\(\int \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx\) [120]

Optimal result

Integrand size = 36, antiderivative size = 156 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {4 \sqrt [4]{-1} a^2 (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}+\frac {4 a^2 (A-i B) \sqrt {\tan (c+d x)}}{d}+\frac {4 a^2 (i A+B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 A-9 i B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 i B \tan ^{\frac {5}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d} \]

[Out]

4*(-1)^(1/4)*a^2*(A-I*B)*arctan((-1)^(3/4)*tan(d*x+c)^(1/2))/d+4*a^2*(A-I*B)*tan(d*x+c)^(1/2)/d+4/3*a^2*(I*A+B
)*tan(d*x+c)^(3/2)/d-2/35*a^2*(7*A-9*I*B)*tan(d*x+c)^(5/2)/d+2/7*I*B*tan(d*x+c)^(5/2)*(a^2+I*a^2*tan(d*x+c))/d

Rubi [A] (verified)

Time = 0.50 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.139, Rules used = {3675, 3673, 3609, 3614, 211} \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {4 \sqrt [4]{-1} a^2 (A-i B) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )}{d}-\frac {2 a^2 (7 A-9 i B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {4 a^2 (B+i A) \tan ^{\frac {3}{2}}(c+d x)}{3 d}+\frac {4 a^2 (A-i B) \sqrt {\tan (c+d x)}}{d}+\frac {2 i B \tan ^{\frac {5}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d} \]

[In]

Int[Tan[c + d*x]^(3/2)*(a + I*a*Tan[c + d*x])^2*(A + B*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 + (4*a^2*(A - I*B)*Sqrt[Tan[c + d*x]])/d
+ (4*a^2*(I*A + B)*Tan[c + d*x]^(3/2))/(3*d) - (2*a^2*(7*A - (9*I)*B)*Tan[c + d*x]^(5/2))/(35*d) + (((2*I)/7)*
B*Tan[c + d*x]^(5/2)*(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 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 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 \tan ^{\frac {5}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d}+\frac {2}{7} \int \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x)) \left (\frac {1}{2} a (7 A-5 i B)+\frac {1}{2} a (7 i A+9 B) \tan (c+d x)\right ) \, dx \\ & = -\frac {2 a^2 (7 A-9 i B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 i B \tan ^{\frac {5}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d}+\frac {2}{7} \int \tan ^{\frac {3}{2}}(c+d x) \left (7 a^2 (A-i B)+7 a^2 (i A+B) \tan (c+d x)\right ) \, dx \\ & = \frac {4 a^2 (i A+B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 A-9 i B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 i B \tan ^{\frac {5}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d}+\frac {2}{7} \int \sqrt {\tan (c+d x)} \left (-7 a^2 (i A+B)+7 a^2 (A-i B) \tan (c+d x)\right ) \, dx \\ & = \frac {4 a^2 (A-i B) \sqrt {\tan (c+d x)}}{d}+\frac {4 a^2 (i A+B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 A-9 i B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 i B \tan ^{\frac {5}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d}+\frac {2}{7} \int \frac {-7 a^2 (A-i B)-7 a^2 (i A+B) \tan (c+d x)}{\sqrt {\tan (c+d x)}} \, dx \\ & = \frac {4 a^2 (A-i B) \sqrt {\tan (c+d x)}}{d}+\frac {4 a^2 (i A+B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 A-9 i B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 i B \tan ^{\frac {5}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d}+\frac {\left (28 a^4 (A-i B)^2\right ) \text {Subst}\left (\int \frac {1}{-7 a^2 (A-i B)+7 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 {4 a^2 (A-i B) \sqrt {\tan (c+d x)}}{d}+\frac {4 a^2 (i A+B) \tan ^{\frac {3}{2}}(c+d x)}{3 d}-\frac {2 a^2 (7 A-9 i B) \tan ^{\frac {5}{2}}(c+d x)}{35 d}+\frac {2 i B \tan ^{\frac {5}{2}}(c+d x) \left (a^2+i a^2 \tan (c+d x)\right )}{7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 2.13 (sec) , antiderivative size = 111, normalized size of antiderivative = 0.71 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {2 a^2 \left ((105+105 i) \sqrt {2} (i A+B) \text {arctanh}\left (\frac {(1+i) \sqrt {\tan (c+d x)}}{\sqrt {2}}\right )+\sqrt {\tan (c+d x)} \left (210 (A-i B)+70 (i A+B) \tan (c+d x)-21 (A-2 i B) \tan ^2(c+d x)-15 B \tan ^3(c+d x)\right )\right )}{105 d} \]

