Integrand size = 14, antiderivative size = 255 \[ \int \frac {1}{\sqrt {b \tan ^3(e+f x)}} \, dx=-\frac {2 \tan (e+f x)}{f \sqrt {b \tan ^3(e+f x)}}+\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {2} f \sqrt {b \tan ^3(e+f x)}}-\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {2} f \sqrt {b \tan ^3(e+f x)}}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)}{2 \sqrt {2} f \sqrt {b \tan ^3(e+f x)}}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)}{2 \sqrt {2} f \sqrt {b \tan ^3(e+f x)}} \]
-2*tan(f*x+e)/f/(b*tan(f*x+e)^3)^(1/2)-1/2*arctan(-1+2^(1/2)*tan(f*x+e)^(1 /2))*tan(f*x+e)^(3/2)/f*2^(1/2)/(b*tan(f*x+e)^3)^(1/2)-1/2*arctan(1+2^(1/2 )*tan(f*x+e)^(1/2))*tan(f*x+e)^(3/2)/f*2^(1/2)/(b*tan(f*x+e)^3)^(1/2)-1/4* ln(1-2^(1/2)*tan(f*x+e)^(1/2)+tan(f*x+e))*tan(f*x+e)^(3/2)/f*2^(1/2)/(b*ta n(f*x+e)^3)^(1/2)+1/4*ln(1+2^(1/2)*tan(f*x+e)^(1/2)+tan(f*x+e))*tan(f*x+e) ^(3/2)/f*2^(1/2)/(b*tan(f*x+e)^3)^(1/2)
Time = 0.49 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.34 \[ \int \frac {1}{\sqrt {b \tan ^3(e+f x)}} \, dx=\frac {\tan (e+f x) \left (-2-\arctan \left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \sqrt [4]{-\tan ^2(e+f x)}+\text {arctanh}\left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \sqrt [4]{-\tan ^2(e+f x)}\right )}{f \sqrt {b \tan ^3(e+f x)}} \]
(Tan[e + f*x]*(-2 - ArcTan[(-Tan[e + f*x]^2)^(1/4)]*(-Tan[e + f*x]^2)^(1/4 ) + ArcTanh[(-Tan[e + f*x]^2)^(1/4)]*(-Tan[e + f*x]^2)^(1/4)))/(f*Sqrt[b*T an[e + f*x]^3])
Time = 0.49 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.70, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.071, Rules used = {3042, 4141, 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.
| \(\displaystyle \int \frac {1}{\sqrt {b \tan ^3(e+f x)}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\sqrt {b \tan (e+f x)^3}}dx\) |
| \(\Big \downarrow \) 4141 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \int \frac {1}{\tan ^{\frac {3}{2}}(e+f x)}dx}{\sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \int \frac {1}{\tan (e+f x)^{3/2}}dx}{\sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\int \sqrt {\tan (e+f x)}dx-\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{\sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\int \sqrt {\tan (e+f x)}dx-\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{\sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\frac {\int \frac {\sqrt {\tan (e+f x)}}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}-\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{\sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\frac {2 \int \frac {\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}}{f}-\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{\sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\frac {2 \left (\frac {1}{2} \int \frac {\tan (e+f x)+1}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}-\frac {1}{2} \int \frac {1-\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}\right )}{f}-\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{\sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}+\frac {1}{2} \int \frac {1}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}\right )-\frac {1}{2} \int \frac {1-\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}\right )}{f}-\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{\sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (e+f x)-1}d\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (e+f x)-1}d\left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}\right )}{f}-\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{\sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}\right )}{f}-\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{\sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (e+f x)}}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )\right )}{f}-\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{\sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (e+f x)}}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )\right )}{f}-\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{\sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (e+f x)}}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (e+f x)}+1}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )\right )}{f}-\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{\sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2}}\right )\right )}{f}-\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{\sqrt {b \tan ^3(e+f x)}}\) |
(((-2*((-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]/Sqrt[2]) + ArcTan[1 + Sqr t[2]*Sqrt[Tan[e + f*x]]]/Sqrt[2])/2 + (Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]/(2*Sqrt[2]))/2))/f - 2/(f*Sqrt[Tan[e + f*x]]))*Tan[e + f*x]^(3/2))/ Sqrt[b*Tan[e + f*x]^3]
3.1.10.3.1 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]
Int[(u_.)*((b_.)*tan[(e_.) + (f_.)*(x_)]^(n_))^(p_), x_Symbol] :> With[{ff = FreeFactors[Tan[e + f*x], x]}, Simp[(b*ff^n)^IntPart[p]*((b*Tan[e + f*x]^ n)^FracPart[p]/(Tan[e + f*x]/ff)^(n*FracPart[p])) Int[ActivateTrig[u]*(Ta n[e + f*x]/ff)^(n*p), x], x]] /; FreeQ[{b, e, f, n, p}, x] && !IntegerQ[p] && IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x])^(m_.) / ; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])
Time = 0.04 (sec) , antiderivative size = 211, normalized size of antiderivative = 0.83
| method | result | size |
| derivativedivides | \(-\frac {\tan \left (f x +e \right ) \left (\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}\, \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}{b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+2 \sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+8 \left (b^{2}\right )^{\frac {1}{4}}\right )}{4 f \sqrt {b \tan \left (f x +e \right )^{3}}\, \left (b^{2}\right )^{\frac {1}{4}}}\) | \(211\) |
| default | \(-\frac {\tan \left (f x +e \right ) \left (\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}\, \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}{b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+2 \sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+2 \sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+8 \left (b^{2}\right )^{\frac {1}{4}}\right )}{4 f \sqrt {b \tan \left (f x +e \right )^{3}}\, \left (b^{2}\right )^{\frac {1}{4}}}\) | \(211\) |
-1/4/f*tan(f*x+e)*(2^(1/2)*(b*tan(f*x+e))^(1/2)*ln(-((b^2)^(1/4)*(b*tan(f* x+e))^(1/2)*2^(1/2)-b*tan(f*x+e)-(b^2)^(1/2))/(b*tan(f*x+e)+(b^2)^(1/4)*(b *tan(f*x+e))^(1/2)*2^(1/2)+(b^2)^(1/2)))+2*2^(1/2)*(b*tan(f*x+e))^(1/2)*ar ctan((2^(1/2)*(b*tan(f*x+e))^(1/2)+(b^2)^(1/4))/(b^2)^(1/4))+2*2^(1/2)*(b* tan(f*x+e))^(1/2)*arctan((2^(1/2)*(b*tan(f*x+e))^(1/2)-(b^2)^(1/4))/(b^2)^ (1/4))+8*(b^2)^(1/4))/(b*tan(f*x+e)^3)^(1/2)/(b^2)^(1/4)
