Integrand size = 14, antiderivative size = 364 \[ \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{5/2}} \, dx=-\frac {2 \cot (c+d x)}{5 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \cot ^3(c+d x)}{9 b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {2 \cot ^5(c+d x)}{13 b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {2 \tan (c+d x)}{b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \tan ^{\frac {3}{2}}(c+d x)}{\sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}+\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}}-\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \tan ^{\frac {3}{2}}(c+d x)}{2 \sqrt {2} b^2 d \sqrt {b \tan ^3(c+d x)}} \]
-2/5*cot(d*x+c)/b^2/d/(b*tan(d*x+c)^3)^(1/2)+2/9*cot(d*x+c)^3/b^2/d/(b*tan (d*x+c)^3)^(1/2)-2/13*cot(d*x+c)^5/b^2/d/(b*tan(d*x+c)^3)^(1/2)+2*tan(d*x+ c)/b^2/d/(b*tan(d*x+c)^3)^(1/2)+1/2*arctan(-1+2^(1/2)*tan(d*x+c)^(1/2))*ta n(d*x+c)^(3/2)/b^2/d*2^(1/2)/(b*tan(d*x+c)^3)^(1/2)+1/2*arctan(1+2^(1/2)*t an(d*x+c)^(1/2))*tan(d*x+c)^(3/2)/b^2/d*2^(1/2)/(b*tan(d*x+c)^3)^(1/2)+1/4 *ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))*tan(d*x+c)^(3/2)/b^2/d*2^(1/2)/ (b*tan(d*x+c)^3)^(1/2)-1/4*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))*tan(d *x+c)^(3/2)/b^2/d*2^(1/2)/(b*tan(d*x+c)^3)^(1/2)
Time = 0.40 (sec) , antiderivative size = 139, normalized size of antiderivative = 0.38 \[ \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{5/2}} \, dx=\frac {-234 \cot (c+d x)+130 \cot ^3(c+d x)-90 \cot ^5(c+d x)+585 \text {arctanh}\left (\sqrt [4]{-\tan ^2(c+d x)}\right ) (-\tan (c+d x))^{5/4} \sqrt [4]{\tan (c+d x)}+1170 \tan (c+d x)+585 \arctan \left (\sqrt [4]{-\tan ^2(c+d x)}\right ) \sqrt [4]{-\tan (c+d x)} \tan ^{\frac {5}{4}}(c+d x)}{585 b^2 d \sqrt {b \tan ^3(c+d x)}} \]
(-234*Cot[c + d*x] + 130*Cot[c + d*x]^3 - 90*Cot[c + d*x]^5 + 585*ArcTanh[ (-Tan[c + d*x]^2)^(1/4)]*(-Tan[c + d*x])^(5/4)*Tan[c + d*x]^(1/4) + 1170*T an[c + d*x] + 585*ArcTan[(-Tan[c + d*x]^2)^(1/4)]*(-Tan[c + d*x])^(1/4)*Ta n[c + d*x]^(5/4))/(585*b^2*d*Sqrt[b*Tan[c + d*x]^3])
Time = 0.74 (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.