[In]

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

[Out]

(2*a^2*((105 + 105*I)*Sqrt[2]*(I*A + B)*ArcTanh[((1 + I)*Sqrt[Tan[c + d*x]])/Sqrt[2]] + Sqrt[Tan[c + d*x]]*(21
0*(A - I*B) + 70*(I*A + B)*Tan[c + d*x] - 21*(A - (2*I)*B)*Tan[c + d*x]^2 - 15*B*Tan[c + d*x]^3)))/(105*d)

Maple [B] (verified)

Both result and optimal contain complex but leaf count of result is larger than twice the leaf count of optimal. 274 vs. \(2 (130 ) = 260\).

Time = 0.04 (sec) , antiderivative size = 275, normalized size of antiderivative = 1.76

method	result	size
derivativedivides	\(\frac {a^{2} \left (-\frac {2 B \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}+\frac {4 i B \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}-\frac {2 A \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {4 i A \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+\frac {4 B \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-4 i B \left (\sqrt {\tan }\left (d x +c \right )\right )+4 A \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\left (2 i B -2 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-2 i A -2 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\)	\(275\)
default	\(\frac {a^{2} \left (-\frac {2 B \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}+\frac {4 i B \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}-\frac {2 A \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}+\frac {4 i A \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+\frac {4 B \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-4 i B \left (\sqrt {\tan }\left (d x +c \right )\right )+4 A \left (\sqrt {\tan }\left (d x +c \right )\right )+\frac {\left (2 i B -2 A \right ) \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}+\frac {\left (-2 i A -2 B \right ) \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\)	\(275\)
parts	\(\frac {\left (2 i A \,a^{2}+B \,a^{2}\right ) \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}+\frac {\left (2 i B \,a^{2}-A \,a^{2}\right ) \left (\frac {2 \left (\tan ^{\frac {5}{2}}\left (d x +c \right )\right )}{5}-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} \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 {B \,a^{2} \left (\frac {2 \left (\tan ^{\frac {7}{2}}\left (d x +c \right )\right )}{7}-\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}\)	\(458\)

[In]

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

[Out]

1/d*a^2*(-2/7*B*tan(d*x+c)^(7/2)+4/5*I*B*tan(d*x+c)^(5/2)-2/5*A*tan(d*x+c)^(5/2)+4/3*I*A*tan(d*x+c)^(3/2)+4/3*
B*tan(d*x+c)^(3/2)-4*I*B*tan(d*x+c)^(1/2)+4*A*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)*t
an(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. 501 vs. \(2 (124) = 248\).

Time = 0.29 (sec) , antiderivative size = 501, normalized size of antiderivative = 3.21 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {105 \, \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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 ) - 105 \, \sqrt {-\frac {{\left (i \, A^{2} + 2 \, A B - i \, B^{2}\right )} a^{4}}{d^{2}}} {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, 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 (301 \, A - 337 i \, B\right )} a^{2} e^{\left (6 i \, d x + 6 i \, c\right )} + {\left (679 \, A - 613 i \, B\right )} a^{2} e^{\left (4 i \, d x + 4 i \, c\right )} + {\left (539 \, A - 563 i \, B\right )} a^{2} e^{\left (2 i \, d x + 2 i \, c\right )} + {\left (161 \, A - 167 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}}}{105 \, {\left (d e^{\left (6 i \, d x + 6 i \, c\right )} + 3 \, d e^{\left (4 i \, d x + 4 i \, c\right )} + 3 \, d e^{\left (2 i \, d x + 2 i \, c\right )} + d\right )}} \]

