Integrand size = 21, antiderivative size = 184 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\frac {(a+b) \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}-\frac {(a+b) \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2} d}+\frac {(a-b) \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}-\frac {(a-b) \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2} d}+\frac {2 a \sqrt {\tan (c+d x)}}{d}+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d} \]
-1/2*(a+b)*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)-1/2*(a+b)*arctan( 1+2^(1/2)*tan(d*x+c)^(1/2))/d*2^(1/2)+1/4*(a-b)*ln(1-2^(1/2)*tan(d*x+c)^(1 /2)+tan(d*x+c))/d*2^(1/2)-1/4*(a-b)*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+ c))/d*2^(1/2)+2*a*tan(d*x+c)^(1/2)/d+2/3*b*tan(d*x+c)^(3/2)/d
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.51 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\frac {3 \sqrt [4]{-1} (a-i b) \arctan \left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+3 \sqrt [4]{-1} (a+i b) \text {arctanh}\left ((-1)^{3/4} \sqrt {\tan (c+d x)}\right )+2 \sqrt {\tan (c+d x)} (3 a+b \tan (c+d x))}{3 d} \]
(3*(-1)^(1/4)*(a - I*b)*ArcTan[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 3*(-1)^(1/ 4)*(a + I*b)*ArcTanh[(-1)^(3/4)*Sqrt[Tan[c + d*x]]] + 2*Sqrt[Tan[c + d*x]] *(3*a + b*Tan[c + d*x]))/(3*d)
Time = 0.54 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.98, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.714, Rules used = {3042, 4011, 3042, 4011, 3042, 4017, 25, 1482, 1476, 1082, 217, 1479, 25, 27, 1103}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \tan (c+d x)^{3/2} (a+b \tan (c+d x))dx\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \sqrt {\tan (c+d x)} (a \tan (c+d x)-b)dx+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \sqrt {\tan (c+d x)} (a \tan (c+d x)-b)dx+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \int \frac {-a-b \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx+\frac {2 a \sqrt {\tan (c+d x)}}{d}+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {-a-b \tan (c+d x)}{\sqrt {\tan (c+d x)}}dx+\frac {2 a \sqrt {\tan (c+d x)}}{d}+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 4017 |
| \(\displaystyle \frac {2 \int -\frac {a+b \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}+\frac {2 a \sqrt {\tan (c+d x)}}{d}+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \int \frac {a+b \tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}+\frac {2 a \sqrt {\tan (c+d x)}}{d}+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 1482 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} (a-b) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} (a+b) \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}+\frac {2 a \sqrt {\tan (c+d x)}}{d}+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} (a-b) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} (a+b) \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d}+\frac {2 a \sqrt {\tan (c+d x)}}{d}+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} (a-b) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} (a+b) \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \sqrt {\tan (c+d x)}}{d}+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} (a-b) \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} (a+b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \sqrt {\tan (c+d x)}}{d}+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} (a-b) \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \sqrt {\tan (c+d x)}}{d}+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} (a-b) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )-\frac {1}{2} (a+b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \sqrt {\tan (c+d x)}}{d}+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} (a-b) \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )-\frac {1}{2} (a+b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 a \sqrt {\tan (c+d x)}}{d}+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 \left (-\frac {1}{2} (a+b) \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} (a-b) \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}+\frac {2 a \sqrt {\tan (c+d x)}}{d}+\frac {2 b \tan ^{\frac {3}{2}}(c+d x)}{3 d}\) |
(2*(-1/2*((a + b)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]) + Arc Tan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2])) - ((a - b)*(-1/2*Log[1 - Sqr t[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Tan [c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2])))/2))/d + (2*a*Sqrt[Tan[c + d*x]])/ d + (2*b*Tan[c + d*x]^(3/2))/(3*d)
3.6.56.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
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])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*S implify[a*(c/b^2)]}, Simp[-2/b Subst[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])] /; Fre eQ[{a, b, c}, x]
Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> S imp[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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ 2*(d/e), 2]}, Simp[e/(2*c) Int[1/Simp[d/e + q*x + x^2, x], x], x] + Simp[ 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]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ -2*(d/e), 2]}, Simp[e/(2*c*q) Int[(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Simp[e/(2*c*q) Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /; F reeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[ a*c, 2]}, Simp[(d*q + a*e)/(2*a*c) Int[(q + c*x^2)/(a + c*x^4), x], x] + Simp[(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]
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]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[2/f Subst[Int[(b*c + d*x^2)/(b^2 + x^4), x], x, Sq rt[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]
Time = 0.03 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.09
| method | result | size |
| derivativedivides | \(\frac {\frac {2 b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 a \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {a \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}-\frac {b \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(200\) |
| default | \(\frac {\frac {2 b \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}+2 a \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {a \sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}-\frac {b \sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}}{d}\) | \(200\) |
| parts | \(\frac {a \left (2 \left (\sqrt {\tan }\left (d x +c \right )\right )-\frac {\sqrt {2}\, \left (\ln \left (\frac {1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}+\frac {b \left (\frac {2 \left (\tan ^{\frac {3}{2}}\left (d x +c \right )\right )}{3}-\frac {\sqrt {2}\, \left (\ln \left (\frac {1-\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}{1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )+\tan \left (d x +c \right )}\right )+2 \arctan \left (1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )+2 \arctan \left (-1+\sqrt {2}\, \left (\sqrt {\tan }\left (d x +c \right )\right )\right )\right )}{4}\right )}{d}\) | \(204\) |
1/d*(2/3*b*tan(d*x+c)^(3/2)+2*a*tan(d*x+c)^(1/2)-1/4*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)) )-1/4*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))))
Leaf count of result is larger than twice the leaf count of optimal. 592 vs. \(2 (150) = 300\).