| \(\displaystyle \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{5/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (b \tan (c+d x)^3\right )^{5/2}}dx\) |
| \(\Big \downarrow \) 4141 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \int \frac {1}{\tan ^{\frac {15}{2}}(c+d x)}dx}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \int \frac {1}{\tan (c+d x)^{15/2}}dx}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (-\int \frac {1}{\tan ^{\frac {11}{2}}(c+d x)}dx-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (-\int \frac {1}{\tan (c+d x)^{11/2}}dx-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\int \frac {1}{\tan ^{\frac {7}{2}}(c+d x)}dx+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\int \frac {1}{\tan (c+d x)^{7/2}}dx+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (-\int \frac {1}{\tan ^{\frac {3}{2}}(c+d x)}dx-\frac {2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (-\int \frac {1}{\tan (c+d x)^{3/2}}dx-\frac {2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\int \sqrt {\tan (c+d x)}dx-\frac {2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}+\frac {2}{d \sqrt {\tan (c+d x)}}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\int \sqrt {\tan (c+d x)}dx-\frac {2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}+\frac {2}{d \sqrt {\tan (c+d x)}}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {\int \frac {\sqrt {\tan (c+d x)}}{\tan ^2(c+d x)+1}d\tan (c+d x)}{d}-\frac {2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}+\frac {2}{d \sqrt {\tan (c+d x)}}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \int \frac {\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}-\frac {2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}+\frac {2}{d \sqrt {\tan (c+d x)}}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}-\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}-\frac {2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}+\frac {2}{d \sqrt {\tan (c+d x)}}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )-\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}-\frac {2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}+\frac {2}{d \sqrt {\tan (c+d x)}}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}-\frac {2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}+\frac {2}{d \sqrt {\tan (c+d x)}}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )-\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}-\frac {2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}+\frac {2}{d \sqrt {\tan (c+d x)}}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}+\frac {2}{d \sqrt {\tan (c+d x)}}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}+\frac {2}{d \sqrt {\tan (c+d x)}}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}-\frac {2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}+\frac {2}{d \sqrt {\tan (c+d x)}}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {\tan ^{\frac {3}{2}}(c+d x) \left (\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}-\frac {2}{5 d \tan ^{\frac {5}{2}}(c+d x)}+\frac {2}{9 d \tan ^{\frac {9}{2}}(c+d x)}-\frac {2}{13 d \tan ^{\frac {13}{2}}(c+d x)}+\frac {2}{d \sqrt {\tan (c+d x)}}\right )}{b^2 \sqrt {b \tan ^3(c+d x)}}\) |
(((2*((-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2]) + ArcTan[1 + Sqrt [2]*Sqrt[Tan[c + d*x]]]/Sqrt[2])/2 + (Log[1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]) - Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]))/2))/d - 2/(13*d*Tan[c + d*x]^(13/2)) + 2/(9*d*Tan[c + d *x]^(9/2)) - 2/(5*d*Tan[c + d*x]^(5/2)) + 2/(d*Sqrt[Tan[c + d*x]]))*Tan[c + d*x]^(3/2))/(b^2*Sqrt[b*Tan[c + d*x]^3])
3.1.35.3.1 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 = 0.08 (sec) , antiderivative size = 272, normalized size of antiderivative = 0.75
| method | result | size |
