Integrand size = 14, antiderivative size = 298 \[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{3/2}} \, dx=\frac {2}{3 b f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot ^2(e+f x)}{7 b f \sqrt {b \tan ^3(e+f x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {2} b f \sqrt {b \tan ^3(e+f x)}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {2} b f \sqrt {b \tan ^3(e+f x)}}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)}{2 \sqrt {2} b f \sqrt {b \tan ^3(e+f x)}}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)}{2 \sqrt {2} b f \sqrt {b \tan ^3(e+f x)}} \]
2/3/b/f/(b*tan(f*x+e)^3)^(1/2)-2/7*cot(f*x+e)^2/b/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)/b/f*2^(1/2)/(b*t an(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) /b/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)/b/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)/b/f*2^(1/2)/(b*tan(f*x+e) ^3)^(1/2)
Time = 0.39 (sec) , antiderivative size = 98, normalized size of antiderivative = 0.33 \[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{3/2}} \, dx=\frac {14-6 \cot ^2(e+f x)-21 \arctan \left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \left (-\tan ^2(e+f x)\right )^{3/4}-21 \text {arctanh}\left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \left (-\tan ^2(e+f x)\right )^{3/4}}{21 b f \sqrt {b \tan ^3(e+f x)}} \]
(14 - 6*Cot[e + f*x]^2 - 21*ArcTan[(-Tan[e + f*x]^2)^(1/4)]*(-Tan[e + f*x] ^2)^(3/4) - 21*ArcTanh[(-Tan[e + f*x]^2)^(1/4)]*(-Tan[e + f*x]^2)^(3/4))/( 21*b*f*Sqrt[b*Tan[e + f*x]^3])
Time = 0.56 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.67, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.214, Rules used = {3042, 4141, 3042, 3955, 3042, 3955, 3042, 3957, 266, 755, 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.
| \(\displaystyle \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (b \tan (e+f x)^3\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4141 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \int \frac {1}{\tan ^{\frac {9}{2}}(e+f x)}dx}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \int \frac {1}{\tan (e+f x)^{9/2}}dx}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\int \frac {1}{\tan ^{\frac {5}{2}}(e+f x)}dx-\frac {2}{7 f \tan ^{\frac {7}{2}}(e+f x)}\right )}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\int \frac {1}{\tan (e+f x)^{5/2}}dx-\frac {2}{7 f \tan ^{\frac {7}{2}}(e+f x)}\right )}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\int \frac {1}{\sqrt {\tan (e+f x)}}dx+\frac {2}{3 f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2}{7 f \tan ^{\frac {7}{2}}(e+f x)}\right )}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\int \frac {1}{\sqrt {\tan (e+f x)}}dx+\frac {2}{3 f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2}{7 f \tan ^{\frac {7}{2}}(e+f x)}\right )}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {\int \frac {1}{\sqrt {\tan (e+f x)} \left (\tan ^2(e+f x)+1\right )}d\tan (e+f x)}{f}+\frac {2}{3 f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2}{7 f \tan ^{\frac {7}{2}}(e+f x)}\right )}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \int \frac {1}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}}{f}+\frac {2}{3 f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2}{7 f \tan ^{\frac {7}{2}}(e+f x)}\right )}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \int \frac {1-\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}+\frac {1}{2} \int \frac {\tan (e+f x)+1}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}\right )}{f}+\frac {2}{3 f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2}{7 f \tan ^{\frac {7}{2}}(e+f x)}\right )}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \int \frac {1-\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}+\frac {1}{2} \int \frac {1}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}\right )\right )}{f}+\frac {2}{3 f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2}{7 f \tan ^{\frac {7}{2}}(e+f x)}\right )}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \int \frac {1-\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}+\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (e+f x)-1}d\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (e+f