Integrand size = 21, antiderivative size = 234 \[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=\frac {d^{3/2} \arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}-\frac {d^{3/2} \arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} f}+\frac {2 d^4}{5 f (d \cot (e+f x))^{5/2}}-\frac {2 d^2}{f \sqrt {d \cot (e+f x)}}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f}+\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} f} \]
2/5*d^4/f/(d*cot(f*x+e))^(5/2)+1/2*d^(3/2)*arctan(1-2^(1/2)*(d*cot(f*x+e)) ^(1/2)/d^(1/2))/f*2^(1/2)-1/2*d^(3/2)*arctan(1+2^(1/2)*(d*cot(f*x+e))^(1/2 )/d^(1/2))/f*2^(1/2)-1/4*d^(3/2)*ln(d^(1/2)+cot(f*x+e)*d^(1/2)-2^(1/2)*(d* cot(f*x+e))^(1/2))/f*2^(1/2)+1/4*d^(3/2)*ln(d^(1/2)+cot(f*x+e)*d^(1/2)+2^( 1/2)*(d*cot(f*x+e))^(1/2))/f*2^(1/2)-2*d^2/f/(d*cot(f*x+e))^(1/2)
Time = 0.56 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.47 \[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=-\frac {(d \cot (e+f x))^{3/2} \left (-2+10 \cot ^2(e+f x)+5 \arctan \left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot (e+f x)} \cot ^{\frac {9}{4}}(e+f x)+5 \text {arctanh}\left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \left (-\cot ^2(e+f x)\right )^{5/4}\right ) \tan ^4(e+f x)}{5 f} \]
-1/5*((d*Cot[e + f*x])^(3/2)*(-2 + 10*Cot[e + f*x]^2 + 5*ArcTan[(-Cot[e + f*x]^2)^(1/4)]*(-Cot[e + f*x])^(1/4)*Cot[e + f*x]^(9/4) + 5*ArcTanh[(-Cot[ e + f*x]^2)^(1/4)]*(-Cot[e + f*x]^2)^(5/4))*Tan[e + f*x]^4)/f
Time = 0.63 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.98, number of steps used = 18, number of rules used = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.810, Rules used = {3042, 25, 2030, 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.
| \(\displaystyle \int \tan ^5(e+f x) (d \cot (e+f x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int -\frac {\left (-d \tan \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}{\tan \left (e+f x+\frac {\pi }{2}\right )^5}dx\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\int \frac {\left (-d \tan \left (\frac {1}{2} (2 e+\pi )+f x\right )\right )^{3/2}}{\tan \left (\frac {1}{2} (2 e+\pi )+f x\right )^5}dx\) |
| \(\Big \downarrow \) 2030 |
| \(\displaystyle d^5 \int \frac {1}{\left (-d \tan \left (\frac {1}{2} (2 e+\pi )+f x\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle d^5 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\int \frac {1}{(d \cot (e+f x))^{3/2}}dx}{d^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d^5 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\int \frac {1}{\left (-d \tan \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{d^2}\right )\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle d^5 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {\int \sqrt {d \cot (e+f x)}dx}{d^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d^5 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {\int \sqrt {-d \tan \left (e+f x+\frac {\pi }{2}\right )}dx}{d^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle d^5 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {\int \frac {\sqrt {d \cot (e+f x)}}{\cot ^2(e+f x) d^2+d^2}d(d \cot (e+f x))}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle d^5 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \int \frac {d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle d^5 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \int \frac {d^2 \cot ^2(e+f x)+d}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle d^5 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}+\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}\right )-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle d^5 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle d^5 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle d^5 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d^5 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d^5 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle d^5 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
d^5*(2/(5*d*f*(d*Cot[e + f*x])^(5/2)) - (2/(d*f*Sqrt[d*Cot[e + f*x]]) + (2 *((-(ArcTan[1 - Sqrt[2]*Sqrt[d]*Cot[e + f*x]]/(Sqrt[2]*Sqrt[d])) + ArcTan[ 1 + Sqrt[2]*Sqrt[d]*Cot[e + f*x]]/(Sqrt[2]*Sqrt[d]))/2 + (Log[d - Sqrt[2]* d^(3/2)*Cot[e + f*x] + d^2*Cot[e + f*x]^2]/(2*Sqrt[2]*Sqrt[d]) - Log[d + S qrt[2]*d^(3/2)*Cot[e + f*x] + d^2*Cot[e + f*x]^2]/(2*Sqrt[2]*Sqrt[d]))/2)) /(d*f))/d^2)
3.2.99.3.1 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[(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*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]
Leaf count of result is larger than twice the leaf count of optimal. \(631\) vs. \(2(181)=362\).
