Integrand size = 21, antiderivative size = 232 \[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}-\frac {\arctan \left (1+\frac {\sqrt {2} \sqrt {d \cot (e+f x)}}{\sqrt {d}}\right )}{\sqrt {2} d^{3/2} f}+\frac {2 d}{5 f (d \cot (e+f x))^{5/2}}-\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)-\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \cot (e+f x)+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{2 \sqrt {2} d^{3/2} f} \]
2/5*d/f/(d*cot(f*x+e))^(5/2)+1/2*arctan(1-2^(1/2)*(d*cot(f*x+e))^(1/2)/d^( 1/2))/d^(3/2)/f*2^(1/2)-1/2*arctan(1+2^(1/2)*(d*cot(f*x+e))^(1/2)/d^(1/2)) /d^(3/2)/f*2^(1/2)-1/4*ln(d^(1/2)+cot(f*x+e)*d^(1/2)-2^(1/2)*(d*cot(f*x+e) )^(1/2))/d^(3/2)/f*2^(1/2)+1/4*ln(d^(1/2)+cot(f*x+e)*d^(1/2)+2^(1/2)*(d*co t(f*x+e))^(1/2))/d^(3/2)/f*2^(1/2)-2/d/f/(d*cot(f*x+e))^(1/2)
Time = 0.22 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.42 \[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\frac {-5 \arctan \left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot ^2(e+f x)}+5 \text {arctanh}\left (\sqrt [4]{-\cot ^2(e+f x)}\right ) \sqrt [4]{-\cot ^2(e+f x)}+2 \left (-5+\tan ^2(e+f x)\right )}{5 d f \sqrt {d \cot (e+f x)}} \]
(-5*ArcTan[(-Cot[e + f*x]^2)^(1/4)]*(-Cot[e + f*x]^2)^(1/4) + 5*ArcTanh[(- Cot[e + f*x]^2)^(1/4)]*(-Cot[e + f*x]^2)^(1/4) + 2*(-5 + Tan[e + f*x]^2))/ (5*d*f*Sqrt[d*Cot[e + f*x]])
Time = 0.63 (sec) , antiderivative size = 229, normalized size of antiderivative = 0.99, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 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.
| \(\displaystyle \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\tan \left (e+f x+\frac {\pi }{2}\right )^2 \left (-d \tan \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 2030 |
| \(\displaystyle d^2 \int \frac {1}{\left (-d \tan \left (\frac {1}{2} (2 e+\pi )+f x\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle d^2 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\int \frac {1}{(d \cot (e+f x))^{3/2}}dx}{d^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d^2 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\int \frac {1}{\left (-d \tan \left (e+f x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{d^2}\right )\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle d^2 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {\int \sqrt {d \cot (e+f x)}dx}{d^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle d^2 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2}{d f \sqrt {d \cot (e+f x)}}-\frac {\int \sqrt {-d \tan \left (e+f x+\frac {\pi }{2}\right )}dx}{d^2}}{d^2}\right )\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle d^2 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {\int \frac {\sqrt {d \cot (e+f x)}}{\cot ^2(e+f x) d^2+d^2}d(d \cot (e+f x))}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle d^2 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \int \frac {d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 826 |
| \(\displaystyle d^2 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \int \frac {d^2 \cot ^2(e+f x)+d}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle d^2 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}+\frac {1}{2} \int \frac {1}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}\right )-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle d^2 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}-\frac {\int \frac {1}{-d^2 \cot ^2(e+f x)-1}d\left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle d^2 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )-\frac {1}{2} \int \frac {d-d^2 \cot ^2(e+f x)}{d^4 \cot ^4(e+f x)+d^2}d\sqrt {d \cot (e+f x)}\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle d^2 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\int -\frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}+\frac {\int -\frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle d^2 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {2} \left (\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}\right )}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle d^2 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (-\frac {\int \frac {\sqrt {2} \sqrt {d}-2 \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)-\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {2} \sqrt {d}}-\frac {\int \frac {\sqrt {d}+\sqrt {2} \sqrt {d \cot (e+f x)}}{d^2 \cot ^2(e+f x)+\sqrt {2} d^{3/2} \cot (e+f x)+d}d\sqrt {d \cot (e+f x)}}{2 \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle d^2 \left (\frac {2}{5 d f (d \cot (e+f x))^{5/2}}-\frac {\frac {2 \left (\frac {1}{2} \left (\frac {\arctan \left (\sqrt {2} \sqrt {d} \cot (e+f x)+1\right )}{\sqrt {2} \sqrt {d}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {d} \cot (e+f x)\right )}{\sqrt {2} \sqrt {d}}\right )+\frac {1}{2} \left (\frac {\log \left (-\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}-\frac {\log \left (\sqrt {2} d^{3/2} \cot (e+f x)+d^2 \cot ^2(e+f x)+d\right )}{2 \sqrt {2} \sqrt {d}}\right )\right )}{d f}+\frac {2}{d f \sqrt {d \cot (e+f x)}}}{d^2}\right )\) |
d^2*(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)
3.3.14.3.1 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. \(733\) vs. \(2(179)=358\).