x)-1}d\left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}\right )\right )}{f}+\frac {2}{3 f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2}{7 f \tan ^{\frac {7}{2}}(e+f x)}\right )}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \int \frac {1-\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )\right )}{f}+\frac {2}{3 f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2}{7 f \tan ^{\frac {7}{2}}(e+f x)}\right )}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (e+f x)}}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )\right )}{f}+\frac {2}{3 f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2}{7 f \tan ^{\frac {7}{2}}(e+f x)}\right )}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (e+f x)}}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )\right )}{f}+\frac {2}{3 f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2}{7 f \tan ^{\frac {7}{2}}(e+f x)}\right )}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (e+f x)}}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (e+f x)}+1}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )\right )}{f}+\frac {2}{3 f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2}{7 f \tan ^{\frac {7}{2}}(e+f x)}\right )}{b \sqrt {b \tan ^3(e+f x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2}}\right )\right )}{f}+\frac {2}{3 f \tan ^{\frac {3}{2}}(e+f x)}-\frac {2}{7 f \tan ^{\frac {7}{2}}(e+f x)}\right )}{b \sqrt {b \tan ^3(e+f x)}}\) |
(((2*((-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt [2]*Sqrt[Tan[e + f*x]]]/Sqrt[2])/2 + (-1/2*Log[1 - Sqrt[2]*Sqrt[Tan[e + f* x]] + Tan[e + f*x]]/Sqrt[2] + Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]]/(2*Sqrt[2]))/2))/f - 2/(7*f*Tan[e + f*x]^(7/2)) + 2/(3*f*Tan[e + f* x]^(3/2)))*Tan[e + f*x]^(3/2))/(b*Sqrt[b*Tan[e + f*x]^3])
3.1.11.3.1 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[((a_) + (b_.)*(x_)^4)^(-1), x_Symbol] :> With[{r = Numerator[Rt[a/b, 2] ], s = Denominator[Rt[a/b, 2]]}, Simp[1/(2*r) Int[(r - s*x^2)/(a + b*x^4) , x], x] + Simp[1/(2*r) 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 = 0.03 (sec) , antiderivative size = 236, normalized size of antiderivative = 0.79
| method | result | size |
| derivativedivides | \(\frac {\tan \left (f x +e \right ) \left (21 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {7}{2}} \ln \left (-\frac {b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}{\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}\right )+42 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+42 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+56 b^{4} \tan \left (f x +e \right )^{2}-24 b^{4}\right )}{84 f \,b^{4} \left (b \tan \left (f x +e \right )^{3}\right )^{\frac {3}{2}}}\) | \(236\) |
| default | \(\frac {\tan \left (f x +e \right ) \left (21 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {7}{2}} \ln \left (-\frac {b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}{\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}\right )+42 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+42 \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+56 b^{4} \tan \left (f x +e \right )^{2}-24 b^{4}\right )}{84 f \,b^{4} \left (b \tan \left (f x +e \right )^{3}\right )^{\frac {3}{2}}}\) | \(236\) |
1/84/f*tan(f*x+e)/b^4*(21*(b^2)^(1/4)*2^(1/2)*(b*tan(f*x+e))^(7/2)*ln(-(b* tan(f*x+e)+(b^2)^(1/4)*(b*tan(f*x+e))^(1/2)*2^(1/2)+(b^2)^(1/2))/((b^2)^(1 /4)*(b*tan(f*x+e))^(1/2)*2^(1/2)-b*tan(f*x+e)-(b^2)^(1/2)))+42*(b^2)^(1/4) *2^(1/2)*(b*tan(f*x+e))^(7/2)*arctan((2^(1/2)*(b*tan(f*x+e))^(1/2)+(b^2)^( 1/4))/(b^2)^(1/4))+42*(b^2)^(1/4)*2^(1/2)*(b*tan(f*x+e))^(7/2)*arctan((2^( 1/2)*(b*tan(f*x+e))^(1/2)-(b^2)^(1/4))/(b^2)^(1/4))+56*b^4*tan(f*x+e)^2-24 *b^4)/(b*tan(f*x+e)^3)^(3/2)
