Integrand size = 12, antiderivative size = 212 \[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b d^{3/2}}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (a+b x)}}{\sqrt {d}}\right )}{\sqrt {2} b d^{3/2}}-\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)-\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b d^{3/2}}+\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (a+b x)+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{2 \sqrt {2} b d^{3/2}}-\frac {2}{b d \sqrt {d \tan (a+b x)}} \]
1/2*arctan(1-2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))/b/d^(3/2)*2^(1/2)-1/2*a rctan(1+2^(1/2)*(d*tan(b*x+a))^(1/2)/d^(1/2))/b/d^(3/2)*2^(1/2)-1/4*ln(d^( 1/2)-2^(1/2)*(d*tan(b*x+a))^(1/2)+d^(1/2)*tan(b*x+a))/b/d^(3/2)*2^(1/2)+1/ 4*ln(d^(1/2)+2^(1/2)*(d*tan(b*x+a))^(1/2)+d^(1/2)*tan(b*x+a))/b/d^(3/2)*2^ (1/2)-2/b/d/(d*tan(b*x+a))^(1/2)
Time = 0.09 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.39 \[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=\frac {-2-\arctan \left (\sqrt [4]{-\tan ^2(a+b x)}\right ) \sqrt [4]{-\tan ^2(a+b x)}+\text {arctanh}\left (\sqrt [4]{-\tan ^2(a+b x)}\right ) \sqrt [4]{-\tan ^2(a+b x)}}{b d \sqrt {d \tan (a+b x)}} \]
(-2 - ArcTan[(-Tan[a + b*x]^2)^(1/4)]*(-Tan[a + b*x]^2)^(1/4) + ArcTanh[(- Tan[a + b*x]^2)^(1/4)]*(-Tan[a + b*x]^2)^(1/4))/(b*d*Sqrt[d*Tan[a + b*x]])
Time = 0.46 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.93, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.083, Rules used = {3042, 3955, 3042, 3957, 266, 826, 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.
| \(\displaystyle \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{(d \tan (a+b x))^{3/2}}dx\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle -\frac {\int \sqrt {d \tan (a+b x)}dx}{d^2}-\frac {2}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle -\frac {\int \sqrt {d \tan (a+b x)}dx}{d^2}-\frac {2}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle -\frac {\int \frac {\sqrt {d \tan (a+b x)}}{\tan ^2(a+b x) d^2+d^2}d(d \tan (a+b x))}{b d}-\frac {2}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle -\frac {2 \int \frac {d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}}{b d}-\frac {2}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \int \frac {d^2 \tan ^2(a+b x)+d}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )}{b d}-\frac {2}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}+\frac {1}{2} \int \frac {1}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )}{b d}-\frac {2}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-d^2 \tan ^2(a+b x)-1}d\left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \tan ^2(a+b x)-1}d\left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )}{b d}-\frac {2}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \tan ^2(a+b x)}{d^4 \tan ^4(a+b x)+d^2}d\sqrt {d \tan (a+b x)}\right )}{b d}-\frac {2}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{b d}-\frac {2}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}\right )}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{b d}-\frac {2}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)-\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \tan (a+b x)}}{d^2 \tan ^2(a+b x)+\sqrt {2} d^{3/2} \tan (a+b x)+d}d\sqrt {d \tan (a+b x)}}{2 \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{b d}-\frac {2}{b d \sqrt {d \tan (a+b x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle -\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (a+b x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (a+b x)\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} d^{3/2} \tan (a+b x)+d^2 \tan ^2(a+b x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (\sqrt {2} d^{3/2} \tan (a+b x)+d^2 \tan ^2(a+b x)+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )}{b d}-\frac {2}{b d \sqrt {d \tan (a+b x)}}\) |
(-2*((-(ArcTan[1 - Sqrt[2]*Sqrt[d]*Tan[a + b*x]]/(Sqrt[2]*Sqrt[d])) + ArcT an[1 + Sqrt[2]*Sqrt[d]*Tan[a + b*x]]/(Sqrt[2]*Sqrt[d]))/2 + (Log[d - Sqrt[ 2]*d^(3/2)*Tan[a + b*x] + d^2*Tan[a + b*x]^2]/(2*Sqrt[2]*Sqrt[d]) - Log[d + Sqrt[2]*d^(3/2)*Tan[a + b*x] + d^2*Tan[a + b*x]^2]/(2*Sqrt[2]*Sqrt[d]))/ 2))/(b*d) - 2/(b*d*Sqrt[d*Tan[a + b*x]])
3.3.61.3.1 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[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> With[{k = De nominator[m]}, Simp[k/c Subst[Int[x^(k*(m + 1) - 1)*(a + b*(x^(2*k)/c^2)) ^p, x], x, (c*x)^(1/k)], x]] /; FreeQ[{a, b, c, p}, x] && FractionQ[m] && I ntBinomialQ[a, b, c, 2, m, p, x]
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2]], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*s) Int[(r + s*x^2)/(a + b*x^ 4), x], x] - Simp[1/(2*s) Int[(r - s*x^2)/(a + b*x^4), x], x]] /; FreeQ[{ a, b}, x] && (GtQ[a/b, 0] || (PosQ[a/b] && AtomQ[SplitProduct[SumBaseQ, a]] && AtomQ[SplitProduct[SumBaseQ, b]]))
