Integrand size = 14, antiderivative size = 364 \[ \int \left (b \tan ^3(c+d x)\right )^{5/2} \, dx=-\frac {2 b^2 \cot (c+d x) \sqrt {b \tan ^3(c+d x)}}{d}-\frac {b^2 \arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {b \tan ^3(c+d x)}}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b^2 \arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right ) \sqrt {b \tan ^3(c+d x)}}{\sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}-\frac {b^2 \log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {b \tan ^3(c+d x)}}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {b^2 \log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right ) \sqrt {b \tan ^3(c+d x)}}{2 \sqrt {2} d \tan ^{\frac {3}{2}}(c+d x)}+\frac {2 b^2 \tan (c+d x) \sqrt {b \tan ^3(c+d x)}}{5 d}-\frac {2 b^2 \tan ^3(c+d x) \sqrt {b \tan ^3(c+d x)}}{9 d}+\frac {2 b^2 \tan ^5(c+d x) \sqrt {b \tan ^3(c+d x)}}{13 d} \]
-2*b^2*cot(d*x+c)*(b*tan(d*x+c)^3)^(1/2)/d+1/2*b^2*arctan(-1+2^(1/2)*tan(d *x+c)^(1/2))*(b*tan(d*x+c)^3)^(1/2)/d*2^(1/2)/tan(d*x+c)^(3/2)+1/2*b^2*arc tan(1+2^(1/2)*tan(d*x+c)^(1/2))*(b*tan(d*x+c)^3)^(1/2)/d*2^(1/2)/tan(d*x+c )^(3/2)-1/4*b^2*ln(1-2^(1/2)*tan(d*x+c)^(1/2)+tan(d*x+c))*(b*tan(d*x+c)^3) ^(1/2)/d*2^(1/2)/tan(d*x+c)^(3/2)+1/4*b^2*ln(1+2^(1/2)*tan(d*x+c)^(1/2)+ta n(d*x+c))*(b*tan(d*x+c)^3)^(1/2)/d*2^(1/2)/tan(d*x+c)^(3/2)+2/5*b^2*(b*tan (d*x+c)^3)^(1/2)*tan(d*x+c)/d-2/9*b^2*(b*tan(d*x+c)^3)^(1/2)*tan(d*x+c)^3/ d+2/13*b^2*(b*tan(d*x+c)^3)^(1/2)*tan(d*x+c)^5/d
Time = 0.68 (sec) , antiderivative size = 204, normalized size of antiderivative = 0.56 \[ \int \left (b \tan ^3(c+d x)\right )^{5/2} \, dx=\frac {\left (b \tan ^3(c+d x)\right )^{5/2} \left (-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}+\frac {\arctan \left (1+\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\log \left (1-\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2}}+\frac {\log \left (1+\sqrt {2} \sqrt {\tan (c+d x)}+\tan (c+d x)\right )}{2 \sqrt {2}}-2 \sqrt {\tan (c+d x)}+\frac {2}{5} \tan ^{\frac {5}{2}}(c+d x)-\frac {2}{9} \tan ^{\frac {9}{2}}(c+d x)+\frac {2}{13} \tan ^{\frac {13}{2}}(c+d x)\right )}{d \tan ^{\frac {15}{2}}(c+d x)} \]
((b*Tan[c + d*x]^3)^(5/2)*(-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2 ]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[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*Sqrt[Tan[c + d*x]] + (2*Tan[c + d*x] ^(5/2))/5 - (2*Tan[c + d*x]^(9/2))/9 + (2*Tan[c + d*x]^(13/2))/13))/(d*Tan [c + d*x]^(15/2))
Time = 0.79 (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, 3954, 3042, 3954, 3042, 3954, 3042, 3954, 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.