Time = 3.41 (sec) , antiderivative size = 632, normalized size of antiderivative = 2.70
| method | result | size |
| default | \(\frac {\left (\sec ^{3}\left (f x +e \right )\right ) \csc \left (f x +e \right ) \left (5 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (-\frac {\cot \left (f x +e \right ) \cos \left (f x +e \right )-2 \cot \left (f x +e \right )+2 \sin \left (f x +e \right ) \sqrt {-\left (\cot ^{3}\left (f x +e \right )\right )+3 \left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )-3 \left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )+\csc ^{3}\left (f x +e \right )+\cot \left (f x +e \right )-\csc \left (f x +e \right )}-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )+\csc \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\right ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-5 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \ln \left (-\frac {\cot \left (f x +e \right ) \cos \left (f x +e \right )-2 \cot \left (f x +e \right )-2 \sin \left (f x +e \right ) \sqrt {-\left (\cot ^{3}\left (f x +e \right )\right )+3 \left (\cot ^{2}\left (f x +e \right )\right ) \csc \left (f x +e \right )-3 \left (\csc ^{2}\left (f x +e \right )\right ) \cot \left (f x +e \right )+\csc ^{3}\left (f x +e \right )+\cot \left (f x +e \right )-\csc \left (f x +e \right )}-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )+\csc \left (f x +e \right )+2}{\cos \left (f x +e \right )-1}\right ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+10 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \arctan \left (\frac {\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}-10 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \arctan \left (\frac {-\sqrt {2}\, \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}\, \sin \left (f x +e \right )+\cos \left (f x +e \right )-1}{\cos \left (f x +e \right )-1}\right ) \sqrt {-\frac {\sin \left (f x +e \right ) \cos \left (f x +e \right )}{\left (\cos \left (f x +e \right )+1\right )^{2}}}+24 \sqrt {2}\, \left (\cos ^{3}\left (f x +e \right )\right )-24 \sqrt {2}\, \left (\cos ^{2}\left (f x +e \right )\right )-4 \sqrt {2}\, \cos \left (f x +e \right )+4 \sqrt {2}\right ) \left (\cos \left (f x +e \right )+1\right ) d \sqrt {\cot \left (f x +e \right ) d}\, \sqrt {2}}{20 f}\) | \(632\) |
1/20/f*sec(f*x+e)^3*csc(f*x+e)*(5*cos(f*x+e)^2*sin(f*x+e)*ln(-(cot(f*x+e)* cos(f*x+e)-2*cot(f*x+e)+2*sin(f*x+e)*(-cot(f*x+e)^3+3*cot(f*x+e)^2*csc(f*x +e)-3*csc(f*x+e)^2*cot(f*x+e)+csc(f*x+e)^3+cot(f*x+e)-csc(f*x+e))^(1/2)-2* cos(f*x+e)-sin(f*x+e)+csc(f*x+e)+2)/(cos(f*x+e)-1))*(-sin(f*x+e)*cos(f*x+e )/(cos(f*x+e)+1)^2)^(1/2)-5*cos(f*x+e)^2*sin(f*x+e)*ln(-(cot(f*x+e)*cos(f* x+e)-2*cot(f*x+e)-2*sin(f*x+e)*(-cot(f*x+e)^3+3*cot(f*x+e)^2*csc(f*x+e)-3* csc(f*x+e)^2*cot(f*x+e)+csc(f*x+e)^3+cot(f*x+e)-csc(f*x+e))^(1/2)-2*cos(f* x+e)-sin(f*x+e)+csc(f*x+e)+2)/(cos(f*x+e)-1))*(-sin(f*x+e)*cos(f*x+e)/(cos (f*x+e)+1)^2)^(1/2)+10*cos(f*x+e)^2*sin(f*x+e)*arctan((2^(1/2)*(-sin(f*x+e )*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin(f*x+e)+cos(f*x+e)-1)/(cos(f*x+e)- 1))*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)-10*cos(f*x+e)^2*sin(f* x+e)*arctan((-2^(1/2)*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1)^2)^(1/2)*sin( f*x+e)+cos(f*x+e)-1)/(cos(f*x+e)-1))*(-sin(f*x+e)*cos(f*x+e)/(cos(f*x+e)+1 )^2)^(1/2)+24*2^(1/2)*cos(f*x+e)^3-24*2^(1/2)*cos(f*x+e)^2-4*2^(1/2)*cos(f *x+e)+4*2^(1/2))*(cos(f*x+e)+1)*d*(cot(f*x+e)*d)^(1/2)*2^(1/2)