Time = 3.28 (sec) , antiderivative size = 734, normalized size of antiderivative = 3.16
| method | result | size |
| default | \(-\frac {{\left (\left (\csc ^{2}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{2}-1\right )}^{2} \left (-40 \left (\csc ^{7}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{7}+5 \ln \left (\frac {\csc \left (f x +e \right ) \left (1-\cos \left (f x +e \right )\right )^{2}+2 \sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}+2-2 \cos \left (f x +e \right )-\sin \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right ) {\left (\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}^{\frac {5}{2}}-10 \arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}+1-\cos \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right ) {\left (\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}^{\frac {5}{2}}-5 \ln \left (-\frac {-\csc \left (f x +e \right ) \left (1-\cos \left (f x +e \right )\right )^{2}+2 \sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-2+2 \cos \left (f x +e \right )+\sin \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right ) {\left (\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}^{\frac {5}{2}}-10 \arctan \left (\frac {\sin \left (f x +e \right ) \sqrt {\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )}-1+\cos \left (f x +e \right )}{1-\cos \left (f x +e \right )}\right ) {\left (\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}^{\frac {5}{2}}+112 \left (\csc ^{5}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{5}-40 \left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}\right ) \sin \left (f x +e \right ) \sqrt {2}}{20 f {\left (-\frac {d \left (\csc \left (f x +e \right ) \left (1-\cos \left (f x +e \right )\right )^{2}-\sin \left (f x +e \right )\right )}{1-\cos \left (f x +e \right )}\right )}^{\frac {3}{2}} \left (1-\cos \left (f x +e \right )\right ) \sqrt {\csc \left (f x +e \right ) \left (\left (\csc ^{2}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{2}-1\right ) \left (1-\cos \left (f x +e \right )\right )}\, {\left (\left (\csc ^{3}\left (f x +e \right )\right ) \left (1-\cos \left (f x +e \right )\right )^{3}-\csc \left (f x +e \right )+\cot \left (f x +e \right )\right )}^{\frac {5}{2}}}\) | \(734\) |
-1/20/f*(csc(f*x+e)^2*(1-cos(f*x+e))^2-1)^2*(-40*csc(f*x+e)^7*(1-cos(f*x+e ))^7+5*ln(1/(1-cos(f*x+e))*(csc(f*x+e)*(1-cos(f*x+e))^2+2*sin(f*x+e)*(csc( f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(1/2)+2-2*cos(f*x+e)-sin( f*x+e)))*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(5/2)-10*ar ctan(1/(1-cos(f*x+e))*(sin(f*x+e)*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+e )+cot(f*x+e))^(1/2)+1-cos(f*x+e)))*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+ e)+cot(f*x+e))^(5/2)-5*ln(-1/(1-cos(f*x+e))*(-csc(f*x+e)*(1-cos(f*x+e))^2+ 2*sin(f*x+e)*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(1/2)-2 +2*cos(f*x+e)+sin(f*x+e)))*(csc(f*x+e)^3*(1-cos(f*x+e))^3-csc(f*x+e)+cot(f *x+e))^(5/2)-10*arctan(1/(1-cos(f*x+e))*(sin(f*x+e)*(csc(f*x+e)^3*(1-cos(f *x+e))^3-csc(f*x+e)+cot(f*x+e))^(1/2)-1+cos(f*x+e)))*(csc(f*x+e)^3*(1-cos( f*x+e))^3-csc(f*x+e)+cot(f*x+e))^(5/2)+112*csc(f*x+e)^5*(1-cos(f*x+e))^5-4 0*csc(f*x+e)^3*(1-cos(f*x+e))^3)/(-d/(1-cos(f*x+e))*(csc(f*x+e)*(1-cos(f*x +e))^2-sin(f*x+e)))^(3/2)/(1-cos(f*x+e))*sin(f*x+e)/(csc(f*x+e)*(csc(f*x+e )^2*(1-cos(f*x+e))^2-1)*(1-cos(f*x+e)))^(1/2)/(csc(f*x+e)^3*(1-cos(f*x+e)) ^3-csc(f*x+e)+cot(f*x+e))^(5/2)*2^(1/2)