[In]

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

[Out]

-1/105*(105*sqrt(-(I*A^2 + 2*A*B - I*B^2)*a^4/d^2)*(d*e^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*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)) - 105*sqrt(-(I*A^2 + 2*A*B - I*B^2)*a^4/d^2)*(d*e^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c)
+ 3*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*((301*A - 337*I*B)*a^2*e^(6*I*d*x + 6*I*c) + (679*A - 613*I*B)*a^2*e^(4*I*d*x
+ 4*I*c) + (539*A - 563*I*B)*a^2*e^(2*I*d*x + 2*I*c) + (161*A - 167*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^(6*I*d*x + 6*I*c) + 3*d*e^(4*I*d*x + 4*I*c) + 3*d*e^(2*I*d*x + 2*I*c) + d)

Sympy [F]

\[ \int \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=- a^{2} \left (\int \left (- A \tan ^{\frac {3}{2}}{\left (c + d x \right )}\right )\, dx + \int A \tan ^{\frac {7}{2}}{\left (c + d x \right )}\, dx + \int \left (- B \tan ^{\frac {5}{2}}{\left (c + d x \right )}\right )\, dx + \int B \tan ^{\frac {9}{2}}{\left (c + d x \right )}\, dx + \int \left (- 2 i A \tan ^{\frac {5}{2}}{\left (c + d x \right )}\right )\, dx + \int \left (- 2 i B \tan ^{\frac {7}{2}}{\left (c + d x \right )}\right )\, dx\right ) \]

[In]

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

[Out]

-a**2*(Integral(-A*tan(c + d*x)**(3/2), x) + Integral(A*tan(c + d*x)**(7/2), x) + Integral(-B*tan(c + d*x)**(5
/2), x) + Integral(B*tan(c + d*x)**(9/2), x) + Integral(-2*I*A*tan(c + d*x)**(5/2), x) + Integral(-2*I*B*tan(c
+ d*x)**(7/2), x))

Maxima [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.36 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=-\frac {60 \, B a^{2} \tan \left (d x + c\right )^{\frac {7}{2}} + 84 \, {\left (A - 2 i \, B\right )} a^{2} \tan \left (d x + c\right )^{\frac {5}{2}} + 280 \, {\left (-i \, A - B\right )} a^{2} \tan \left (d x + c\right )^{\frac {3}{2}} - 840 \, {\left (A - i \, B\right )} a^{2} \sqrt {\tan \left (d x + c\right )} - 105 \, {\left (2 \, \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} {\left (-\left (i + 1\right ) \, A + \left (i - 1\right ) \, B\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} {\left (\left (i - 1\right ) \, A + \left (i + 1\right ) \, B\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} a^{2}}{210 \, d} \]

[In]

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

[Out]

-1/210*(60*B*a^2*tan(d*x + c)^(7/2) + 84*(A - 2*I*B)*a^2*tan(d*x + c)^(5/2) + 280*(-I*A - B)*a^2*tan(d*x + c)^
(3/2) - 840*(A - I*B)*a^2*sqrt(tan(d*x + c)) - 105*(2*sqrt(2)*(-(I + 1)*A + (I - 1)*B)*arctan(1/2*sqrt(2)*(sqr
t(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 = 0.96 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.03 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {\left (2 i - 2\right ) \, \sqrt {2} {\left (i \, A a^{2} + B a^{2}\right )} \arctan \left (-\left (\frac {1}{2} i - \frac {1}{2}\right ) \, \sqrt {2} \sqrt {\tan \left (d x + c\right )}\right )}{d} - \frac {2 \, {\left (15 \, B a^{2} d^{6} \tan \left (d x + c\right )^{\frac {7}{2}} + 21 \, A a^{2} d^{6} \tan \left (d x + c\right )^{\frac {5}{2}} - 42 i \, B a^{2} d^{6} \tan \left (d x + c\right )^{\frac {5}{2}} - 70 i \, A a^{2} d^{6} \tan \left (d x + c\right )^{\frac {3}{2}} - 70 \, B a^{2} d^{6} \tan \left (d x + c\right )^{\frac {3}{2}} - 210 \, A a^{2} d^{6} \sqrt {\tan \left (d x + c\right )} + 210 i \, B a^{2} d^{6} \sqrt {\tan \left (d x + c\right )}\right )}}{105 \, d^{7}} \]