| \(\displaystyle \int \left (b \tan ^3(c+d x)\right )^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (b \tan (c+d x)^3\right )^{5/2}dx\) |
| \(\Big \downarrow \) 4141 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \int \tan ^{\frac {15}{2}}(c+d x)dx}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \int \tan (c+d x)^{15/2}dx}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\int \tan ^{\frac {11}{2}}(c+d x)dx\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\int \tan (c+d x)^{11/2}dx\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\int \tan ^{\frac {7}{2}}(c+d x)dx+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\int \tan (c+d x)^{7/2}dx+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (-\int \tan ^{\frac {3}{2}}(c+d x)dx+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (-\int \tan (c+d x)^{3/2}dx+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\int \frac {1}{\sqrt {\tan (c+d x)}}dx+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 \sqrt {\tan (c+d x)}}{d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\int \frac {1}{\sqrt {\tan (c+d x)}}dx+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 \sqrt {\tan (c+d x)}}{d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\frac {\int \frac {1}{\sqrt {\tan (c+d x)} \left (\tan ^2(c+d x)+1\right )}d\tan (c+d x)}{d}+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 \sqrt {\tan (c+d x)}}{d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\frac {2 \int \frac {1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}}{d}+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 \sqrt {\tan (c+d x)}}{d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\frac {2 \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {\tan (c+d x)+1}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}\right )}{d}+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 \sqrt {\tan (c+d x)}}{d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\frac {2 \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \int \frac {1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )\right )}{d}+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 \sqrt {\tan (c+d x)}}{d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\frac {2 \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}-\frac {\int \frac {1}{-\tan (c+d x)-1}d\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 \sqrt {\tan (c+d x)}}{d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\frac {2 \left (\frac {1}{2} \int \frac {1-\tan (c+d x)}{\tan ^2(c+d x)+1}d\sqrt {\tan (c+d x)}+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 \sqrt {\tan (c+d x)}}{d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\frac {2 \left (\frac {1}{2} \left (-\frac {\int -\frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 \sqrt {\tan (c+d x)}}{d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 \sqrt {\tan (c+d x)}}{d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {\sqrt {2}-2 \sqrt {\tan (c+d x)}}{\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}}{2 \sqrt {2}}+\frac {1}{2} \int \frac {\sqrt {2} \sqrt {\tan (c+d x)}+1}{\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1}d\sqrt {\tan (c+d x)}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )\right )}{d}+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 \sqrt {\tan (c+d x)}}{d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {b^2 \sqrt {b \tan ^3(c+d x)} \left (\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{\sqrt {2}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )}{\sqrt {2}}\right )+\frac {1}{2} \left (\frac {\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}-\frac {\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )}{2 \sqrt {2}}\right )\right )}{d}+\frac {2 \tan ^{\frac {13}{2}}(c+d x)}{13 d}-\frac {2 \tan ^{\frac {9}{2}}(c+d x)}{9 d}+\frac {2 \tan ^{\frac {5}{2}}(c+d x)}{5 d}-\frac {2 \sqrt {\tan (c+d x)}}{d}\right )}{\tan ^{\frac {3}{2}}(c+d x)}\) |
(b^2*Sqrt[b*Tan[c + d*x]^3]*((2*((-(ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] /Sqrt[2]) + ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]]/Sqrt[2])/2 + (-1/2*Log[ 1 - Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]/Sqrt[2] + Log[1 + Sqrt[2]*S qrt[Tan[c + d*x]] + Tan[c + d*x]]/(2*Sqrt[2]))/2))/d - (2*Sqrt[Tan[c + d*x ]])/d + (2*Tan[c + d*x]^(5/2))/(5*d) - (2*Tan[c + d*x]^(9/2))/(9*d) + (2*T an[c + d*x]^(13/2))/(13*d)))/Tan[c + d*x]^(3/2)
