Integrand size = 25, antiderivative size = 186 \[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^3 \, dx=-\frac {2 \sqrt {2} a^3 d^{5/2} \arctan \left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{f}-\frac {4 a^3 d^2 \sqrt {d \tan (e+f x)}}{f}-\frac {4 a^3 d (d \tan (e+f x))^{3/2}}{3 f}+\frac {4 a^3 (d \tan (e+f x))^{5/2}}{5 f}+\frac {40 a^3 (d \tan (e+f x))^{7/2}}{63 d f}+\frac {2 (d \tan (e+f x))^{7/2} \left (a^3+a^3 \tan (e+f x)\right )}{9 d f} \] Output:
-2*2^(1/2)*a^3*d^(5/2)*arctan(1/2*(d^(1/2)-d^(1/2)*tan(f*x+e))*2^(1/2)/(d* tan(f*x+e))^(1/2))/f-4*a^3*d^2*(d*tan(f*x+e))^(1/2)/f-4/3*a^3*d*(d*tan(f*x +e))^(3/2)/f+4/5*a^3*(d*tan(f*x+e))^(5/2)/f+40/63*a^3*(d*tan(f*x+e))^(7/2) /d/f+2/9*(d*tan(f*x+e))^(7/2)*(a^3+a^3*tan(f*x+e))/d/f
Leaf count is larger than twice the leaf count of optimal. \(401\) vs. \(2(186)=372\).
Time = 5.29 (sec) , antiderivative size = 401, normalized size of antiderivative = 2.16 \[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^3 \, dx=\frac {a^3 d (d \tan (e+f x))^{3/2} (1+\tan (e+f x))^3 \left (1260 \arctan \left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \cos ^3(e+f x) \sqrt [4]{-\tan (e+f x)}-1260 \text {arctanh}\left (\sqrt [4]{-\tan ^2(e+f x)}\right ) \cos ^3(e+f x) \sqrt [4]{-\tan (e+f x)}+\cos (e+f x) \sqrt [4]{\tan (e+f x)} \left (-630 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x)+630 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right ) \cos ^2(e+f x)-315 \sqrt {2} \cos ^2(e+f x) \log \left (1-\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )+315 \sqrt {2} \cos ^2(e+f x) \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )-2520 \cos ^2(e+f x) \sqrt {\tan (e+f x)}-840 \cos ^2(e+f x) \tan ^{\frac {3}{2}}(e+f x)+504 \cos ^2(e+f x) \tan ^{\frac {5}{2}}(e+f x)+140 \sin ^2(e+f x) \tan ^{\frac {5}{2}}(e+f x)+270 \sin (2 (e+f x)) \tan ^{\frac {5}{2}}(e+f x)\right )\right )}{630 f (\cos (e+f x)+\sin (e+f x))^3 \tan ^{\frac {7}{4}}(e+f x)} \] Input:
Integrate[(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3,x]
Output:
(a^3*d*(d*Tan[e + f*x])^(3/2)*(1 + Tan[e + f*x])^3*(1260*ArcTan[(-Tan[e + f*x]^2)^(1/4)]*Cos[e + f*x]^3*(-Tan[e + f*x])^(1/4) - 1260*ArcTanh[(-Tan[e + f*x]^2)^(1/4)]*Cos[e + f*x]^3*(-Tan[e + f*x])^(1/4) + Cos[e + f*x]*Tan[ e + f*x]^(1/4)*(-630*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x]^2 + 630*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Tan[e + f*x]]]*Cos[e + f*x] ^2 - 315*Sqrt[2]*Cos[e + f*x]^2*Log[1 - Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] + 315*Sqrt[2]*Cos[e + f*x]^2*Log[1 + Sqrt[2]*Sqrt[Tan[e + f*x]] + Tan[e + f*x]] - 2520*Cos[e + f*x]^2*Sqrt[Tan[e + f*x]] - 840*Cos[e + f*x] ^2*Tan[e + f*x]^(3/2) + 504*Cos[e + f*x]^2*Tan[e + f*x]^(5/2) + 140*Sin[e + f*x]^2*Tan[e + f*x]^(5/2) + 270*Sin[2*(e + f*x)]*Tan[e + f*x]^(5/2))))/( 630*f*(Cos[e + f*x] + Sin[e + f*x])^3*Tan[e + f*x]^(7/4))
Time = 1.05 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.09, number of steps used = 14, number of rules used = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.520, Rules used = {3042, 4049, 3042, 4113, 3042, 4011, 3042, 4011, 3042, 4011, 3042, 4015, 218}
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.