| derivativedivides | \(\frac {\tan \left (d x +c \right ) \left (585 \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}} \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}-b \tan \left (d x +c \right )-\sqrt {b^{2}}}{b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+1170 \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+1170 \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+4680 \left (b^{2}\right )^{\frac {1}{4}} b^{6} \left (\tan ^{6}\left (d x +c \right )\right )-936 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \left (\tan ^{4}\left (d x +c \right )\right )+520 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \left (\tan ^{2}\left (d x +c \right )\right )-360 b^{6} \left (b^{2}\right )^{\frac {1}{4}}\right )}{2340 d \,b^{6} {\left (b \left (\tan ^{3}\left (d x +c \right )\right )\right )}^{\frac {5}{2}} \left (b^{2}\right )^{\frac {1}{4}}}\) | \(272\) |
| default | \(\frac {\tan \left (d x +c \right ) \left (585 \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}} \ln \left (-\frac {\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}-b \tan \left (d x +c \right )-\sqrt {b^{2}}}{b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+1170 \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+1170 \sqrt {2}\, \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}} \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+4680 \left (b^{2}\right )^{\frac {1}{4}} b^{6} \left (\tan ^{6}\left (d x +c \right )\right )-936 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \left (\tan ^{4}\left (d x +c \right )\right )+520 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \left (\tan ^{2}\left (d x +c \right )\right )-360 b^{6} \left (b^{2}\right )^{\frac {1}{4}}\right )}{2340 d \,b^{6} {\left (b \left (\tan ^{3}\left (d x +c \right )\right )\right )}^{\frac {5}{2}} \left (b^{2}\right )^{\frac {1}{4}}}\) | \(272\) |
1/2340/d*tan(d*x+c)/b^6*(585*2^(1/2)*(b*tan(d*x+c))^(13/2)*ln(-((b^2)^(1/4 )*(b*tan(d*x+c))^(1/2)*2^(1/2)-b*tan(d*x+c)-(b^2)^(1/2))/(b*tan(d*x+c)+(b^ 2)^(1/4)*(b*tan(d*x+c))^(1/2)*2^(1/2)+(b^2)^(1/2)))+1170*2^(1/2)*(b*tan(d* x+c))^(13/2)*arctan((2^(1/2)*(b*tan(d*x+c))^(1/2)+(b^2)^(1/4))/(b^2)^(1/4) )+1170*2^(1/2)*(b*tan(d*x+c))^(13/2)*arctan((2^(1/2)*(b*tan(d*x+c))^(1/2)- (b^2)^(1/4))/(b^2)^(1/4))+4680*(b^2)^(1/4)*b^6*tan(d*x+c)^6-936*b^6*(b^2)^ (1/4)*tan(d*x+c)^4+520*b^6*(b^2)^(1/4)*tan(d*x+c)^2-360*b^6*(b^2)^(1/4))/( b*tan(d*x+c)^3)^(5/2)/(b^2)^(1/4)
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 348, normalized size of antiderivative = 0.96 \[ \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{5/2}} \, dx=\frac {585 \, b^{3} d \left (-\frac {1}{b^{10} d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {b^{8} d^{3} \left (-\frac {1}{b^{10} d^{4}}\right )^{\frac {3}{4}} \tan \left (d x + c\right ) + \sqrt {b \tan \left (d x + c\right )^{3}}}{\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{8} - 585 \, b^{3} d \left (-\frac {1}{b^{10} d^{4}}\right )^{\frac {1}{4}} \log \left (-\frac {b^{8} d^{3} \left (-\frac {1}{b^{10} d^{4}}\right )^{\frac {3}{4}} \tan \left (d x + c\right ) - \sqrt {b \tan \left (d x + c\right )^{3}}}{\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{8} - 585 i \, b^{3} d \left (-\frac {1}{b^{10} d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {i \, b^{8} d^{3} \left (-\frac {1}{b^{10} d^{4}}\right )^{\frac {3}{4}} \tan \left (d x + c\right ) + \sqrt {b \tan \left (d x + c\right )^{3}}}{\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{8} + 585 i \, b^{3} d \left (-\frac {1}{b^{10} d^{4}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, b^{8} d^{3} \left (-\frac {1}{b^{10} d^{4}}\right )^{\frac {3}{4}} \tan \left (d x + c\right ) + \sqrt {b \tan \left (d x + c\right )^{3}}}{\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right )^{8} + 4 \, {\left (585 \, \tan \left (d x + c\right )^{6} - 117 \, \tan \left (d x + c\right )^{4} + 65 \, \tan \left (d x + c\right )^{2} - 45\right )} \sqrt {b \tan \left (d x + c\right )^{3}}}{1170 \, b^{3} d \tan \left (d x + c\right )^{8}} \]