3.1.30.3.1 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*((b*Tan[c + d *x])^(n - 1)/(d*(n - 1))), x] - Simp[b^2 Int[(b*Tan[c + d*x])^(n - 2), x] , x] /; FreeQ[{b, c, d}, x] && GtQ[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 = 0.17 (sec) , antiderivative size = 263, normalized size of antiderivative = 0.72
| method | result | size |
| derivativedivides | \(\frac {{\left (b \left (\tan ^{3}\left (d x +c \right )\right )\right )}^{\frac {5}{2}} \left (360 \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}}-520 b^{2} \left (b \tan \left (d x +c \right )\right )^{\frac {9}{2}}+585 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+1170 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+1170 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+936 b^{4} \left (b \tan \left (d x +c \right )\right )^{\frac {5}{2}}-4680 b^{6} \sqrt {b \tan \left (d x +c \right )}\right )}{2340 d \tan \left (d x +c \right )^{5} \left (b \tan \left (d x +c \right )\right )^{\frac {5}{2}} b^{4}}\) | \(263\) |
| default | \(\frac {{\left (b \left (\tan ^{3}\left (d x +c \right )\right )\right )}^{\frac {5}{2}} \left (360 \left (b \tan \left (d x +c \right )\right )^{\frac {13}{2}}-520 b^{2} \left (b \tan \left (d x +c \right )\right )^{\frac {9}{2}}+585 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {b \tan \left (d x +c \right )+\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}{b \tan \left (d x +c \right )-\left (b^{2}\right )^{\frac {1}{4}} \sqrt {b \tan \left (d x +c \right )}\, \sqrt {2}+\sqrt {b^{2}}}\right )+1170 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}+\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+1170 b^{6} \left (b^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {b \tan \left (d x +c \right )}-\left (b^{2}\right )^{\frac {1}{4}}}{\left (b^{2}\right )^{\frac {1}{4}}}\right )+936 b^{4} \left (b \tan \left (d x +c \right )\right )^{\frac {5}{2}}-4680 b^{6} \sqrt {b \tan \left (d x +c \right )}\right )}{2340 d \tan \left (d x +c \right )^{5} \left (b \tan \left (d x +c \right )\right )^{\frac {5}{2}} b^{4}}\) | \(263\) |
1/2340/d*(b*tan(d*x+c)^3)^(5/2)*(360*(b*tan(d*x+c))^(13/2)-520*b^2*(b*tan( d*x+c))^(9/2)+585*b^6*(b^2)^(1/4)*2^(1/2)*ln((b*tan(d*x+c)+(b^2)^(1/4)*(b* tan(d*x+c))^(1/2)*2^(1/2)+(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*b^6*(b^2)^(1/4)*2^(1/2)*arctan((2^( 1/2)*(b*tan(d*x+c))^(1/2)+(b^2)^(1/4))/(b^2)^(1/4))+1170*b^6*(b^2)^(1/4)*2 ^(1/2)*arctan((2^(1/2)*(b*tan(d*x+c))^(1/2)-(b^2)^(1/4))/(b^2)^(1/4))+936* b^4*(b*tan(d*x+c))^(5/2)-4680*b^6*(b*tan(d*x+c))^(1/2))/tan(d*x+c)^5/(b*ta n(d*x+c))^(5/2)/b^4
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 332, normalized size of antiderivative = 0.91 \[ \int \left (b \tan ^3(c+d x)\right )^{5/2} \, dx=\frac {585 \, \left (-\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} d \log \left (\frac {\sqrt {b \tan \left (d x + c\right )^{3}} b^{2} + \left (-\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} d \tan \left (d x + c\right )}{\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right ) + 585 i \, \left (-\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} d \log \left (\frac {\sqrt {b \tan \left (d x + c\right )^{3}} b^{2} + i \, \left (-\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} d \tan \left (d x + c\right )}{\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right ) - 585 i \, \left (-\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} d \log \left (\frac {\sqrt {b \tan \left (d x + c\right )^{3}} b^{2} - i \, \left (-\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} d \tan \left (d x + c\right )}{\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right ) - 585 \, \left (-\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} d \log \left (\frac {\sqrt {b \tan \left (d x + c\right )^{3}} b^{2} - \left (-\frac {b^{10}}{d^{4}}\right )^{\frac {1}{4}} d \tan \left (d x + c\right )}{\tan \left (d x + c\right )}\right ) \tan \left (d x + c\right ) + 4 \, {\left (45 \, b^{2} \tan \left (d x + c\right )^{6} - 65 \, b^{2} \tan \left (d x + c\right )^{4} + 117 \, b^{2} \tan \left (d x + c\right )^{2} - 585 \, b^{2}\right )} \sqrt {b \tan \left (d x + c\right )^{3}}}{1170 \, d \tan \left (d x + c\right )} \]
