i B) \sqrt {\tan (c+d x)}}{3 d}+\frac {2 i B \sqrt {\tan (c+d x)} \left (a^2+i a^2 \tan (c+d x)\right )}{3 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 1.97 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.72 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=-\frac {2 a^2 \left ((3+3 i) \sqrt {2} (i A+B) \text {arctanh}\left (\frac {(1+i) \sqrt {\tan (c+d x)}}{\sqrt {2}}\right )+\sqrt {\tan (c+d x)} (3 A-6 i B+B \tan (c+d x))\right )}{3 d} \]

[In]

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

[Out]

(-2*a^2*((3 + 3*I)*Sqrt[2]*(I*A + B)*ArcTanh[((1 + I)*Sqrt[Tan[c + d*x]])/Sqrt[2]] + Sqrt[Tan[c + d*x]]*(3*A -
(6*I)*B + B*Tan[c + d*x])))/(3*d)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 228 vs. \(2 (86 ) = 172\).

Time = 0.03 (sec) , antiderivative size = 229, normalized size of antiderivative = 2.20

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\frac {2 B \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-2 A \left (\sqrt {\tan }\left (d x +c \right )\right )+4 i B \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\left (-2 i B +2 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (2 i A +2 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\)	\(229\)
default	\(\frac {a^{2} \left (-\frac {2 B \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-2 A \left (\sqrt {\tan }\left (d x +c \right )\right )+4 i B \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\left (-2 i B +2 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (2 i A +2 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\)	\(229\)
parts	\(\frac {\left (2 i A \,a^{2}+B \,a^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4 d}+\frac {\left (2 i B \,a^{2}-A \,a^{2}\right ) \left (2 \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}+\frac {A \,a^{2} \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4 d}-\frac {B \,a^{2} \left (\frac {2 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\)	\(414\)

[In]

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

[Out]

1/d*a^2*(-2/3*B*tan(d*x+c)^(3/2)-2*A*tan(d*x+c)^(1/2)+4*I*B*tan(d*x+c)^(1/2)+1/4*(2*A-2*I*B)*2^(1/2)*(ln((1+2^
(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2
))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2)))+1/4*(2*B+2*I*A)*2^(1/2)*(ln((1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))/
(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c)))+2*arctan(1+2^(1/2)*tan(d*x+c)^(1/2))+2*arctan(-1+2^(1/2)*tan(d*x+c)^(
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. 389 vs. \(2 (82) = 164\).

Time = 0.25 (sec) , antiderivative size = 389, normalized size of antiderivative = 3.74 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=\frac {3 \, \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (-\frac {2 \, {\left ({\left (A - i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (i \, d e^{\left (2 i \, d x + 2 i \, c\right )} + i \, d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{2}}\right ) - 3 \, \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )} \log \left (-\frac {2 \, {\left ({\left (A - i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (-i \, d e^{\left (2 i \, d x + 2 i \, c\right )} - i \, d\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}\right )} e^{\left (-2 i \, d x - 2 i \, c\right )}}{{\left (-i \, A - B\right )} a^{2}}\right ) - 2 \, {\left ({\left (3 \, A - 7 i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (3 \, A - 5 i \, B\right )} a^{2}\right )} \sqrt {\frac {-i \, e^{\left (2 i \, d x + 2 i \, c\right )} + i}{e^{\left (2 i \, d x + 2 i \, c\right )} + 1}}}{3 \, {\left (d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

integrate((a+I*a*tan(d*x+c))^2*(A+B*tan(d*x+c))/tan(d*x+c)^(1/2),x, algorithm="fricas")

[Out]

1/3*(3*sqrt(-(I*A^2 + 2*A*B - I*B^2)*a^4/d^2)*(d*e^(2*I*d*x + 2*I*c) + d)*log(-2*((A - I*B)*a^2*e^(2*I*d*x + 2
*I*c) + sqrt(-(I*A^2 + 2*A*B - I*B^2)*a^4/d^2)*(I*d*e^(2*I*d*x + 2*I*c) + I*d)*sqrt((-I*e^(2*I*d*x + 2*I*c) +
I)/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-2*I*d*x - 2*I*c)/((-I*A - B)*a^2)) - 3*sqrt(-(I*A^2 + 2*A*B - I*B^2)*a^4/d^
2)*(d*e^(2*I*d*x + 2*I*c) + d)*log(-2*((A - I*B)*a^2*e^(2*I*d*x + 2*I*c) + sqrt(-(I*A^2 + 2*A*B - I*B^2)*a^4/d
^2)*(-I*d*e^(2*I*d*x + 2*I*c) - I*d)*sqrt((-I*e^(2*I*d*x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1)))*e^(-2*I*d*x
- 2*I*c)/((-I*A - B)*a^2)) - 2*((3*A - 7*I*B)*a^2*e^(2*I*d*x + 2*I*c) + (3*A - 5*I*B)*a^2)*sqrt((-I*e^(2*I*d*
x + 2*I*c) + I)/(e^(2*I*d*x + 2*I*c) + 1)))/(d*e^(2*I*d*x + 2*I*c) + d)

Sympy [F]

\[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=- a^{2} \left (\int \left (- \frac {A}{\sqrt {\tan {\left (c + d x \right )}}}\right )\, dx + \int A \tan ^{\frac {3}{2}}{\left (c + d x \right )}\, dx + \int \left (- B \sqrt {\tan {\left (c + d x \right )}}\right )\, dx + \int B \tan ^{\frac {5}{2}}{\left (c + d x \right )}\, dx + \int \left (- 2 i A \sqrt {\tan {\left (c + d x \right )}}\right )\, dx + \int \left (- 2 i B \tan ^{\frac {3}{2}}{\left (c + d x \right )}\right )\, dx\right ) \]

[In]

integrate((a+I*a*tan(d*x+c))**2*(A+B*tan(d*x+c))/tan(d*x+c)**(1/2),x)

[Out]

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

Maxima [B] (verification not implemented)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 174 vs. \(2 (82) = 164\).