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 320, normalized size of antiderivative = 1.07 \[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{3/2}} \, dx=\frac {21 \, b^{2} f \left (-\frac {1}{b^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {b^{2} f \left (-\frac {1}{b^{6} f^{4}}\right )^{\frac {1}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{5} - 21 \, b^{2} f \left (-\frac {1}{b^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {b^{2} f \left (-\frac {1}{b^{6} f^{4}}\right )^{\frac {1}{4}} \tan \left (f x + e\right ) - \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{5} + 21 i \, b^{2} f \left (-\frac {1}{b^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, b^{2} f \left (-\frac {1}{b^{6} f^{4}}\right )^{\frac {1}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{5} - 21 i \, b^{2} f \left (-\frac {1}{b^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, b^{2} f \left (-\frac {1}{b^{6} f^{4}}\right )^{\frac {1}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{5} + 4 \, \sqrt {b \tan \left (f x + e\right )^{3}} {\left (7 \, \tan \left (f x + e\right )^{2} - 3\right )}}{42 \, b^{2} f \tan \left (f x + e\right )^{5}} \]
1/42*(21*b^2*f*(-1/(b^6*f^4))^(1/4)*log((b^2*f*(-1/(b^6*f^4))^(1/4)*tan(f* x + e) + sqrt(b*tan(f*x + e)^3))/tan(f*x + e))*tan(f*x + e)^5 - 21*b^2*f*( -1/(b^6*f^4))^(1/4)*log(-(b^2*f*(-1/(b^6*f^4))^(1/4)*tan(f*x + e) - sqrt(b *tan(f*x + e)^3))/tan(f*x + e))*tan(f*x + e)^5 + 21*I*b^2*f*(-1/(b^6*f^4)) ^(1/4)*log((I*b^2*f*(-1/(b^6*f^4))^(1/4)*tan(f*x + e) + sqrt(b*tan(f*x + e )^3))/tan(f*x + e))*tan(f*x + e)^5 - 21*I*b^2*f*(-1/(b^6*f^4))^(1/4)*log(( -I*b^2*f*(-1/(b^6*f^4))^(1/4)*tan(f*x + e) + sqrt(b*tan(f*x + e)^3))/tan(f *x + e))*tan(f*x + e)^5 + 4*sqrt(b*tan(f*x + e)^3)*(7*tan(f*x + e)^2 - 3)) /(b^2*f*tan(f*x + e)^5)
\[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{\left (b \tan ^{3}{\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.34 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.55 \[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{3/2}} \, dx=\frac {\frac {21 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) + \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right ) - \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right )\right )}}{b^{\frac {3}{2}}} + \frac {8 \, {\left (21 \, \sqrt {\tan \left (f x + e\right )} + \frac {7}{\tan \left (f x + e\right )^{\frac {3}{2}}} - \frac {3}{\tan \left (f x + e\right )^{\frac {7}{2}}}\right )}}{b^{\frac {3}{2}}} - \frac {168 \, \sqrt {\tan \left (f x + e\right )}}{b^{\frac {3}{2}}}}{84 \, f} \]
1/84*(21*(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) *log(sqrt(2)*sqrt(tan(f*x + e)) + tan(f*x + e) + 1) - sqrt(2)*log(-sqrt(2) *sqrt(tan(f*x + e)) + tan(f*x + e) + 1))/b^(3/2) + 8*(21*sqrt(tan(f*x + e) ) + 7/tan(f*x + e)^(3/2) - 3/tan(f*x + e)^(7/2))/b^(3/2) - 168*sqrt(tan(f* x + e))/b^(3/2))/f
Time = 0.51 (sec) , antiderivative size = 279, normalized size of antiderivative = 0.94 \[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{3/2}} \, dx=\frac {1}{84} \, b^{4} {\left (\frac {42 \, \sqrt {2} \sqrt {{\left | b \right |}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{6} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {42 \, \sqrt {2} \sqrt {{\left | b \right |}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{6} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {21 \, \sqrt {2} \sqrt {{\left | b \right |}} \log \left (b \tan \left (f x + e\right ) + \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{6} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} - \frac {21 \, \sqrt {2} \sqrt {{\left | b \right |}} \log \left (b \tan \left (f x + e\right ) - \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{6} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {8 \, {\left (7 \, b^{2} \tan \left (f x + e\right )^{2} - 3 \, b^{2}\right )}}{\sqrt {b \tan \left (f x + e\right )} b^{7} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right ) \tan \left (f x + e\right )^{3}}\right )} \]
1/84*b^4*(42*sqrt(2)*sqrt(abs(b))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(b)) + 2*sqrt(b*tan(f*x + e)))/sqrt(abs(b)))/(b^6*f*sgn(tan(f*x + e))) + 42*sq rt(2)*sqrt(abs(b))*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(b)) - 2*sqrt(b*ta n(f*x + e)))/sqrt(abs(b)))/(b^6*f*sgn(tan(f*x + e))) + 21*sqrt(2)*sqrt(abs (b))*log(b*tan(f*x + e) + sqrt(2)*sqrt(b*tan(f*x + e))*sqrt(abs(b)) + abs( b))/(b^6*f*sgn(tan(f*x + e))) - 21*sqrt(2)*sqrt(abs(b))*log(b*tan(f*x + e) - sqrt(2)*sqrt(b*tan(f*x + e))*sqrt(abs(b)) + abs(b))/(b^6*f*sgn(tan(f*x + e))) + 8*(7*b^2*tan(f*x + e)^2 - 3*b^2)/(sqrt(b*tan(f*x + e))*b^7*f*sgn( tan(f*x + e))*tan(f*x + e)^3))
Timed out. \[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{3/2}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^3\right )}^{3/2}} \,d x \]