Integrand size = 25, antiderivative size = 269 \[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2 \, dx=-\frac {\sqrt {2} a^2 d^{5/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}+\frac {\sqrt {2} a^2 d^{5/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{f}-\frac {a^2 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} f}+\frac {a^2 d^{5/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{\sqrt {2} f}-\frac {4 a^2 d^2 \sqrt {d \tan (e+f x)}}{f}+\frac {4 a^2 (d \tan (e+f x))^{5/2}}{5 f}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f} \]
-1/2*a^2*d^(5/2)*ln(d^(1/2)-2^(1/2)*(d*tan(f*x+e))^(1/2)+d^(1/2)*tan(f*x+e ))/f*2^(1/2)+1/2*a^2*d^(5/2)*ln(d^(1/2)+2^(1/2)*(d*tan(f*x+e))^(1/2)+d^(1/ 2)*tan(f*x+e))/f*2^(1/2)-a^2*d^(5/2)*arctan(1-2^(1/2)*(d*tan(f*x+e))^(1/2) /d^(1/2))*2^(1/2)/f+a^2*d^(5/2)*arctan(1+2^(1/2)*(d*tan(f*x+e))^(1/2)/d^(1 /2))*2^(1/2)/f-4*a^2*d^2*(d*tan(f*x+e))^(1/2)/f+4/5*a^2*(d*tan(f*x+e))^(5/ 2)/f+2/7*a^2*(d*tan(f*x+e))^(7/2)/d/f
Time = 1.34 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.70 \[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2 \, dx=\frac {a^2 (d \tan (e+f x))^{5/2} \left (-70 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )+70 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right )-35 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )+35 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-280 \sqrt {\tan (e+f x)}+56 \tan ^{\frac {5}{2}}(e+f x)+20 \tan ^{\frac {7}{2}}(e+f x)\right )}{70 f \tan ^{\frac {5}{2}}(e+f x)} \]
(a^2*(d*Tan[e + f*x])^(5/2)*(-70*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f *x]]] + 70*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]] - 35*Sqrt[2]*Log [1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] + 35*Sqrt[2]*Log[1 + Sqrt[ 2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 280*Sqrt[Tan[e + f*x]] + 56*Tan[e + f*x]^(5/2) + 20*Tan[e + f*x]^(7/2)))/(70*f*Tan[e + f*x]^(5/2))
Time = 0.75 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.97, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.760, Rules used = {3042, 4026, 27, 2030, 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.
| \(\displaystyle \int (a \tan (e+f x)+a)^2 (d \tan (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \tan (e+f x)+a)^2 (d \tan (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 4026 |
| \(\displaystyle \int 2 a^2 \tan (e+f x) (d \tan (e+f x))^{5/2}dx+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle 2 a^2 \int \tan (e+f x) (d \tan (e+f x))^{5/2}dx+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 2030 |
| \(\displaystyle \frac {2 a^2 \int (d \tan (e+f x))^{7/2}dx}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 a^2 \int (d \tan (e+f x))^{7/2}dx}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 d (d \tan (e+f x))^{5/2}}{5 f}-d^2 \int (d \tan (e+f x))^{3/2}dx\right )}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 d (d \tan (e+f x))^{5/2}}{5 f}-d^2 \int (d \tan (e+f x))^{3/2}dx\right )}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 3954 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 d (d \tan (e+f x))^{5/2}}{5 f}-d^2 \left (\frac {2 d \sqrt {d \tan (e+f x)}}{f}-d^2 \int \frac {1}{\sqrt {d \tan (e+f x)}}dx\right )\right )}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 d (d \tan (e+f x))^{5/2}}{5 f}-d^2 \left (\frac {2 d \sqrt {d \tan (e+f x)}}{f}-d^2 \int \frac {1}{\sqrt {d \tan (e+f x)}}dx\right )\right )}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 d (d \tan (e+f x))^{5/2}}{5 f}-d^2 \left (\frac {2 d \sqrt {d \tan (e+f x)}}{f}-\frac {d^3 \int \frac {1}{\sqrt {d \tan (e+f x)} \left (\tan ^2(e+f x) d^2+d^2\right )}d(d \tan (e+f x))}{f}\right )\right )}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 d (d \tan (e+f x))^{5/2}}{5 f}-d^2 \left (\frac {2 d \sqrt {d \tan (e+f x)}}{f}-\frac {2 d^3 \int \frac {1}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}}{f}\right )\right )}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 d (d \tan (e+f x))^{5/2}}{5 f}-d^2 \left (\frac {2 d \sqrt {d \tan (e+f x)}}{f}-\frac {2 d^3 \left (\frac {\int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}}{2 d}+\frac {\int \frac {d^2 \tan ^2(e+f x)+d}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}}{2 d}\right )}{f}\right )\right )}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 d (d \tan (e+f x))^{5/2}}{5 f}-d^2 \left (\frac {2 d \sqrt {d \tan (e+f x)}}{f}-\frac {2 d^3 \left (\frac {\frac {1}{2} \int \frac {1}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}+\frac {1}{2} \int \frac {1}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 d}+\frac {\int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}}{2 d}\right )}{f}\right )\right )}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 d (d \tan (e+f x))^{5/2}}{5 f}-d^2 \left (\frac {2 d \sqrt {d \tan (e+f x)}}{f}-\frac {2 d^3 \left (\frac {\frac {\int \frac {1}{-d^2 \tan ^2(e+f x)-1}d\left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \tan ^2(e+f x)-1}d\left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}}{2 d}\right )}{f}\right )\right )}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 d (d \tan (e+f x))^{5/2}}{5 f}-d^2 \left (\frac {2 d \sqrt {d \tan (e+f x)}}{f}-\frac {2 d^3 \left (\frac {\int \frac {d-d^2 \tan ^2(e+f x)}{d^4 \tan ^4(e+f x)+d^2}d\sqrt {d \tan (e+f x)}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\right )\right )}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 d (d \tan (e+f x))^{5/2}}{5 f}-d^2 \left (\frac {2 d \sqrt {d \tan (e+f x)}}{f}-\frac {2 d^3 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\right )\right )}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 d (d \tan (e+f x))^{5/2}}{5 f}-d^2 \left (\frac {2 d \sqrt {d \tan (e+f x)}}{f}-\frac {2 d^3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\right )\right )}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 d (d \tan (e+f x))^{5/2}}{5 f}-d^2 \left (\frac {2 d \sqrt {d \tan (e+f x)}}{f}-\frac {2 d^3 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)-\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \tan (e+f x)}}{d^2 \tan ^2(e+f x)+\sqrt {2} d^{3/2} \tan (e+f x)+d}d\sqrt {d \tan (e+f x)}}{2 \sqrt {d}}}{2 d}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\right )\right )}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 d (d \tan (e+f x))^{5/2}}{5 f}-d^2 \left (\frac {2 d \sqrt {d \tan (e+f x)}}{f}-\frac {2 d^3 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {d} \tan (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \tan (e+f x)\right )}{\sqrt {2} \sqrt {d}}}{2 d}+\frac {\frac {\log \left (\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (-\sqrt {2} d^{3/2} \tan (e+f x)+d^2 \tan ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}}{2 d}\right )}{f}\right )\right )}{d}+\frac {2 a^2 (d \tan (e+f x))^{7/2}}{7 d f}\) |