[In]

integrate(tan(d*x+c)^(3/2)*(a+I*a*tan(d*x+c))^2*(A+B*tan(d*x+c)),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 - 2/105*(15*B*a^2*d^6*
tan(d*x + c)^(7/2) + 21*A*a^2*d^6*tan(d*x + c)^(5/2) - 42*I*B*a^2*d^6*tan(d*x + c)^(5/2) - 70*I*A*a^2*d^6*tan(
d*x + c)^(3/2) - 70*B*a^2*d^6*tan(d*x + c)^(3/2) - 210*A*a^2*d^6*sqrt(tan(d*x + c)) + 210*I*B*a^2*d^6*sqrt(tan
(d*x + c)))/d^7

Mupad [B] (verification not implemented)

Time = 10.81 (sec) , antiderivative size = 291, normalized size of antiderivative = 1.87 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+i a \tan (c+d x))^2 (A+B \tan (c+d x)) \, dx=\frac {4\,A\,a^2\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}}{d}+\frac {A\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}\,4{}\mathrm {i}}{3\,d}-\frac {2\,A\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^{5/2}}{5\,d}-\frac {B\,a^2\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,4{}\mathrm {i}}{d}+\frac {4\,B\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}}{3\,d}+\frac {B\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^{5/2}\,4{}\mathrm {i}}{5\,d}-\frac {2\,B\,a^2\,{\mathrm {tan}\left (c+d\,x\right )}^{7/2}}{7\,d}+\frac {\sqrt {2}\,A\,a^2\,\ln \left (-A\,a^2\,d\,4{}\mathrm {i}+\sqrt {2}\,A\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\left (-2+2{}\mathrm {i}\right )\right )\,\left (1-\mathrm {i}\right )}{d}-\frac {\sqrt {-4{}\mathrm {i}}\,A\,a^2\,\ln \left (-A\,a^2\,d\,4{}\mathrm {i}+2\,\sqrt {-4{}\mathrm {i}}\,A\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{d}+\frac {\sqrt {2}\,B\,a^2\,\ln \left (-4\,B\,a^2\,d+\sqrt {2}\,B\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\,\left (-2-2{}\mathrm {i}\right )\right )\,\left (1+1{}\mathrm {i}\right )}{d}-\frac {\sqrt {4{}\mathrm {i}}\,B\,a^2\,\ln \left (-4\,B\,a^2\,d+2\,\sqrt {4{}\mathrm {i}}\,B\,a^2\,d\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{d} \]

[In]

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

[Out]

(4*A*a^2*tan(c + d*x)^(1/2))/d + (A*a^2*tan(c + d*x)^(3/2)*4i)/(3*d) - (2*A*a^2*tan(c + d*x)^(5/2))/(5*d) - (B
*a^2*tan(c + d*x)^(1/2)*4i)/d + (4*B*a^2*tan(c + d*x)^(3/2))/(3*d) + (B*a^2*tan(c + d*x)^(5/2)*4i)/(5*d) - (2*
B*a^2*tan(c + d*x)^(7/2))/(7*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(2*(-4i)^(1/2)*A*a^2*d*tan(c + d*x)^(1/2) - A*a^2*d*4i))/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(2*4i^(1/
2)*B*a^2*d*tan(c + d*x)^(1/2) - 4*B*a^2*d))/d