| \(\displaystyle \int (a \tan (e+f x)+a)^3 (d \tan (e+f x))^{5/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int (a \tan (e+f x)+a)^3 (d \tan (e+f x))^{5/2}dx\) |
| \(\Big \downarrow \) 4049 |
| \(\displaystyle \frac {2 \int (d \tan (e+f x))^{5/2} \left (10 d \tan ^2(e+f x) a^3+d a^3+9 d \tan (e+f x) a^3\right )dx}{9 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) (d \tan (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int (d \tan (e+f x))^{5/2} \left (10 d \tan (e+f x)^2 a^3+d a^3+9 d \tan (e+f x) a^3\right )dx}{9 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) (d \tan (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {2 \left (\int (d \tan (e+f x))^{5/2} \left (9 a^3 d \tan (e+f x)-9 a^3 d\right )dx+\frac {20 a^3 (d \tan (e+f x))^{7/2}}{7 f}\right )}{9 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) (d \tan (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\int (d \tan (e+f x))^{5/2} \left (9 a^3 d \tan (e+f x)-9 a^3 d\right )dx+\frac {20 a^3 (d \tan (e+f x))^{7/2}}{7 f}\right )}{9 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) (d \tan (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {2 \left (\int (d \tan (e+f x))^{3/2} \left (-9 d^2 a^3-9 d^2 \tan (e+f x) a^3\right )dx+\frac {20 a^3 (d \tan (e+f x))^{7/2}}{7 f}+\frac {18 a^3 d (d \tan (e+f x))^{5/2}}{5 f}\right )}{9 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) (d \tan (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\int (d \tan (e+f x))^{3/2} \left (-9 d^2 a^3-9 d^2 \tan (e+f x) a^3\right )dx+\frac {20 a^3 (d \tan (e+f x))^{7/2}}{7 f}+\frac {18 a^3 d (d \tan (e+f x))^{5/2}}{5 f}\right )}{9 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) (d \tan (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {2 \left (\int \sqrt {d \tan (e+f x)} \left (9 a^3 d^3-9 a^3 d^3 \tan (e+f x)\right )dx-\frac {6 a^3 d^2 (d \tan (e+f x))^{3/2}}{f}+\frac {20 a^3 (d \tan (e+f x))^{7/2}}{7 f}+\frac {18 a^3 d (d \tan (e+f x))^{5/2}}{5 f}\right )}{9 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) (d \tan (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\int \sqrt {d \tan (e+f x)} \left (9 a^3 d^3-9 a^3 d^3 \tan (e+f x)\right )dx-\frac {6 a^3 d^2 (d \tan (e+f x))^{3/2}}{f}+\frac {20 a^3 (d \tan (e+f x))^{7/2}}{7 f}+\frac {18 a^3 d (d \tan (e+f x))^{5/2}}{5 f}\right )}{9 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) (d \tan (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {2 \left (\int \frac {9 a^3 d^4+9 a^3 \tan (e+f x) d^4}{\sqrt {d \tan (e+f x)}}dx-\frac {18 a^3 d^3 \sqrt {d \tan (e+f x)}}{f}-\frac {6 a^3 d^2 (d \tan (e+f x))^{3/2}}{f}+\frac {20 a^3 (d \tan (e+f x))^{7/2}}{7 f}+\frac {18 a^3 d (d \tan (e+f x))^{5/2}}{5 f}\right )}{9 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) (d \tan (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\int \frac {9 a^3 d^4+9 a^3 \tan (e+f x) d^4}{\sqrt {d \tan (e+f x)}}dx-\frac {18 a^3 d^3 \sqrt {d \tan (e+f x)}}{f}-\frac {6 a^3 d^2 (d \tan (e+f x))^{3/2}}{f}+\frac {20 a^3 (d \tan (e+f x))^{7/2}}{7 f}+\frac {18 a^3 d (d \tan (e+f x))^{5/2}}{5 f}\right )}{9 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) (d \tan (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 4015 |