(2*a^2*(d*Tan[e + f*x])^(7/2))/(7*d*f) + (2*a^2*((2*d*(d*Tan[e + f*x])^(5/ 2))/(5*f) - d^2*((-2*d^3*((-(ArcTan[1 - Sqrt[2]*Sqrt[d]*Tan[e + f*x]]/(Sqr t[2]*Sqrt[d])) + ArcTan[1 + Sqrt[2]*Sqrt[d]*Tan[e + f*x]]/(Sqrt[2]*Sqrt[d] ))/(2*d) + (-1/2*Log[d - Sqrt[2]*d^(3/2)*Tan[e + f*x] + d^2*Tan[e + f*x]^2 ]/(Sqrt[2]*Sqrt[d]) + Log[d + Sqrt[2]*d^(3/2)*Tan[e + f*x] + d^2*Tan[e + f *x]^2]/(2*Sqrt[2]*Sqrt[d]))/(2*d)))/f + (2*d*Sqrt[d*Tan[e + f*x]])/f)))/d
3.4.43.3.1 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[(Fx_.)*(v_)^(m_.)*((b_)*(v_))^(n_), x_Symbol] :> Simp[1/b^m Int[(b*v) ^(m + n)*Fx, x], x] /; FreeQ[{b, n}, x] && IntegerQ[m]
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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[d^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*( m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Simp[c^2 - d^2 + 2*c*d*Tan[e + f* x], x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b*c - a*d, 0] && !LeQ [m, -1] && !(EqQ[m, 2] && EqQ[a, 0])
Time = 0.97 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(\frac {2 a^{2} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {2 d \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-2 d^{3} \sqrt {d \tan \left (f x +e \right )}+\frac {d^{3} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4}\right )}{f d}\) | \(187\) |
| default | \(\frac {2 a^{2} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {2 d \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-2 d^{3} \sqrt {d \tan \left (f x +e \right )}+\frac {d^{3} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4}\right )}{f d}\) | \(187\) |
| parts | \(\frac {2 a^{2} d \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-\frac {d^{2} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f}+\frac {2 a^{2} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}} d^{2}}{3}+\frac {d^{4} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f d}+\frac {2 a^{2} \left (\frac {2 \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-2 d^{2} \sqrt {d \tan \left (f x +e \right )}+\frac {d^{2} \left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4}\right )}{f}\) | \(501\) |
2/f*a^2/d*(1/7*(d*tan(f*x+e))^(7/2)+2/5*d*(d*tan(f*x+e))^(5/2)-2*d^3*(d*ta n(f*x+e))^(1/2)+1/4*d^3*(d^2)^(1/4)*2^(1/2)*(ln((d*tan(f*x+e)+(d^2)^(1/4)* (d*tan(f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2))/(d*tan(f*x+e)-(d^2)^(1/4)*(d*tan (f*x+e))^(1/2)*2^(1/2)+(d^2)^(1/2)))+2*arctan(2^(1/2)/(d^2)^(1/4)*(d*tan(f *x+e))^(1/2)+1)-2*arctan(-2^(1/2)/(d^2)^(1/4)*(d*tan(f*x+e))^(1/2)+1)))
Result contains complex when optimal does not.
Time = 0.24 (sec) , antiderivative size = 262, normalized size of antiderivative = 0.97 \[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2 \, dx=\frac {35 \, \left (-\frac {a^{8} d^{10}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (\sqrt {d \tan \left (f x + e\right )} a^{2} d^{2} + \left (-\frac {a^{8} d^{10}}{f^{4}}\right )^{\frac {1}{4}} f\right ) + 35 i \, \left (-\frac {a^{8} d^{10}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (\sqrt {d \tan \left (f x + e\right )} a^{2} d^{2} + i \, \left (-\frac {a^{8} d^{10}}{f^{4}}\right )^{\frac {1}{4}} f\right ) - 35 i \, \left (-\frac {a^{8} d^{10}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (\sqrt {d \tan \left (f x + e\right )} a^{2} d^{2} - i \, \left (-\frac {a^{8} d^{10}}{f^{4}}\right )^{\frac {1}{4}} f\right ) - 35 \, \left (-\frac {a^{8} d^{10}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (\sqrt {d \tan \left (f x + e\right )} a^{2} d^{2} - \left (-\frac {a^{8} d^{10}}{f^{4}}\right )^{\frac {1}{4}} f\right ) + 2 \, {\left (5 \, a^{2} d^{2} \tan \left (f x + e\right )^{3} + 14 \, a^{2} d^{2} \tan \left (f x + e\right )^{2} - 70 \, a^{2} d^{2}\right )} \sqrt {d \tan \left (f x + e\right )}}{35 \, f} \]