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[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[(b*Tan[c + d*x] )^(n + 1)/(b*d*(n + 1)), x] - Simp[1/b^2 Int[(b*Tan[c + d*x])^(n + 2), x] , x] /; FreeQ[{b, c, d}, x] && LtQ[n, -1]
Int[((b_.)*tan[(c_.) + (d_.)*(x_)])^(n_), x_Symbol] :> Simp[b/d Subst[Int [x^n/(b^2 + x^2), x], x, b*Tan[c + d*x]], x] /; FreeQ[{b, c, d, n}, x] && !IntegerQ[n]
Time = 0.16 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.74
| method | result | size |
| derivativedivides | \(\frac {2 d \left (-\frac {1}{d^{2} \sqrt {d \tan \left (b x +a \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (b x +a \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (b x +a \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (b x +a \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (b x +a \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d^{2} \left (d^{2}\right )^{\frac {1}{4}}}\right )}{b}\) | \(157\) |
| default | \(\frac {2 d \left (-\frac {1}{d^{2} \sqrt {d \tan \left (b x +a \right )}}-\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (b x +a \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (b x +a \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (b x +a \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (b x +a \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (b x +a \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d^{2} \left (d^{2}\right )^{\frac {1}{4}}}\right )}{b}\) | \(157\) |
2/b*d*(-1/d^2/(d*tan(b*x+a))^(1/2)-1/8/d^2/(d^2)^(1/4)*2^(1/2)*(ln((d*tan( b*x+a)-(d^2)^(1/4)*(d*tan(b*x+a))^(1/2)*2^(1/2)+(d^2)^(1/2))/(d*tan(b*x+a) +(d^2)^(1/4)*(d*tan(b*x+a))^(1/2)*2^(1/2)+(d^2)^(1/2)))+2*arctan(2^(1/2)/( d^2)^(1/4)*(d*tan(b*x+a))^(1/2)+1)-2*arctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(b* x+a))^(1/2)+1)))
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 235, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {b d^{2} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} \log \left (b^{3} d^{5} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {3}{4}} + \sqrt {d \tan \left (b x + a\right )}\right ) \tan \left (b x + a\right ) - i \, b d^{2} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} \log \left (i \, b^{3} d^{5} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {3}{4}} + \sqrt {d \tan \left (b x + a\right )}\right ) \tan \left (b x + a\right ) + i \, b d^{2} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} \log \left (-i \, b^{3} d^{5} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {3}{4}} + \sqrt {d \tan \left (b x + a\right )}\right ) \tan \left (b x + a\right ) - b d^{2} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {1}{4}} \log \left (-b^{3} d^{5} \left (-\frac {1}{b^{4} d^{6}}\right )^{\frac {3}{4}} + \sqrt {d \tan \left (b x + a\right )}\right ) \tan \left (b x + a\right ) + 4 \, \sqrt {d \tan \left (b x + a\right )}}{2 \, b d^{2} \tan \left (b x + a\right )} \]
-1/2*(b*d^2*(-1/(b^4*d^6))^(1/4)*log(b^3*d^5*(-1/(b^4*d^6))^(3/4) + sqrt(d *tan(b*x + a)))*tan(b*x + a) - I*b*d^2*(-1/(b^4*d^6))^(1/4)*log(I*b^3*d^5* (-1/(b^4*d^6))^(3/4) + sqrt(d*tan(b*x + a)))*tan(b*x + a) + I*b*d^2*(-1/(b ^4*d^6))^(1/4)*log(-I*b^3*d^5*(-1/(b^4*d^6))^(3/4) + sqrt(d*tan(b*x + a))) *tan(b*x + a) - b*d^2*(-1/(b^4*d^6))^(1/4)*log(-b^3*d^5*(-1/(b^4*d^6))^(3/ 4) + sqrt(d*tan(b*x + a)))*tan(b*x + a) + 4*sqrt(d*tan(b*x + a)))/(b*d^2*t an(b*x + a))
\[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=\int \frac {1}{\left (d \tan {\left (a + b x \right )}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.58 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.79 \[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=-\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (b x + a\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) + \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (d \tan \left (b x + a\right ) - \sqrt {2} \sqrt {d \tan \left (b x + a\right )} \sqrt {d} + d\right )}{\sqrt {d}} + \frac {8}{\sqrt {d \tan \left (b x + a\right )}}}{4 \, b d} \]
-1/4*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d*tan(b*x + a )))/sqrt(d))/sqrt(d) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(d) - 2* sqrt(d*tan(b*x + a)))/sqrt(d))/sqrt(d) - sqrt(2)*log(d*tan(b*x + a) + sqrt (2)*sqrt(d*tan(b*x + a))*sqrt(d) + d)/sqrt(d) + sqrt(2)*log(d*tan(b*x + a) - sqrt(2)*sqrt(d*tan(b*x + a))*sqrt(d) + d)/sqrt(d) + 8/sqrt(d*tan(b*x + a)))/(b*d)
Timed out. \[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=\text {Timed out} \]
Time = 2.83 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.36 \[ \int \frac {1}{(d \tan (a+b x))^{3/2}} \, dx=\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}}{\sqrt {d}}\right )}{b\,d^{3/2}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}}{\sqrt {d}}\right )}{b\,d^{3/2}}-\frac {2}{b\,d\,\sqrt {d\,\mathrm {tan}\left (a+b\,x\right )}} \]