| \(\displaystyle \frac {2 \left (-\frac {162 a^6 d^8 \int \frac {1}{162 a^6 d^8+81 \cot (e+f x) \left (a^3 d^4-a^3 d^4 \tan (e+f x)\right )^2}d\frac {9 \left (a^3 d^4-a^3 d^4 \tan (e+f x)\right )}{\sqrt {d \tan (e+f x)}}}{f}-\frac {18 a^3 d^3 \sqrt {d \tan (e+f x)}}{f}-\frac {6 a^3 d^2 (d \tan (e+f x))^{3/2}}{f}+\frac {18 a^3 d (d \tan (e+f x))^{5/2}}{5 f}+\frac {20 a^3 (d \tan (e+f x))^{7/2}}{7 f}\right )}{9 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) (d \tan (e+f x))^{7/2}}{9 d f}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 \left (-\frac {9 \sqrt {2} a^3 d^{7/2} \arctan \left (\frac {a^3 d^4-a^3 d^4 \tan (e+f x)}{\sqrt {2} a^3 d^{7/2} \sqrt {d \tan (e+f x)}}\right )}{f}-\frac {18 a^3 d^3 \sqrt {d \tan (e+f x)}}{f}-\frac {6 a^3 d^2 (d \tan (e+f x))^{3/2}}{f}+\frac {18 a^3 d (d \tan (e+f x))^{5/2}}{5 f}+\frac {20 a^3 (d \tan (e+f x))^{7/2}}{7 f}\right )}{9 d}+\frac {2 \left (a^3 \tan (e+f x)+a^3\right ) (d \tan (e+f x))^{7/2}}{9 d f}\) |
Input:
Int[(d*Tan[e + f*x])^(5/2)*(a + a*Tan[e + f*x])^3,x]
Output:
(2*(d*Tan[e + f*x])^(7/2)*(a^3 + a^3*Tan[e + f*x]))/(9*d*f) + (2*((-9*Sqrt [2]*a^3*d^(7/2)*ArcTan[(a^3*d^4 - a^3*d^4*Tan[e + f*x])/(Sqrt[2]*a^3*d^(7/ 2)*Sqrt[d*Tan[e + f*x]])])/f - (18*a^3*d^3*Sqrt[d*Tan[e + f*x]])/f - (6*a^ 3*d^2*(d*Tan[e + f*x])^(3/2))/f + (18*a^3*d*(d*Tan[e + f*x])^(5/2))/(5*f) + (20*a^3*(d*Tan[e + f*x])^(7/2))/(7*f)))/(9*d)
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_ )]], x_Symbol] :> Simp[-2*(d^2/f) Subst[Int[1/(2*c*d + b*x^2), x], x, (c - d*Tan[e + f*x])/Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && EqQ[c^2 - d^2, 0]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2 , 0] && NeQ[c^2 + d^2, 0] && IntegerQ[2*m] && GtQ[m, 2] && (GeQ[n, -1] || I ntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) )
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(353\) vs. \(2(157)=314\).
Time = 1.82 (sec) , antiderivative size = 354, normalized size of antiderivative = 1.90
| method | result | size |
| derivativedivides | \(\frac {2 a^{3} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {3 d \left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {2 d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {2 d^{3} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-2 d^{4} \sqrt {d \tan \left (f x +e \right )}+2 d^{5} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\) | \(354\) |
| default | \(\frac {2 a^{3} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}+\frac {3 d \left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}+\frac {2 d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}-\frac {2 d^{3} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{3}-2 d^{4} \sqrt {d \tan \left (f x +e \right )}+2 d^{5} \left (\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}\right )\right )}{f \,d^{2}}\) | \(354\) |
| parts | \(\frac {2 a^{3} 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^{3} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {9}{2}}}{9}-\frac {d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {5}{2}}}{5}+d^{4} \sqrt {d \tan \left (f x +e \right )}-\frac {d^{4} \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 )}{8}\right )}{f \,d^{2}}+\frac {3 a^{3} \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}+\frac {6 a^{3} \left (\frac {\left (d \tan \left (f x +e \right )\right )^{\frac {7}{2}}}{7}-\frac {d^{2} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{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}\) | \(688\) |
Input:
int((d*tan(f*x+e))^(5/2)*(a+a*tan(f*x+e))^3,x,method=_RETURNVERBOSE)