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 223, normalized size of antiderivative = 0.95 \[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=-\frac {5 \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (d^{4} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {3}{4}} f^{3}\right ) - 5 i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (d^{4} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {3}{4}} f^{3}\right ) + 5 i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (d^{4} \sqrt {\frac {d}{\tan \left (f x + e\right )}} - i \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {3}{4}} f^{3}\right ) - 5 \, \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {1}{4}} f \log \left (d^{4} \sqrt {\frac {d}{\tan \left (f x + e\right )}} - \left (-\frac {d^{6}}{f^{4}}\right )^{\frac {3}{4}} f^{3}\right ) - 4 \, {\left (d \tan \left (f x + e\right )^{3} - 5 \, d \tan \left (f x + e\right )\right )} \sqrt {\frac {d}{\tan \left (f x + e\right )}}}{10 \, f} \]
-1/10*(5*(-d^6/f^4)^(1/4)*f*log(d^4*sqrt(d/tan(f*x + e)) + (-d^6/f^4)^(3/4 )*f^3) - 5*I*(-d^6/f^4)^(1/4)*f*log(d^4*sqrt(d/tan(f*x + e)) + I*(-d^6/f^4 )^(3/4)*f^3) + 5*I*(-d^6/f^4)^(1/4)*f*log(d^4*sqrt(d/tan(f*x + e)) - I*(-d ^6/f^4)^(3/4)*f^3) - 5*(-d^6/f^4)^(1/4)*f*log(d^4*sqrt(d/tan(f*x + e)) - ( -d^6/f^4)^(3/4)*f^3) - 4*(d*tan(f*x + e)^3 - 5*d*tan(f*x + e))*sqrt(d/tan( f*x + e)))/f
\[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=\int \left (d \cot {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{5}{\left (e + f x \right )}\, dx \]
Time = 0.47 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.88 \[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=-\frac {d^{6} {\left (\frac {5 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}}\right )}}{d^{4}} - \frac {8 \, {\left (d^{2} - \frac {5 \, d^{2}}{\tan \left (f x + e\right )^{2}}\right )}}{d^{4} \left (\frac {d}{\tan \left (f x + e\right )}\right )^{\frac {5}{2}}}\right )}}{20 \, f} \]
-1/20*d^6*(5*(2*sqrt(2)*arctan(1/2*sqrt(2)*(sqrt(2)*sqrt(d) + 2*sqrt(d/tan (f*x + e)))/sqrt(d))/sqrt(d) + 2*sqrt(2)*arctan(-1/2*sqrt(2)*(sqrt(2)*sqrt (d) - 2*sqrt(d/tan(f*x + e)))/sqrt(d))/sqrt(d) - sqrt(2)*log(sqrt(2)*sqrt( d)*sqrt(d/tan(f*x + e)) + d + d/tan(f*x + e))/sqrt(d) + sqrt(2)*log(-sqrt( 2)*sqrt(d)*sqrt(d/tan(f*x + e)) + d + d/tan(f*x + e))/sqrt(d))/d^4 - 8*(d^ 2 - 5*d^2/tan(f*x + e)^2)/(d^4*(d/tan(f*x + e))^(5/2)))/f
\[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=\int { \left (d \cot \left (f x + e\right )\right )^{\frac {3}{2}} \tan \left (f x + e\right )^{5} \,d x } \]
Time = 2.78 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.41 \[ \int (d \cot (e+f x))^{3/2} \tan ^5(e+f x) \, dx=\frac {\frac {2\,d^4}{5}-\frac {2\,d^4}{{\mathrm {tan}\left (e+f\,x\right )}^2}}{f\,{\left (\frac {d}{\mathrm {tan}\left (e+f\,x\right )}\right )}^{5/2}}-\frac {{\left (-1\right )}^{1/4}\,d^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{f}+\frac {{\left (-1\right )}^{1/4}\,d^{3/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{f} \]