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 307, normalized size of antiderivative = 1.20 \[ \int \frac {1}{\sqrt {b \tan ^3(e+f x)}} \, dx=-\frac {b f \left (-\frac {1}{b^{2} f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {b^{2} f^{3} \left (-\frac {1}{b^{2} f^{4}}\right )^{\frac {3}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{2} - b f \left (-\frac {1}{b^{2} f^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {b^{2} f^{3} \left (-\frac {1}{b^{2} f^{4}}\right )^{\frac {3}{4}} \tan \left (f x + e\right ) - \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{2} - i \, b f \left (-\frac {1}{b^{2} f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, b^{2} f^{3} \left (-\frac {1}{b^{2} f^{4}}\right )^{\frac {3}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{2} + i \, b f \left (-\frac {1}{b^{2} f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, b^{2} f^{3} \left (-\frac {1}{b^{2} f^{4}}\right )^{\frac {3}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{2} + 4 \, \sqrt {b \tan \left (f x + e\right )^{3}}}{2 \, b f \tan \left (f x + e\right )^{2}} \]
-1/2*(b*f*(-1/(b^2*f^4))^(1/4)*log((b^2*f^3*(-1/(b^2*f^4))^(3/4)*tan(f*x + e) + sqrt(b*tan(f*x + e)^3))/tan(f*x + e))*tan(f*x + e)^2 - b*f*(-1/(b^2* f^4))^(1/4)*log(-(b^2*f^3*(-1/(b^2*f^4))^(3/4)*tan(f*x + e) - sqrt(b*tan(f *x + e)^3))/tan(f*x + e))*tan(f*x + e)^2 - I*b*f*(-1/(b^2*f^4))^(1/4)*log( (I*b^2*f^3*(-1/(b^2*f^4))^(3/4)*tan(f*x + e) + sqrt(b*tan(f*x + e)^3))/tan (f*x + e))*tan(f*x + e)^2 + I*b*f*(-1/(b^2*f^4))^(1/4)*log((-I*b^2*f^3*(-1 /(b^2*f^4))^(3/4)*tan(f*x + e) + sqrt(b*tan(f*x + e)^3))/tan(f*x + e))*tan (f*x + e)^2 + 4*sqrt(b*tan(f*x + e)^3))/(b*f*tan(f*x + e)^2)
\[ \int \frac {1}{\sqrt {b \tan ^3(e+f x)}} \, dx=\int \frac {1}{\sqrt {b \tan ^{3}{\left (e + f x \right )}}}\, dx \]
Time = 0.35 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.49 \[ \int \frac {1}{\sqrt {b \tan ^3(e+f x)}} \, dx=-\frac {\frac {2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right )}{\sqrt {b}} + \frac {8}{\sqrt {b} \sqrt {\tan \left (f x + e\right )}}}{4 \, f} \]
-1/4*((2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(f*x + e)))) + 2* sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(f*x + e)))) - sqrt(2)*lo g(sqrt(2)*sqrt(tan(f*x + e)) + tan(f*x + e) + 1) + sqrt(2)*log(-sqrt(2)*sq rt(tan(f*x + e)) + tan(f*x + e) + 1))/sqrt(b) + 8/(sqrt(b)*sqrt(tan(f*x + e))))/f
Time = 0.40 (sec) , antiderivative size = 251, normalized size of antiderivative = 0.98 \[ \int \frac {1}{\sqrt {b \tan ^3(e+f x)}} \, dx=-\frac {1}{4} \, b^{2} {\left (\frac {2 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{4} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {2 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{4} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} - \frac {\sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (f x + e\right ) + \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{4} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {\sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (f x + e\right ) - \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{4} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {8}{\sqrt {b \tan \left (f x + e\right )} b^{2} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )}\right )} \]
-1/4*b^2*(2*sqrt(2)*abs(b)^(3/2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(b)) + 2*sqrt(b*tan(f*x + e)))/sqrt(abs(b)))/(b^4*f*sgn(tan(f*x + e))) + 2*sqrt (2)*abs(b)^(3/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(b)) - 2*sqrt(b*tan( f*x + e)))/sqrt(abs(b)))/(b^4*f*sgn(tan(f*x + e))) - sqrt(2)*abs(b)^(3/2)* log(b*tan(f*x + e) + sqrt(2)*sqrt(b*tan(f*x + e))*sqrt(abs(b)) + abs(b))/( b^4*f*sgn(tan(f*x + e))) + sqrt(2)*abs(b)^(3/2)*log(b*tan(f*x + e) - sqrt( 2)*sqrt(b*tan(f*x + e))*sqrt(abs(b)) + abs(b))/(b^4*f*sgn(tan(f*x + e))) + 8/(sqrt(b*tan(f*x + e))*b^2*f*sgn(tan(f*x + e))))
Timed out. \[ \int \frac {1}{\sqrt {b \tan ^3(e+f x)}} \, dx=\int \frac {1}{\sqrt {b\,{\mathrm {tan}\left (e+f\,x\right )}^3}} \,d x \]