1/35*(35*(-a^8*d^10/f^4)^(1/4)*f*log(sqrt(d*tan(f*x + e))*a^2*d^2 + (-a^8* d^10/f^4)^(1/4)*f) + 35*I*(-a^8*d^10/f^4)^(1/4)*f*log(sqrt(d*tan(f*x + e)) *a^2*d^2 + I*(-a^8*d^10/f^4)^(1/4)*f) - 35*I*(-a^8*d^10/f^4)^(1/4)*f*log(s qrt(d*tan(f*x + e))*a^2*d^2 - I*(-a^8*d^10/f^4)^(1/4)*f) - 35*(-a^8*d^10/f ^4)^(1/4)*f*log(sqrt(d*tan(f*x + e))*a^2*d^2 - (-a^8*d^10/f^4)^(1/4)*f) + 2*(5*a^2*d^2*tan(f*x + e)^3 + 14*a^2*d^2*tan(f*x + e)^2 - 70*a^2*d^2)*sqrt (d*tan(f*x + e)))/f
\[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2 \, dx=a^{2} \left (\int \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx + \int 2 \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \tan {\left (e + f x \right )}\, dx + \int \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \tan ^{2}{\left (e + f x \right )}\, dx\right ) \]
a**2*(Integral((d*tan(e + f*x))**(5/2), x) + Integral(2*(d*tan(e + f*x))** (5/2)*tan(e + f*x), x) + Integral((d*tan(e + f*x))**(5/2)*tan(e + f*x)**2, x))
Time = 0.39 (sec) , antiderivative size = 210, normalized size of antiderivative = 0.78 \[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2 \, dx=\frac {20 \, \left (d \tan \left (f x + e\right )\right )^{\frac {7}{2}} a^{2} + 56 \, \left (d \tan \left (f x + e\right )\right )^{\frac {5}{2}} a^{2} d - 280 \, \sqrt {d \tan \left (f x + e\right )} a^{2} d^{3} + 35 \, {\left (2 \, \sqrt {2} d^{\frac {7}{2}} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right ) + 2 \, \sqrt {2} d^{\frac {7}{2}} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right ) + \sqrt {2} d^{\frac {7}{2}} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right ) - \sqrt {2} d^{\frac {7}{2}} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )\right )} a^{2}}{70 \, d f} \]
1/70*(20*(d*tan(f*x + e))^(7/2)*a^2 + 56*(d*tan(f*x + e))^(5/2)*a^2*d - 28 0*sqrt(d*tan(f*x + e))*a^2*d^3 + 35*(2*sqrt(2)*d^(7/2)*arctan(1/2*sqrt(2)* (sqrt(2)*sqrt(d) + 2*sqrt(d*tan(f*x + e)))/sqrt(d)) + 2*sqrt(2)*d^(7/2)*ar ctan(-1/2*sqrt(2)*(sqrt(2)*sqrt(d) - 2*sqrt(d*tan(f*x + e)))/sqrt(d)) + sq rt(2)*d^(7/2)*log(d*tan(f*x + e) + sqrt(2)*sqrt(d*tan(f*x + e))*sqrt(d) + d) - sqrt(2)*d^(7/2)*log(d*tan(f*x + e) - sqrt(2)*sqrt(d*tan(f*x + e))*sqr t(d) + d))*a^2)/(d*f)
Timed out. \[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2 \, dx=\text {Timed out} \]
Time = 6.29 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.46 \[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2 \, dx=\frac {4\,a^2\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}{5\,f}-\frac {4\,a^2\,d^2\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{f}+\frac {2\,a^2\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{7/2}}{7\,d\,f}-\frac {{\left (-1\right )}^{1/4}\,a^2\,d^{5/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )\,2{}\mathrm {i}}{f}-\frac {2\,{\left (-1\right )}^{1/4}\,a^2\,d^{5/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,1{}\mathrm {i}}{\sqrt {d}}\right )}{f} \]