Output:
2/f*a^3/d^2*(1/9*(d*tan(f*x+e))^(9/2)+3/7*d*(d*tan(f*x+e))^(7/2)+2/5*d^2*( d*tan(f*x+e))^(5/2)-2/3*d^3*(d*tan(f*x+e))^(3/2)-2*d^4*(d*tan(f*x+e))^(1/2 )+2*d^5*(1/8/d*(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))+1/8/(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))))
Time = 0.09 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.63 \[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^3 \, dx=\left [\frac {315 \, \sqrt {2} a^{3} \sqrt {-d} d^{2} \log \left (\frac {d \tan \left (f x + e\right )^{2} + 2 \, \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {-d} {\left (\tan \left (f x + e\right ) - 1\right )} - 4 \, d \tan \left (f x + e\right ) + d}{\tan \left (f x + e\right )^{2} + 1}\right ) + 2 \, {\left (35 \, a^{3} d^{2} \tan \left (f x + e\right )^{4} + 135 \, a^{3} d^{2} \tan \left (f x + e\right )^{3} + 126 \, a^{3} d^{2} \tan \left (f x + e\right )^{2} - 210 \, a^{3} d^{2} \tan \left (f x + e\right ) - 630 \, a^{3} d^{2}\right )} \sqrt {d \tan \left (f x + e\right )}}{315 \, f}, \frac {2 \, {\left (315 \, \sqrt {2} a^{3} d^{\frac {5}{2}} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )} {\left (\tan \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {d} \tan \left (f x + e\right )}\right ) + {\left (35 \, a^{3} d^{2} \tan \left (f x + e\right )^{4} + 135 \, a^{3} d^{2} \tan \left (f x + e\right )^{3} + 126 \, a^{3} d^{2} \tan \left (f x + e\right )^{2} - 210 \, a^{3} d^{2} \tan \left (f x + e\right ) - 630 \, a^{3} d^{2}\right )} \sqrt {d \tan \left (f x + e\right )}\right )}}{315 \, f}\right ] \] Input:
integrate((d*tan(f*x+e))^(5/2)*(a+a*tan(f*x+e))^3,x, algorithm="fricas")
Output:
[1/315*(315*sqrt(2)*a^3*sqrt(-d)*d^2*log((d*tan(f*x + e)^2 + 2*sqrt(2)*sqr t(d*tan(f*x + e))*sqrt(-d)*(tan(f*x + e) - 1) - 4*d*tan(f*x + e) + d)/(tan (f*x + e)^2 + 1)) + 2*(35*a^3*d^2*tan(f*x + e)^4 + 135*a^3*d^2*tan(f*x + e )^3 + 126*a^3*d^2*tan(f*x + e)^2 - 210*a^3*d^2*tan(f*x + e) - 630*a^3*d^2) *sqrt(d*tan(f*x + e)))/f, 2/315*(315*sqrt(2)*a^3*d^(5/2)*arctan(1/2*sqrt(2 )*sqrt(d*tan(f*x + e))*(tan(f*x + e) - 1)/(sqrt(d)*tan(f*x + e))) + (35*a^ 3*d^2*tan(f*x + e)^4 + 135*a^3*d^2*tan(f*x + e)^3 + 126*a^3*d^2*tan(f*x + e)^2 - 210*a^3*d^2*tan(f*x + e) - 630*a^3*d^2)*sqrt(d*tan(f*x + e)))/f]
\[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^3 \, dx=a^{3} \left (\int \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}\, dx + \int 3 \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \tan {\left (e + f x \right )}\, dx + \int 3 \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \tan ^{2}{\left (e + f x \right )}\, dx + \int \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \tan ^{3}{\left (e + f x \right )}\, dx\right ) \] Input:
integrate((d*tan(f*x+e))**(5/2)*(a+a*tan(f*x+e))**3,x)
Output:
a**3*(Integral((d*tan(e + f*x))**(5/2), x) + Integral(3*(d*tan(e + f*x))** (5/2)*tan(e + f*x), x) + Integral(3*(d*tan(e + f*x))**(5/2)*tan(e + f*x)** 2, x) + Integral((d*tan(e + f*x))**(5/2)*tan(e + f*x)**3, x))
Time = 0.12 (sec) , antiderivative size = 180, normalized size of antiderivative = 0.97 \[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^3 \, dx=\frac {2 \, {\left (315 \, a^{3} d^{4} {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}}\right )} + \frac {35 \, \left (d \tan \left (f x + e\right )\right )^{\frac {9}{2}} a^{3} + 135 \, \left (d \tan \left (f x + e\right )\right )^{\frac {7}{2}} a^{3} d + 126 \, \left (d \tan \left (f x + e\right )\right )^{\frac {5}{2}} a^{3} d^{2} - 210 \, \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a^{3} d^{3} - 630 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{4}}{d}\right )}}{315 \, d f} \] Input:
