Integrand size = 14, antiderivative size = 364 \[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{5/2}} \, dx=-\frac {2 \cot (e+f x)}{5 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \cot ^3(e+f x)}{9 b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {2 \cot ^5(e+f x)}{13 b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {2 \tan (e+f x)}{b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {2} b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \tan ^{\frac {3}{2}}(e+f x)}{\sqrt {2} b^2 f \sqrt {b \tan ^3(e+f x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)}{2 \sqrt {2} b^2 f \sqrt {b \tan ^3(e+f x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right ) \tan ^{\frac {3}{2}}(e+f x)}{2 \sqrt {2} b^2 f \sqrt {b \tan ^3(e+f x)}} \]
-2/5*cot(f*x+e)/b^2/f/(b*tan(f*x+e)^3)^(1/2)+2/9*cot(f*x+e)^3/b^2/f/(b*tan (f*x+e)^3)^(1/2)-2/13*cot(f*x+e)^5/b^2/f/(b*tan(f*x+e)^3)^(1/2)+2*tan(f*x+ e)/b^2/f/(b*tan(f*x+e)^3)^(1/2)+1/2*arctan(-1+2^(1/2)*tan(f*x+e)^(1/2))*ta n(f*x+e)^(3/2)/b^2/f*2^(1/2)/(b*tan(f*x+e)^3)^(1/2)+1/2*arctan(1+2^(1/2)*t an(f*x+e)^(1/2))*tan(f*x+e)^(3/2)/b^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)/b^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)/b^2/f*2^(1/2)/(b*tan(f*x+e)^3)^(1/2)
Time = 0.71 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.38 \[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{5/2}} \, dx=\frac {-234 \cot (e+f x)+130 \cot ^3(e+f x)-90 \cot ^5(e+f x)+585 \text {arctanh}\left (\sqrt [4]{-\tan ^2(e+f x)}\right ) (-\tan (e+f x))^{5/4} \sqrt [4]{\tan (e+f x)}+1170 \tan (e+f x)+585 \arctan \left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \sqrt [4]{-\tan (e+f x)} \tan ^{\frac {5}{4}}(e+f x)}{585 b^2 f \sqrt {b \tan ^3(e+f x)}} \]
(-234*Cot[e + f*x] + 130*Cot[e + f*x]^3 - 90*Cot[e + f*x]^5 + 585*ArcTanh[ (-Tan[e + f*x]^2)^(1/4)]*(-Tan[e + f*x])^(5/4)*Tan[e + f*x]^(1/4) + 1170*T an[e + f*x] + 585*ArcTan[(-Tan[e + f*x]^2)^(1/4)]*(-Tan[e + f*x])^(1/4)*Ta n[e + f*x]^(5/4))/(585*b^2*f*Sqrt[b*Tan[e + f*x]^3])
Time = 0.75 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.64, number of steps used = 22, number of rules used = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 1.500, Rules used = {3042, 4141, 3042, 3955, 3042, 3955, 3042, 3955, 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.