1/1170*(585*b^3*d*(-1/(b^10*d^4))^(1/4)*log((b^8*d^3*(-1/(b^10*d^4))^(3/4) *tan(d*x + c) + sqrt(b*tan(d*x + c)^3))/tan(d*x + c))*tan(d*x + c)^8 - 585 *b^3*d*(-1/(b^10*d^4))^(1/4)*log(-(b^8*d^3*(-1/(b^10*d^4))^(3/4)*tan(d*x + c) - sqrt(b*tan(d*x + c)^3))/tan(d*x + c))*tan(d*x + c)^8 - 585*I*b^3*d*( -1/(b^10*d^4))^(1/4)*log((I*b^8*d^3*(-1/(b^10*d^4))^(3/4)*tan(d*x + c) + s qrt(b*tan(d*x + c)^3))/tan(d*x + c))*tan(d*x + c)^8 + 585*I*b^3*d*(-1/(b^1 0*d^4))^(1/4)*log((-I*b^8*d^3*(-1/(b^10*d^4))^(3/4)*tan(d*x + c) + sqrt(b* tan(d*x + c)^3))/tan(d*x + c))*tan(d*x + c)^8 + 4*(585*tan(d*x + c)^6 - 11 7*tan(d*x + c)^4 + 65*tan(d*x + c)^2 - 45)*sqrt(b*tan(d*x + c)^3))/(b^3*d* tan(d*x + c)^8)
\[ \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{5/2}} \, dx=\int \frac {1}{\left (b \tan ^{3}{\left (c + d x \right )}\right )^{\frac {5}{2}}}\, dx \]
Time = 0.42 (sec) , antiderivative size = 172, normalized size of antiderivative = 0.47 \[ \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{5/2}} \, dx=\frac {\frac {585 \, {\left (2 \, \sqrt {2} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) - \sqrt {2} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) + \sqrt {2} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )}}{b^{\frac {5}{2}}} + \frac {8 \, {\left (\frac {585 \, \sqrt {b}}{\sqrt {\tan \left (d x + c\right )}} - \frac {117 \, \sqrt {b}}{\tan \left (d x + c\right )^{\frac {5}{2}}} + \frac {65 \, \sqrt {b}}{\tan \left (d x + c\right )^{\frac {9}{2}}} - \frac {45 \, \sqrt {b}}{\tan \left (d x + c\right )^{\frac {13}{2}}}\right )}}{b^{3}}}{2340 \, d} \]
1/2340*(585*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2) + 2*sqrt(tan(d*x + c))) ) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2*sqrt(tan(d*x + c)))) - sqrt (2)*log(sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1) + sqrt(2)*log(-sqrt (2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))/b^(5/2) + 8*(585*sqrt(b)/sqrt( tan(d*x + c)) - 117*sqrt(b)/tan(d*x + c)^(5/2) + 65*sqrt(b)/tan(d*x + c)^( 9/2) - 45*sqrt(b)/tan(d*x + c)^(13/2))/b^3)/d
Time = 0.86 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.84 \[ \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{5/2}} \, dx=\frac {1}{2340} \, b^{6} {\left (\frac {1170 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{10} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} + \frac {1170 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{b^{10} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} - \frac {585 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{10} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} + \frac {585 \, \sqrt {2} {\left | b \right |}^{\frac {3}{2}} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{b^{10} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right )} + \frac {8 \, {\left (585 \, b^{6} \tan \left (d x + c\right )^{6} - 117 \, b^{6} \tan \left (d x + c\right )^{4} + 65 \, b^{6} \tan \left (d x + c\right )^{2} - 45 \, b^{6}\right )}}{\sqrt {b \tan \left (d x + c\right )} b^{14} d \mathrm {sgn}\left (\tan \left (d x + c\right )\right ) \tan \left (d x + c\right )^{6}}\right )} \]
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(d*x + c)))/sqrt(abs(b)))/(b^10*d*sgn(tan(d*x + c))) + 1170*sqrt(2)*abs(b)^(3/2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(b)) - 2*sq rt(b*tan(d*x + c)))/sqrt(abs(b)))/(b^10*d*sgn(tan(d*x + c))) - 585*sqrt(2) *abs(b)^(3/2)*log(b*tan(d*x + c) + sqrt(2)*sqrt(b*tan(d*x + c))*sqrt(abs(b )) + abs(b))/(b^10*d*sgn(tan(d*x + c))) + 585*sqrt(2)*abs(b)^(3/2)*log(b*t an(d*x + c) - sqrt(2)*sqrt(b*tan(d*x + c))*sqrt(abs(b)) + abs(b))/(b^10*d* sgn(tan(d*x + c))) + 8*(585*b^6*tan(d*x + c)^6 - 117*b^6*tan(d*x + c)^4 + 65*b^6*tan(d*x + c)^2 - 45*b^6)/(sqrt(b*tan(d*x + c))*b^14*d*sgn(tan(d*x + c))*tan(d*x + c)^6))
Timed out. \[ \int \frac {1}{\left (b \tan ^3(c+d x)\right )^{5/2}} \, dx=\int \frac {1}{{\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}^{5/2}} \,d x \]