1/1170*(585*(-b^10/d^4)^(1/4)*d*log((sqrt(b*tan(d*x + c)^3)*b^2 + (-b^10/d ^4)^(1/4)*d*tan(d*x + c))/tan(d*x + c))*tan(d*x + c) + 585*I*(-b^10/d^4)^( 1/4)*d*log((sqrt(b*tan(d*x + c)^3)*b^2 + I*(-b^10/d^4)^(1/4)*d*tan(d*x + c ))/tan(d*x + c))*tan(d*x + c) - 585*I*(-b^10/d^4)^(1/4)*d*log((sqrt(b*tan( d*x + c)^3)*b^2 - I*(-b^10/d^4)^(1/4)*d*tan(d*x + c))/tan(d*x + c))*tan(d* x + c) - 585*(-b^10/d^4)^(1/4)*d*log((sqrt(b*tan(d*x + c)^3)*b^2 - (-b^10/ d^4)^(1/4)*d*tan(d*x + c))/tan(d*x + c))*tan(d*x + c) + 4*(45*b^2*tan(d*x + c)^6 - 65*b^2*tan(d*x + c)^4 + 117*b^2*tan(d*x + c)^2 - 585*b^2)*sqrt(b* tan(d*x + c)^3))/(d*tan(d*x + c))
\[ \int \left (b \tan ^3(c+d x)\right )^{5/2} \, dx=\int \left (b \tan ^{3}{\left (c + d x \right )}\right )^{\frac {5}{2}}\, dx \]
Time = 0.34 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.49 \[ \int \left (b \tan ^3(c+d x)\right )^{5/2} \, dx=\frac {360 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{\frac {13}{2}} - 520 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{\frac {9}{2}} + 936 \, b^{\frac {5}{2}} \tan \left (d x + c\right )^{\frac {5}{2}} + 585 \, {\left (2 \, \sqrt {2} \sqrt {b} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + 2 \, \sqrt {2} \sqrt {b} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - 2 \, \sqrt {\tan \left (d x + c\right )}\right )}\right ) + \sqrt {2} \sqrt {b} \log \left (\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right ) - \sqrt {2} \sqrt {b} \log \left (-\sqrt {2} \sqrt {\tan \left (d x + c\right )} + \tan \left (d x + c\right ) + 1\right )\right )} b^{2} - 4680 \, b^{\frac {5}{2}} \sqrt {\tan \left (d x + c\right )}}{2340 \, d} \]
1/2340*(360*b^(5/2)*tan(d*x + c)^(13/2) - 520*b^(5/2)*tan(d*x + c)^(9/2) + 936*b^(5/2)*tan(d*x + c)^(5/2) + 585*(2*sqrt(2)*sqrt(b)*arctan(1/2*sqrt(2 )*(sqrt(2) + 2*sqrt(tan(d*x + c)))) + 2*sqrt(2)*sqrt(b)*arctan(-1/2*sqrt(2 )*(sqrt(2) - 2*sqrt(tan(d*x + c)))) + sqrt(2)*sqrt(b)*log(sqrt(2)*sqrt(tan (d*x + c)) + tan(d*x + c) + 1) - sqrt(2)*sqrt(b)*log(-sqrt(2)*sqrt(tan(d*x + c)) + tan(d*x + c) + 1))*b^2 - 4680*b^(5/2)*sqrt(tan(d*x + c)))/d
Time = 0.50 (sec) , antiderivative size = 291, normalized size of antiderivative = 0.80 \[ \int \left (b \tan ^3(c+d x)\right )^{5/2} \, dx=\frac {1}{2340} \, {\left (\frac {1170 \, \sqrt {2} b \sqrt {{\left | b \right |}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} + 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{d} + \frac {1170 \, \sqrt {2} b \sqrt {{\left | b \right |}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | b \right |}} - 2 \, \sqrt {b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt {{\left | b \right |}}}\right )}{d} + \frac {585 \, \sqrt {2} b \sqrt {{\left | b \right |}} \log \left (b \tan \left (d x + c\right ) + \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{d} - \frac {585 \, \sqrt {2} b \sqrt {{\left | b \right |}} \log \left (b \tan \left (d x + c\right ) - \sqrt {2} \sqrt {b \tan \left (d x + c\right )} \sqrt {{\left | b \right |}} + {\left | b \right |}\right )}{d} + \frac {8 \, {\left (45 \, \sqrt {b \tan \left (d x + c\right )} b^{66} d^{12} \tan \left (d x + c\right )^{6} - 65 \, \sqrt {b \tan \left (d x + c\right )} b^{66} d^{12} \tan \left (d x + c\right )^{4} + 117 \, \sqrt {b \tan \left (d x + c\right )} b^{66} d^{12} \tan \left (d x + c\right )^{2} - 585 \, \sqrt {b \tan \left (d x + c\right )} b^{66} d^{12}\right )}}{b^{65} d^{13}}\right )} b \mathrm {sgn}\left (\tan \left (d x + c\right )\right ) \]
1/2340*(1170*sqrt(2)*b*sqrt(abs(b))*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(abs(b )) + 2*sqrt(b*tan(d*x + c)))/sqrt(abs(b)))/d + 1170*sqrt(2)*b*sqrt(abs(b)) *arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(abs(b)) - 2*sqrt(b*tan(d*x + c)))/sqrt( abs(b)))/d + 585*sqrt(2)*b*sqrt(abs(b))*log(b*tan(d*x + c) + sqrt(2)*sqrt( b*tan(d*x + c))*sqrt(abs(b)) + abs(b))/d - 585*sqrt(2)*b*sqrt(abs(b))*log( b*tan(d*x + c) - sqrt(2)*sqrt(b*tan(d*x + c))*sqrt(abs(b)) + abs(b))/d + 8 *(45*sqrt(b*tan(d*x + c))*b^66*d^12*tan(d*x + c)^6 - 65*sqrt(b*tan(d*x + c ))*b^66*d^12*tan(d*x + c)^4 + 117*sqrt(b*tan(d*x + c))*b^66*d^12*tan(d*x + c)^2 - 585*sqrt(b*tan(d*x + c))*b^66*d^12)/(b^65*d^13))*b*sgn(tan(d*x + c ))
Timed out. \[ \int \left (b \tan ^3(c+d x)\right )^{5/2} \, dx=\int {\left (b\,{\mathrm {tan}\left (c+d\,x\right )}^3\right )}^{5/2} \,d x \]