Integrand size = 25, antiderivative size = 249 \[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=-\frac {\sqrt {2} a^2 \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{7/2}}+\frac {\sqrt {2} a^2 \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{d e^{7/2}}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}+\frac {4 a^2}{3 d e^2 (e \cot (c+d x))^{3/2}}-\frac {a^2 \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d e^{7/2}}+\frac {a^2 \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{\sqrt {2} d e^{7/2}} \]
2/5*a^2/d/e/(e*cot(d*x+c))^(5/2)+4/3*a^2/d/e^2/(e*cot(d*x+c))^(3/2)-1/2*a^ 2*ln(e^(1/2)+cot(d*x+c)*e^(1/2)-2^(1/2)*(e*cot(d*x+c))^(1/2))/d/e^(7/2)*2^ (1/2)+1/2*a^2*ln(e^(1/2)+cot(d*x+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/ d/e^(7/2)*2^(1/2)-a^2*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))*2^(1/ 2)/d/e^(7/2)+a^2*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))*2^(1/2)/d/ e^(7/2)
Time = 1.55 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.69 \[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=-\frac {a^2 \sqrt {e \cot (c+d x)} (1+\cot (c+d x))^2 \left (-20 \cos ^2(c+d x)+30 \arctan \left (\sqrt [4]{-\cot ^2(c+d x)}\right ) (-\cot (c+d x))^{3/4} \cot ^{\frac {11}{4}}(c+d x) \sin ^2(c+d x)+30 \text {arctanh}\left (\sqrt [4]{-\cot ^2(c+d x)}\right ) (-\cot (c+d x))^{3/4} \cot ^{\frac {11}{4}}(c+d x) \sin ^2(c+d x)-3 \sin (2 (c+d x))\right ) \tan ^4(c+d x)}{15 d e^4 (\cos (c+d x)+\sin (c+d x))^2} \]
-1/15*(a^2*Sqrt[e*Cot[c + d*x]]*(1 + Cot[c + d*x])^2*(-20*Cos[c + d*x]^2 + 30*ArcTan[(-Cot[c + d*x]^2)^(1/4)]*(-Cot[c + d*x])^(3/4)*Cot[c + d*x]^(11 /4)*Sin[c + d*x]^2 + 30*ArcTanh[(-Cot[c + d*x]^2)^(1/4)]*(-Cot[c + d*x])^( 3/4)*Cot[c + d*x]^(11/4)*Sin[c + d*x]^2 - 3*Sin[2*(c + d*x)])*Tan[c + d*x] ^4)/(d*e^4*(Cos[c + d*x] + Sin[c + d*x])^2)
Time = 0.62 (sec) , antiderivative size = 239, normalized size of antiderivative = 0.96, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.640, Rules used = {3042, 4025, 27, 3042, 3955, 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.
| \(\displaystyle \int \frac {(a \cot (c+d x)+a)^2}{(e \cot (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 4025 |
| \(\displaystyle \frac {\int \frac {2 a^2 e}{(e \cot (c+d x))^{5/2}}dx}{e^2}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a^2 \int \frac {1}{(e \cot (c+d x))^{5/2}}dx}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 a^2 \int \frac {1}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3955 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2}{3 d e (e \cot (c+d x))^{3/2}}-\frac {\int \frac {1}{\sqrt {e \cot (c+d x)}}dx}{e^2}\right )}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2}{3 d e (e \cot (c+d x))^{3/2}}-\frac {\int \frac {1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{e^2}\right )}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3957 |
| \(\displaystyle \frac {2 a^2 \left (\frac {\int \frac {1}{\sqrt {e \cot (c+d x)} \left (\cot ^2(c+d x) e^2+e^2\right )}d(e \cot (c+d x))}{d e}+\frac {2}{3 d e (e \cot (c+d x))^{3/2}}\right )}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 266 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 \int \frac {1}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{d e}+\frac {2}{3 d e (e \cot (c+d x))^{3/2}}\right )}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 755 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 \left (\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}+\frac {\int \frac {e^2 \cot ^2(c+d x)+e}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}\right )}{d e}+\frac {2}{3 d e (e \cot (c+d x))^{3/2}}\right )}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 1476 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 \left (\frac {\frac {1}{2} \int \frac {1}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}+\frac {1}{2} \int \frac {1}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 e}+\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}\right )}{d e}+\frac {2}{3 d e (e \cot (c+d x))^{3/2}}\right )}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 1082 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 \left (\frac {\frac {\int \frac {1}{-e^2 \cot ^2(c+d x)-1}d\left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}-\frac {\int \frac {1}{-e^2 \cot ^2(c+d x)-1}d\left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}\right )}{d e}+\frac {2}{3 d e (e \cot (c+d