\(\displaystyle \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{5/2}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {1}{\left (b \tan (e+f x)^3\right )^{5/2}}dx\) |
\(\Big \downarrow \) 4141 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \int \frac {1}{\tan ^{\frac {15}{2}}(e+f x)}dx}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \int \frac {1}{\tan (e+f x)^{15/2}}dx}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 3955 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\int \frac {1}{\tan ^{\frac {11}{2}}(e+f x)}dx-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\int \frac {1}{\tan (e+f x)^{11/2}}dx-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 3955 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\int \frac {1}{\tan ^{\frac {7}{2}}(e+f x)}dx+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\int \frac {1}{\tan (e+f x)^{7/2}}dx+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 3955 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\int \frac {1}{\tan ^{\frac {3}{2}}(e+f x)}dx-\frac {2}{5 f \tan ^{\frac {5}{2}}(e+f x)}+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (-\int \frac {1}{\tan (e+f x)^{3/2}}dx-\frac {2}{5 f \tan ^{\frac {5}{2}}(e+f x)}+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 3955 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\int \sqrt {\tan (e+f x)}dx-\frac {2}{5 f \tan ^{\frac {5}{2}}(e+f x)}+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}+\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\int \sqrt {\tan (e+f x)}dx-\frac {2}{5 f \tan ^{\frac {5}{2}}(e+f x)}+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}+\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 3957 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {\int \frac {\sqrt {\tan (e+f x)}}{\tan ^2(e+f x)+1}d\tan (e+f x)}{f}-\frac {2}{5 f \tan ^{\frac {5}{2}}(e+f x)}+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}+\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 266 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \int \frac {\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}}{f}-\frac {2}{5 f \tan ^{\frac {5}{2}}(e+f x)}+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}+\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 826 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \int \frac {\tan (e+f x)+1}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}-\frac {1}{2} \int \frac {1-\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}\right )}{f}-\frac {2}{5 f \tan ^{\frac {5}{2}}(e+f x)}+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}+\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 1476 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}+\frac {1}{2} \int \frac {1}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}\right )-\frac {1}{2} \int \frac {1-\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}\right )}{f}-\frac {2}{5 f \tan ^{\frac {5}{2}}(e+f x)}+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}+\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 1082 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (e+f x)-1}d\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (e+f x)-1}d\left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}\right )}{f}-\frac {2}{5 f \tan ^{\frac {5}{2}}(e+f x)}+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}+\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 217 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (e+f x)}{\tan ^2(e+f x)+1}d\sqrt {\tan (e+f x)}\right )}{f}-\frac {2}{5 f \tan ^{\frac {5}{2}}(e+f x)}+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}+\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 1479 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (e+f x)}}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )\right )}{f}-\frac {2}{5 f \tan ^{\frac {5}{2}}(e+f x)}+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}+\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (e+f x)}}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )\right )}{f}-\frac {2}{5 f \tan ^{\frac {5}{2}}(e+f x)}+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}+\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 27 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (e+f x)}}{\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (e+f x)}+1}{\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1}d\sqrt {\tan (e+f x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )\right )}{f}-\frac {2}{5 f \tan ^{\frac {5}{2}}(e+f x)}+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}+\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