Time = 0.28 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.67 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=-\frac {4 \, B a^{2} \tan \left (d x + c\right )^{\frac {3}{2}} + 12 \, {\left (A - 2 i \, B\right )} a^{2} \sqrt {\tan \left (d x + c\right )} + 3 \, {\left (2 \, \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} a^{2}}{6 \, d} \]

[In]

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

[Out]

-1/6*(4*B*a^2*tan(d*x + c)^(3/2) + 12*(A - 2*I*B)*a^2*sqrt(tan(d*x + c)) + 3*(2*sqrt(2)*(-(I + 1)*A + (I - 1)*
B)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*(-(I + 1)*A + (I - 1)*B)*arctan(-1/2*sqrt(
2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) + sqrt(2)*((I - 1)*A + (I + 1)*B)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*
x + c) + 1) - sqrt(2)*((I - 1)*A + (I + 1)*B)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))*a^2)/d

Giac [A] (verification not implemented)

none

Time = 0.76 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.88 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=\frac {\left (2 i - 2\right ) \, \sqrt {2} {\left (-i \, A a^{2} - B a^{2}\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{d} - \frac {2 \, {\left (B a^{2} d^{2} \tan \left (d x + c\right )^{\frac {3}{2}} + 3 \, A a^{2} d^{2} \sqrt {\tan \left (d x + c\right )} - 6 i \, B a^{2} d^{2} \sqrt {\tan \left (d x + c\right )}\right )}}{3 \, d^{3}} \]

[In]

integrate((a+I*a*tan(d*x+c))^2*(A+B*tan(d*x+c))/tan(d*x+c)^(1/2),x, algorithm="giac")

[Out]

(2*I - 2)*sqrt(2)*(-I*A*a^2 - B*a^2)*arctan(-(1/2*I - 1/2)*sqrt(2)*sqrt(tan(d*x + c)))/d - 2/3*(B*a^2*d^2*tan(
d*x + c)^(3/2) + 3*A*a^2*d^2*sqrt(tan(d*x + c)) - 6*I*B*a^2*d^2*sqrt(tan(d*x + c)))/d^3

Mupad [B] (verification not implemented)

Time = 8.85 (sec) , antiderivative size = 221, normalized size of antiderivative = 2.12 \[ \int \frac {(a+i a \tan (c+d x))^2 (A+B \tan (c+d x))}{\sqrt {\tan (c+d x)}} \, dx=-\frac {2\,A\,a^2\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}}{d}+\frac {B\,a^2\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,4{}\mathrm {i}}{d}-\frac {2\,B\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}}{3\,d}+\frac {\sqrt {2}\,A\,a^2\,\ln \left (A\,a^2\,d\,4{}\mathrm {i}+\sqrt {2}\,A\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\left (-2+2{}\mathrm {i}\right )\right )\,\left (1-\mathrm {i}\right )}{d}-\frac {\sqrt {-4{}\mathrm {i}}\,A\,a^2\,\ln \left (A\,a^2\,d\,4{}\mathrm {i}+2\,\sqrt {-4{}\mathrm {i}}\,A\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{d}+\frac {\sqrt {2}\,B\,a^2\,\ln \left (4\,B\,a^2\,d+\sqrt {2}\,B\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\left (-2-2{}\mathrm {i}\right )\right )\,\left (1+1{}\mathrm {i}\right )}{d}-\frac {\sqrt {4{}\mathrm {i}}\,B\,a^2\,\ln \left (4\,B\,a^2\,d+2\,\sqrt {4{}\mathrm {i}}\,B\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{d} \]

[In]

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

[Out]

(B*a^2*tan(c + d*x)^(1/2)*4i)/d - (2*A*a^2*tan(c + d*x)^(1/2))/d - (2*B*a^2*tan(c + d*x)^(3/2))/(3*d) + (2^(1/
2)*A*a^2*log(A*a^2*d*4i - 2^(1/2)*A*a^2*d*tan(c + d*x)^(1/2)*(2 - 2i))*(1 - 1i))/d - ((-4i)^(1/2)*A*a^2*log(A*
a^2*d*4i + 2*(-4i)^(1/2)*A*a^2*d*tan(c + d*x)^(1/2)))/d + (2^(1/2)*B*a^2*log(4*B*a^2*d - 2^(1/2)*B*a^2*d*tan(c
+ d*x)^(1/2)*(2 + 2i))*(1 + 1i))/d - (4i^(1/2)*B*a^2*log(4*B*a^2*d + 2*4i^(1/2)*B*a^2*d*tan(c + d*x)^(1/2)))/
d