x))^{3/2}}\right )}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 \left (\frac {\int \frac {e-e^2 \cot ^2(c+d x)}{e^4 \cot ^4(c+d x)+e^2}d\sqrt {e \cot (c+d x)}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d e}+\frac {2}{3 d e (e \cot (c+d x))^{3/2}}\right )}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 1479 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 \left (\frac {-\frac {\int -\frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}-\frac {\int -\frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d e}+\frac {2}{3 d e (e \cot (c+d x))^{3/2}}\right )}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {2} \left (\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d e}+\frac {2}{3 d e (e \cot (c+d x))^{3/2}}\right )}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 \left (\frac {\frac {\int \frac {\sqrt {2} \sqrt {e}-2 \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)-\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {2} \sqrt {e}}+\frac {\int \frac {\sqrt {e}+\sqrt {2} \sqrt {e \cot (c+d x)}}{e^2 \cot ^2(c+d x)+\sqrt {2} e^{3/2} \cot (c+d x)+e}d\sqrt {e \cot (c+d x)}}{2 \sqrt {e}}}{2 e}+\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}\right )}{d e}+\frac {2}{3 d e (e \cot (c+d x))^{3/2}}\right )}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 1103 |
| \(\displaystyle \frac {2 a^2 \left (\frac {2 \left (\frac {\frac {\arctan \left (\sqrt {2} \sqrt {e} \cot (c+d x)+1\right )}{\sqrt {2} \sqrt {e}}-\frac {\arctan \left (1-\sqrt {2} \sqrt {e} \cot (c+d x)\right )}{\sqrt {2} \sqrt {e}}}{2 e}+\frac {\frac {\log \left (\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}-\frac {\log \left (-\sqrt {2} e^{3/2} \cot (c+d x)+e^2 \cot ^2(c+d x)+e\right )}{2 \sqrt {2} \sqrt {e}}}{2 e}\right )}{d e}+\frac {2}{3 d e (e \cot (c+d x))^{3/2}}\right )}{e}+\frac {2 a^2}{5 d e (e \cot (c+d x))^{5/2}}\) |
(2*a^2)/(5*d*e*(e*Cot[c + d*x])^(5/2)) + (2*a^2*(2/(3*d*e*(e*Cot[c + d*x]) ^(3/2)) + (2*((-(ArcTan[1 - Sqrt[2]*Sqrt[e]*Cot[c + d*x]]/(Sqrt[2]*Sqrt[e] )) + ArcTan[1 + Sqrt[2]*Sqrt[e]*Cot[c + d*x]]/(Sqrt[2]*Sqrt[e]))/(2*e) + ( -1/2*Log[e - Sqrt[2]*e^(3/2)*Cot[c + d*x] + e^2*Cot[c + d*x]^2]/(Sqrt[2]*S qrt[e]) + Log[e + Sqrt[2]*e^(3/2)*Cot[c + d*x] + e^2*Cot[c + d*x]^2]/(2*Sq rt[2]*Sqrt[e]))/(2*e)))/(d*e)))/e
3.1.14.3.1 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[((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*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]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^2, x_Symbol] :> Simp[(b*c - a*d)^2*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1)*(a^2 + b^2))), x] + Simp[1/(a^2 + b^2) Int[(a + b*Tan[e + f*x])^(m + 1)*Simp[a*c^2 + 2*b*c*d - a*d^2 - (b*c^2 - 2*a*c*d - b*d^2)*Ta n[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0]
Time = 0.07 (sec) , antiderivative size = 174, normalized size of antiderivative = 0.70
| method | result | size |
| derivativedivides | \(-\frac {2 a^{2} \left (-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e^{3}}-\frac {1}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {2}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d e}\) | \(174\) |
| default | \(-\frac {2 a^{2} \left (-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e^{3}}-\frac {1}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {2}{3 e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d e}\) | \(174\) |
| parts | \(-\frac {2 a^{2} e \left (-\frac {1}{5 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}+\frac {1}{e^{4} \sqrt {e \cot \left (d x +c \right )}}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{4} \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d}-\frac {2 a^{2} \left (-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2} \left (e^{2}\right )^{\frac {1}{4}}}-\frac {1}{e^{2} \sqrt {e \cot \left (d x +c \right )}}\right )}{d e}+\frac {2 a^{2} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e^{4}}+\frac {2}{3 e^{2} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}\right )}{d}\) | \(494\) |
-2/d*a^2/e*(-1/4/e^3*(e^2)^(1/4)*2^(1/2)*(ln((e*cot(d*x+c)+(e^2)^(1/4)*(e* cot(d*x+c))^(1/2)*2^(1/2)+(e^2)^(1/2))/(e*cot(d*x+c)-(e^2)^(1/4)*(e*cot(d* x+c))^(1/2)*2^(1/2)+(e^2)^(1/2)))+2*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+ c))^(1/2)+1)-2*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1))-1/5/(e *cot(d*x+c))^(5/2)-2/3/e/(e*cot(d*x+c))^(3/2))
Result contains complex when optimal does not.