\(\Big \downarrow \) 1103 |
\(\displaystyle \frac {\tan ^{\frac {3}{2}}(e+f x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (e+f x)-\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (e+f x)+\sqrt {2} \sqrt {\tan (e+f x)}+1\right )}{2 \sqrt {2}}\right )\right )}{f}-\frac {2}{5 f \tan ^{\frac {5}{2}}(e+f x)}+\frac {2}{9 f \tan ^{\frac {9}{2}}(e+f x)}-\frac {2}{13 f \tan ^{\frac {13}{2}}(e+f x)}+\frac {2}{f \sqrt {\tan (e+f x)}}\right )}{b^2 \sqrt {b \tan ^3(e+f x)}}\) |
(((2*((-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt [2]*Sqrt[Tan[e + f*x]]]/Sqrt[2])/2 + (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/(13*f*Tan[e + f*x]^(13/2)) + 2/(9*f*Tan[e + f *x]^(9/2)) - 2/(5*f*Tan[e + f*x]^(5/2)) + 2/(f*Sqrt[Tan[e + f*x]]))*Tan[e + f*x]^(3/2))/(b^2*Sqrt[b*Tan[e + f*x]^3])
3.1.12.3.1 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 = 0.03 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.75
method | result | size |
derivativedivides | \(\frac {\tan \left (f x +e \right ) \left (585 \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {13}{2}} \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}{b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+1170 \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+1170 \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+4680 \left (b^{2}\right )^{\frac {1}{4}} b^{6} \tan \left (f x +e \right )^{6}-936 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \tan \left (f x +e \right )^{4}+520 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \tan \left (f x +e \right )^{2}-360 b^{6} \left (b^{2}\right )^{\frac {1}{4}}\right )}{2340 f \,b^{6} \left (b \tan \left (f x +e \right )^{3}\right )^{\frac {5}{2}} \left (b^{2}\right )^{\frac {1}{4}}}\) | \(272\) |
default | \(\frac {\tan \left (f x +e \right ) \left (585 \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {13}{2}} \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}-b \tan \left (f x +e \right )-\sqrt {b^{2}}}{b \tan \left (f x +e \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+1170 \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+1170 \sqrt {2}\, \left (b \tan \left (f x +e \right )\right )^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (f x +e \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+4680 \left (b^{2}\right )^{\frac {1}{4}} b^{6} \tan \left (f x +e \right )^{6}-936 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \tan \left (f x +e \right )^{4}+520 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \tan \left (f x +e \right )^{2}-360 b^{6} \left (b^{2}\right )^{\frac {1}{4}}\right )}{2340 f \,b^{6} \left (b \tan \left (f x +e \right )^{3}\right )^{\frac {5}{2}} \left (b^{2}\right )^{\frac {1}{4}}}\) | \(272\) |
1/2340/f*tan(f*x+e)/b^6*(585*2^(1/2)*(b*tan(f*x+e))^(13/2)*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)))+1170*2^(1/2)*(b*tan(f* x+e))^(13/2)*arctan((2^(1/2)*(b*tan(f*x+e))^(1/2)+(b^2)^(1/4))/(b^2)^(1/4) )+1170*2^(1/2)*(b*tan(f*x+e))^(13/2)*arctan((2^(1/2)*(b*tan(f*x+e))^(1/2)- (b^2)^(1/4))/(b^2)^(1/4))+4680*(b^2)^(1/4)*b^6*tan(f*x+e)^6-936*b^6*(b^2)^ (1/4)*tan(f*x+e)^4+520*b^6*(b^2)^(1/4)*tan(f*x+e)^2-360*b^6*(b^2)^(1/4))/( b*tan(f*x+e)^3)^(5/2)/(b^2)^(1/4)
Result contains complex when optimal does not.
Time = 0.28 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{5/2}} \, dx=\frac {585 \, b^{3} f \left (-\frac {1}{b^{10} f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {b^{8} f^{3} \left (-\frac {1}{b^{10} f^{4}}\right )^{\frac {3}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{8} - 585 \, b^{3} f \left (-\frac {1}{b^{10} f^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {b^{8} f^{3} \left (-\frac {1}{b^{10} f^{4}}\right )^{\frac {3}{4}} \tan \left (f x + e\right ) - \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{8} - 585 i \, b^{3} f \left (-\frac {1}{b^{10} f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, b^{8} f^{3} \left (-\frac {1}{b^{10} f^{4}}\right )^{\frac {3}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{8} + 585 i \, b^{3} f \left (-\frac {1}{b^{10} f^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, b^{8} f^{3} \left (-\frac {1}{b^{10} f^{4}}\right )^{\frac {3}{4}} \tan \left (f x + e\right ) + \sqrt {b \tan \left (f x + e\right )^{3}}}{\tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{8} + 4 \, {\left (585 \, \tan \left (f x + e\right )^{6} - 117 \, \tan \left (f x + e\right )^{4} + 65 \, \tan \left (f x + e\right )^{2} - 45\right )} \sqrt {b \tan \left (f x + e\right )^{3}}}{1170 \, b^{3} f \tan \left (f x + e\right )^{8}} \]
1/1170*(585*b^3*f*(-1/(b^10*f^4))^(1/4)*log((b^8*f^3*(-1/(b^10*f^4))^(3/4) *tan(f*x + e) + sqrt(b*tan(f*x + e)^3))/tan(f*x + e))*tan(f*x + e)^8 - 585 *b^3*f*(-1/(b^10*f^4))^(1/4)*log(-(b^8*f^3*(-1/(b^10*f^4))^(3/4)*tan(f*x + e) - sqrt(b*tan(f*x + e)^3))/tan(f*x + e))*tan(f*x + e)^8 - 585*I*b^3*f*( -1/(b^10*f^4))^(1/4)*log((I*b^8*f^3*(-1/(b^10*f^4))^(3/4)*tan(f*x + e) + s qrt(b*tan(f*x + e)^3))/tan(f*x + e))*tan(f*x + e)^8 + 585*I*b^3*f*(-1/(b^1 0*f^4))^(1/4)*log((-I*b^8*f^3*(-1/(b^10*f^4))^(3/4)*tan(f*x + e) + sqrt(b* tan(f*x + e)^3))/tan(f*x + e))*tan(f*x + e)^8 + 4*(585*tan(f*x + e)^6 - 11 7*tan(f*x + e)^4 + 65*tan(f*x + e)^2 - 45)*sqrt(b*tan(f*x + e)^3))/(b^3*f* tan(f*x + e)^8)
\[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{\left (b \tan ^{3}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
Time = 0.35 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.47 \[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{5/2}} \, dx=\frac {\frac {585 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (f x + e\right )}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (f x + e\right )} + \tan \left (f x + e\right ) + 1\right )\right )}}{b^{\frac {5}{2}}} + \frac {8 \, {\left (\frac {585 \, \sqrt {b}}{\sqrt {\tan \left (f x + e\right )}} - \frac {117 \, \sqrt {b}}{\tan \left (f x + e\right )^{\frac {5}{2}}} + \frac {65 \, \sqrt {b}}{\tan \left (f x + e\right )^{\frac {9}{2}}} - \frac {45 \, \sqrt {b}}{\tan \left (f x + e\right )^{\frac {13}{2}}}\right )}}{b^{3}}}{2340 \, f} \]
1/2340*(585*(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^(5/2) + 8*(585*sqrt(b)/sqrt( tan(f*x + e)) - 117*sqrt(b)/tan(f*x + e)^(5/2) + 65*sqrt(b)/tan(f*x + e)^( 9/2) - 45*sqrt(b)/tan(f*x + e)^(13/2))/b^3)/f
Time = 0.77 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{5/2}} \, dx=\frac {1}{2340} \, b^{6} {\left (\frac {1170 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{10} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {1170 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{10} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} - \frac {585 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (f x + e\right ) + \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{10} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {585 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (f x + e\right ) - \sqrt {2} \sqrt {b \tan \left (f x + e\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{10} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right )} + \frac {8 \, {\left (585 \, b^{6} \tan \left (f x + e\right )^{6} - 117 \, b^{6} \tan \left (f x + e\right )^{4} + 65 \, b^{6} \tan \left (f x + e\right )^{2} - 45 \, b^{6}\right )}}{\sqrt {b \tan \left (f x + e\right )} b^{14} f \mathrm {sgn}\left (\tan \left (f x + e\right )\right ) \tan \left (f x + e\right )^{6}}\right )} \]
1/2340*b^6*(1170*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^10*f*sgn(tan(f*x + e))) + 1170*sqrt(2)*abs(b)^(3/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(b)) - 2*sq rt(b*tan(f*x + e)))/sqrt(abs(b)))/(b^10*f*sgn(tan(f*x + e))) - 585*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^10*f*sgn(tan(f*x + e))) + 585*sqrt(2)*abs(b)^(3/2)*log(b*t an(f*x + e) - sqrt(2)*sqrt(b*tan(f*x + e))*sqrt(abs(b)) + abs(b))/(b^10*f* sgn(tan(f*x + e))) + 8*(585*b^6*tan(f*x + e)^6 - 117*b^6*tan(f*x + e)^4 + 65*b^6*tan(f*x + e)^2 - 45*b^6)/(sqrt(b*tan(f*x + e))*b^14*f*sgn(tan(f*x + e))*tan(f*x + e)^6))
Timed out. \[ \int \frac {1}{\left (b \tan ^3(e+f x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {tan}\left (e+f\,x\right )}^3\right )}^{5/2}} \,d x \]