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 231, normalized size of antiderivative = 1.00 \[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=-\frac {5 \, d^{2} f \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (d^{5} f^{3} \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - 5 i \, d^{2} f \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (i \, d^{5} f^{3} \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) + 5 i \, d^{2} f \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (-i \, d^{5} f^{3} \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - 5 \, d^{2} f \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {1}{4}} \log \left (-d^{5} f^{3} \left (-\frac {1}{d^{6} f^{4}}\right )^{\frac {3}{4}} + \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right ) - 4 \, {\left (\tan \left (f x + e\right )^{3} - 5 \, \tan \left (f x + e\right )\right )} \sqrt {\frac {d}{\tan \left (f x + e\right )}}}{10 \, d^{2} f} \]
-1/10*(5*d^2*f*(-1/(d^6*f^4))^(1/4)*log(d^5*f^3*(-1/(d^6*f^4))^(3/4) + sqr t(d/tan(f*x + e))) - 5*I*d^2*f*(-1/(d^6*f^4))^(1/4)*log(I*d^5*f^3*(-1/(d^6 *f^4))^(3/4) + sqrt(d/tan(f*x + e))) + 5*I*d^2*f*(-1/(d^6*f^4))^(1/4)*log( -I*d^5*f^3*(-1/(d^6*f^4))^(3/4) + sqrt(d/tan(f*x + e))) - 5*d^2*f*(-1/(d^6 *f^4))^(1/4)*log(-d^5*f^3*(-1/(d^6*f^4))^(3/4) + sqrt(d/tan(f*x + e))) - 4 *(tan(f*x + e)^3 - 5*tan(f*x + e))*sqrt(d/tan(f*x + e)))/(d^2*f)
\[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\int \frac {\tan ^{2}{\left (e + f x \right )}}{\left (d \cot {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
Time = 0.34 (sec) , antiderivative size = 207, normalized size of antiderivative = 0.89 \[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=-\frac {d^{3} {\left (\frac {5 \, {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {2 \, \sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {\frac {d}{\tan \left (f x + e\right )}}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}} + \frac {\sqrt {2} \log \left (-\sqrt {2} \sqrt {d} \sqrt {\frac {d}{\tan \left (f x + e\right )}} + d + \frac {d}{\tan \left (f x + e\right )}\right )}{\sqrt {d}}\right )}}{d^{4}} - \frac {8 \, {\left (d^{2} - \frac {5 \, d^{2}}{\tan \left (f x + e\right )^{2}}\right )}}{d^{4} \left (\frac {d}{\tan \left (f x + e\right )}\right )^{\frac {5}{2}}}\right )}}{20 \, f} \]
-1/20*d^3*(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
\[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\int { \frac {\tan \left (f x + e\right )^{2}}{\left (d \cot \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
Time = 3.08 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.40 \[ \int \frac {\tan ^2(e+f x)}{(d \cot (e+f x))^{3/2}} \, dx=\frac {\frac {2\,d}{5}-\frac {2\,d}{{\mathrm {tan}\left (e+f\,x\right )}^2}}{f\,{\left (\frac {d}{\mathrm {tan}\left (e+f\,x\right )}\right )}^{5/2}}-\frac {{\left (-1\right )}^{1/4}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{d^{3/2}\,f}+\frac {{\left (-1\right )}^{1/4}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {\frac {d}{\mathrm {tan}\left (e+f\,x\right )}}}{\sqrt {d}}\right )}{d^{3/2}\,f} \]