Time = 0.29 (sec) , antiderivative size = 542, normalized size of antiderivative = 2.18 \[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\frac {15 \, {\left (d e^{4} \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, d e^{4} \cos \left (2 \, d x + 2 \, c\right ) + d e^{4}\right )} \left (-\frac {a^{8}}{d^{4} e^{14}}\right )^{\frac {1}{4}} \log \left (d e^{4} \left (-\frac {a^{8}}{d^{4} e^{14}}\right )^{\frac {1}{4}} + a^{2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right ) - 15 \, {\left (-i \, d e^{4} \cos \left (2 \, d x + 2 \, c\right )^{2} - 2 i \, d e^{4} \cos \left (2 \, d x + 2 \, c\right ) - i \, d e^{4}\right )} \left (-\frac {a^{8}}{d^{4} e^{14}}\right )^{\frac {1}{4}} \log \left (i \, d e^{4} \left (-\frac {a^{8}}{d^{4} e^{14}}\right )^{\frac {1}{4}} + a^{2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right ) - 15 \, {\left (i \, d e^{4} \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 i \, d e^{4} \cos \left (2 \, d x + 2 \, c\right ) + i \, d e^{4}\right )} \left (-\frac {a^{8}}{d^{4} e^{14}}\right )^{\frac {1}{4}} \log \left (-i \, d e^{4} \left (-\frac {a^{8}}{d^{4} e^{14}}\right )^{\frac {1}{4}} + a^{2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right ) - 15 \, {\left (d e^{4} \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, d e^{4} \cos \left (2 \, d x + 2 \, c\right ) + d e^{4}\right )} \left (-\frac {a^{8}}{d^{4} e^{14}}\right )^{\frac {1}{4}} \log \left (-d e^{4} \left (-\frac {a^{8}}{d^{4} e^{14}}\right )^{\frac {1}{4}} + a^{2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right ) - 2 \, {\left (10 \, a^{2} \cos \left (2 \, d x + 2 \, c\right )^{2} - 10 \, a^{2} + 3 \, {\left (a^{2} \cos \left (2 \, d x + 2 \, c\right ) - a^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{15 \, {\left (d e^{4} \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, d e^{4} \cos \left (2 \, d x + 2 \, c\right ) + d e^{4}\right )}} \]
1/15*(15*(d*e^4*cos(2*d*x + 2*c)^2 + 2*d*e^4*cos(2*d*x + 2*c) + d*e^4)*(-a ^8/(d^4*e^14))^(1/4)*log(d*e^4*(-a^8/(d^4*e^14))^(1/4) + a^2*sqrt((e*cos(2 *d*x + 2*c) + e)/sin(2*d*x + 2*c))) - 15*(-I*d*e^4*cos(2*d*x + 2*c)^2 - 2* I*d*e^4*cos(2*d*x + 2*c) - I*d*e^4)*(-a^8/(d^4*e^14))^(1/4)*log(I*d*e^4*(- a^8/(d^4*e^14))^(1/4) + a^2*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c) )) - 15*(I*d*e^4*cos(2*d*x + 2*c)^2 + 2*I*d*e^4*cos(2*d*x + 2*c) + I*d*e^4 )*(-a^8/(d^4*e^14))^(1/4)*log(-I*d*e^4*(-a^8/(d^4*e^14))^(1/4) + a^2*sqrt( (e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))) - 15*(d*e^4*cos(2*d*x + 2*c)^2 + 2*d*e^4*cos(2*d*x + 2*c) + d*e^4)*(-a^8/(d^4*e^14))^(1/4)*log(-d*e^4*(- a^8/(d^4*e^14))^(1/4) + a^2*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c) )) - 2*(10*a^2*cos(2*d*x + 2*c)^2 - 10*a^2 + 3*(a^2*cos(2*d*x + 2*c) - a^2 )*sin(2*d*x + 2*c))*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*e^ 4*cos(2*d*x + 2*c)^2 + 2*d*e^4*cos(2*d*x + 2*c) + d*e^4)
\[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=a^{2} \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {2 \cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {\cot ^{2}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx\right ) \]
a**2*(Integral((e*cot(c + d*x))**(-7/2), x) + Integral(2*cot(c + d*x)/(e*c ot(c + d*x))**(7/2), x) + Integral(cot(c + d*x)**2/(e*cot(c + d*x))**(7/2) , x))
Exception generated. \[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\text {Exception raised: ValueError} \]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more de tails)Is e
\[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\int { \frac {{\left (a \cot \left (d x + c\right ) + a\right )}^{2}}{\left (e \cot \left (d x + c\right )\right )^{\frac {7}{2}}} \,d x } \]
Time = 13.75 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.40 \[ \int \frac {(a+a \cot (c+d x))^2}{(e \cot (c+d x))^{7/2}} \, dx=\frac {\frac {4\,a^2\,\mathrm {cot}\left (c+d\,x\right )}{3}+\frac {2\,a^2}{5}}{d\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}-\frac {{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,2{}\mathrm {i}}{d\,e^{7/2}}-\frac {{\left (-1\right )}^{1/4}\,a^2\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,2{}\mathrm {i}}{d\,e^{7/2}} \]