Time = 0.24 (sec) , antiderivative size = 592, normalized size of antiderivative = 3.22 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\frac {3 \, d \sqrt {-\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} + 2 \, a b}{d^{2}}} \log \left ({\left (b d^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} + {\left (a^{3} - a b^{2}\right )} d\right )} \sqrt {-\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} + 2 \, a b}{d^{2}}} - {\left (a^{4} - b^{4}\right )} \sqrt {\tan \left (d x + c\right )}\right ) - 3 \, d \sqrt {-\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} + 2 \, a b}{d^{2}}} \log \left (-{\left (b d^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} + {\left (a^{3} - a b^{2}\right )} d\right )} \sqrt {-\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} + 2 \, a b}{d^{2}}} - {\left (a^{4} - b^{4}\right )} \sqrt {\tan \left (d x + c\right )}\right ) - 3 \, d \sqrt {\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} - 2 \, a b}{d^{2}}} \log \left ({\left (b d^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} - {\left (a^{3} - a b^{2}\right )} d\right )} \sqrt {\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} - 2 \, a b}{d^{2}}} - {\left (a^{4} - b^{4}\right )} \sqrt {\tan \left (d x + c\right )}\right ) + 3 \, d \sqrt {\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} - 2 \, a b}{d^{2}}} \log \left (-{\left (b d^{3} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} - {\left (a^{3} - a b^{2}\right )} d\right )} \sqrt {\frac {d^{2} \sqrt {-\frac {a^{4} - 2 \, a^{2} b^{2} + b^{4}}{d^{4}}} - 2 \, a b}{d^{2}}} - {\left (a^{4} - b^{4}\right )} \sqrt {\tan \left (d x + c\right )}\right ) + 4 \, {\left (b \tan \left (d x + c\right ) + 3 \, a\right )} \sqrt {\tan \left (d x + c\right )}}{6 \, d} \]
1/6*(3*d*sqrt(-(d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^4) + 2*a*b)/d^2)*log(( b*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^4) + (a^3 - a*b^2)*d)*sqrt(-(d^2*sqr t(-(a^4 - 2*a^2*b^2 + b^4)/d^4) + 2*a*b)/d^2) - (a^4 - b^4)*sqrt(tan(d*x + c))) - 3*d*sqrt(-(d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^4) + 2*a*b)/d^2)*lo g(-(b*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^4) + (a^3 - a*b^2)*d)*sqrt(-(d^2 *sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^4) + 2*a*b)/d^2) - (a^4 - b^4)*sqrt(tan(d *x + c))) - 3*d*sqrt((d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^4) - 2*a*b)/d^2) *log((b*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^4) - (a^3 - a*b^2)*d)*sqrt((d^ 2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^4) - 2*a*b)/d^2) - (a^4 - b^4)*sqrt(tan( d*x + c))) + 3*d*sqrt((d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^4) - 2*a*b)/d^2 )*log(-(b*d^3*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^4) - (a^3 - a*b^2)*d)*sqrt(( d^2*sqrt(-(a^4 - 2*a^2*b^2 + b^4)/d^4) - 2*a*b)/d^2) - (a^4 - b^4)*sqrt(ta n(d*x + c))) + 4*(b*tan(d*x + c) + 3*a)*sqrt(tan(d*x + c)))/d
\[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\int \left (a + b \tan {\left (c + d x \right )}\right ) \tan ^{\frac {3}{2}}{\left (c + d x \right )}\, dx \]
Time = 0.37 (sec) , antiderivative size = 147, normalized size of antiderivative = 0.80 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=-\frac {6 \, \sqrt {2} {\left (a + b\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 6 \, \sqrt {2} {\left (a + b\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 3 \, \sqrt {2} {\left (a - b\right )} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - 3 \, \sqrt {2} {\left (a - b\right )} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - 8 \, b \tan \left (d x + c\right )^{\frac {3}{2}} - 24 \, a \sqrt {\tan \left (d x + c\right )}}{12 \, d} \]
-1/12*(6*sqrt(2)*(a + b)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c) ))) + 6*sqrt(2)*(a + b)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c) ))) + 3*sqrt(2)*(a - b)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - 3*sqrt(2)*(a - b)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) - 8*b*tan(d*x + c)^(3/2) - 24*a*sqrt(tan(d*x + c)))/d
\[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\int { {\left (b \tan \left (d x + c\right ) + a\right )} \tan \left (d x + c\right )^{\frac {3}{2}} \,d x } \]
Time = 6.02 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.62 \[ \int \tan ^{\frac {3}{2}}(c+d x) (a+b \tan (c+d x)) \, dx=\frac {2\,a\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}}{d}+\frac {2\,b\,{\mathrm {tan}\left (c+d\,x\right )}^{3/2}}{3\,d}-\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{d}+\frac {{\left (-1\right )}^{1/4}\,b\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )}{d}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )\,1{}\mathrm {i}}{d}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left ({\left (-1\right )}^{1/4}\,\sqrt {\mathrm {tan}\left (c+d\,x\right )}\right )\,1{}\mathrm {i}}{d} \]