integrate((d*tan(f*x+e))^(5/2)*(a+a*tan(f*x+e))^3,x, algorithm="maxima")
Output:
2/315*(315*a^3*d^4*(sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d *tan(f*x + e)))/sqrt(d))/sqrt(d) + sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sq rt(d) - 2*sqrt(d*tan(f*x + e)))/sqrt(d))/sqrt(d)) + (35*(d*tan(f*x + e))^( 9/2)*a^3 + 135*(d*tan(f*x + e))^(7/2)*a^3*d + 126*(d*tan(f*x + e))^(5/2)*a ^3*d^2 - 210*(d*tan(f*x + e))^(3/2)*a^3*d^3 - 630*sqrt(d*tan(f*x + e))*a^3 *d^4)/d)/(d*f)
Exception generated. \[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^3 \, dx=\text {Exception raised: TypeError} \] Input:
integrate((d*tan(f*x+e))^(5/2)*(a+a*tan(f*x+e))^3,x, algorithm="giac")
Output:
Exception raised: TypeError >> an error occurred running a Giac command:IN PUT:sage2:=int(sage0,sageVARx):;OUTPUT:Unable to divide, perhaps due to ro unding error%%%{%%%{1,[7,29]%%%}+%%%{12,[7,27]%%%}+%%%{66,[7,25]%%%}+%%%{2 20,[7,23]
Time = 3.32 (sec) , antiderivative size = 176, normalized size of antiderivative = 0.95 \[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^3 \, dx=\frac {4\,a^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}}{5\,f}-\frac {4\,a^3\,d^2\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{f}+\frac {6\,a^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{7/2}}{7\,d\,f}+\frac {2\,a^3\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{9/2}}{9\,d^2\,f}-\frac {4\,a^3\,d\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{3\,f}+\frac {\sqrt {2}\,a^3\,d^{5/2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}+\frac {\sqrt {2}\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{2\,d^{3/2}}\right )\right )}{f} \] Input:
int((d*tan(e + f*x))^(5/2)*(a + a*tan(e + f*x))^3,x)
Output:
(4*a^3*(d*tan(e + f*x))^(5/2))/(5*f) - (4*a^3*d^2*(d*tan(e + f*x))^(1/2))/ f + (6*a^3*(d*tan(e + f*x))^(7/2))/(7*d*f) + (2*a^3*(d*tan(e + f*x))^(9/2) )/(9*d^2*f) - (4*a^3*d*(d*tan(e + f*x))^(3/2))/(3*f) + (2^(1/2)*a^3*d^(5/2 )*(2*atan((2^(1/2)*(d*tan(e + f*x))^(1/2))/(2*d^(1/2))) + 2*atan((2^(1/2)* (d*tan(e + f*x))^(1/2))/(2*d^(1/2)) + (2^(1/2)*(d*tan(e + f*x))^(3/2))/(2* d^(3/2)))))/f
\[ \int (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^3 \, dx=\frac {\sqrt {d}\, a^{3} d^{2} \left (10 \sqrt {\tan \left (f x +e \right )}\, \tan \left (f x +e \right )^{4}+36 \sqrt {\tan \left (f x +e \right )}\, \tan \left (f x +e \right )^{2}-180 \sqrt {\tan \left (f x +e \right )}+90 \left (\int \frac {\sqrt {\tan \left (f x +e \right )}}{\tan \left (f x +e \right )}d x \right ) f +135 \left (\int \sqrt {\tan \left (f x +e \right )}\, \tan \left (f x +e \right )^{4}d x \right ) f +45 \left (\int \sqrt {\tan \left (f x +e \right )}\, \tan \left (f x +e \right )^{2}d x \right ) f \right )}{45 f} \] Input:
int((d*tan(f*x+e))^(5/2)*(a+a*tan(f*x+e))^3,x)
Output:
(sqrt(d)*a**3*d**2*(10*sqrt(tan(e + f*x))*tan(e + f*x)**4 + 36*sqrt(tan(e + f*x))*tan(e + f*x)**2 - 180*sqrt(tan(e + f*x)) + 90*int(sqrt(tan(e + f*x ))/tan(e + f*x),x)*f + 135*int(sqrt(tan(e + f*x))*tan(e + f*x)**4,x)*f + 4 5*int(sqrt(tan(e + f*x))*tan(e + f*x)**2,x)*f))/(45*f)