Integrand size = 25, antiderivative size = 141 \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=-\frac {2 \sqrt {2} a^3 \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d e^{7/2}}+\frac {8 a^3}{5 d e^2 (e \cot (c+d x))^{3/2}}+\frac {4 a^3}{d e^3 \sqrt {e \cot (c+d x)}}+\frac {2 \left (a^3+a^3 \cot (c+d x)\right )}{5 d e (e \cot (c+d x))^{5/2}} \]
8/5*a^3/d/e^2/(e*cot(d*x+c))^(3/2)+2/5*(a^3+a^3*cot(d*x+c))/d/e/(e*cot(d*x +c))^(5/2)-2*a^3*arctan(1/2*(e^(1/2)-cot(d*x+c)*e^(1/2))*2^(1/2)/(e*cot(d* x+c))^(1/2))*2^(1/2)/d/e^(7/2)+4*a^3/d/e^3/(e*cot(d*x+c))^(1/2)
Leaf count is larger than twice the leaf count of optimal. \(415\) vs. \(2(141)=282\).
Time = 5.75 (sec) , antiderivative size = 415, normalized size of antiderivative = 2.94 \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=\frac {a^3 \left (80 \cos ^3(c+d x)+40 \cos ^2(c+d x) \sin (c+d x)+8 \cos (c+d x) \sin ^2(c+d x)+10 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cot ^{\frac {7}{2}}(c+d x) \sin ^3(c+d x)-10 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \cot ^{\frac {7}{2}}(c+d x) \sin ^3(c+d x)-20 \text {arctanh}\left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \left (2 \sqrt [4]{-\cot (c+d x)} \cot ^{\frac {13}{4}}(c+d x)-3 \left (-\cot ^2(c+d x)\right )^{7/4}\right ) \sin ^3(c+d x)+20 \arctan \left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \left (2 \sqrt [4]{-\cot (c+d x)} \cot ^{\frac {13}{4}}(c+d x)+3 \left (-\cot ^2(c+d x)\right )^{7/4}\right ) \sin ^3(c+d x)+5 \sqrt {2} \cot ^{\frac {7}{2}}(c+d x) \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^3(c+d x)-5 \sqrt {2} \cot ^{\frac {7}{2}}(c+d x) \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^3(c+d x)\right ) (1+\tan (c+d x))^3}{20 d e^3 \sqrt {e \cot (c+d x)} (\cos (c+d x)+\sin (c+d x))^3} \]
(a^3*(80*Cos[c + d*x]^3 + 40*Cos[c + d*x]^2*Sin[c + d*x] + 8*Cos[c + d*x]* Sin[c + d*x]^2 + 10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Cot[c + d*x]^(7/2)*Sin[c + d*x]^3 - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d* x]]]*Cot[c + d*x]^(7/2)*Sin[c + d*x]^3 - 20*ArcTanh[(-Cot[c + d*x]^2)^(1/4 )]*(2*(-Cot[c + d*x])^(1/4)*Cot[c + d*x]^(13/4) - 3*(-Cot[c + d*x]^2)^(7/4 ))*Sin[c + d*x]^3 + 20*ArcTan[(-Cot[c + d*x]^2)^(1/4)]*(2*(-Cot[c + d*x])^ (1/4)*Cot[c + d*x]^(13/4) + 3*(-Cot[c + d*x]^2)^(7/4))*Sin[c + d*x]^3 + 5* Sqrt[2]*Cot[c + d*x]^(7/2)*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d* x]]*Sin[c + d*x]^3 - 5*Sqrt[2]*Cot[c + d*x]^(7/2)*Log[1 + Sqrt[2]*Sqrt[Cot [c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^3)*(1 + Tan[c + d*x])^3)/(20*d*e^3 *Sqrt[e*Cot[c + d*x]]*(Cos[c + d*x] + Sin[c + d*x])^3)
Time = 0.79 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.15, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 4048, 25, 3042, 4111, 27, 3042, 4012, 25, 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.
| \(\displaystyle \int \frac {(a \cot (c+d x)+a)^3}{(e \cot (c+d x))^{7/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {\left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{7/2}}dx\) |
| \(\Big \downarrow \) 4048 |
| \(\displaystyle \frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{5 d e (e \cot (c+d x))^{5/2}}-\frac {2 \int -\frac {6 e^2 a^3+e^2 \cot ^2(c+d x) a^3+5 e^2 \cot (c+d x) a^3}{(e \cot (c+d x))^{5/2}}dx}{5 e^3}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \int \frac {6 e^2 a^3+e^2 \cot ^2(c+d x) a^3+5 e^2 \cot (c+d x) a^3}{(e \cot (c+d x))^{5/2}}dx}{5 e^3}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int \frac {6 e^2 a^3+e^2 \tan \left (c+d x+\frac {\pi }{2}\right )^2 a^3-5 e^2 \tan \left (c+d x+\frac {\pi }{2}\right ) a^3}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{5/2}}dx}{5 e^3}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4111 |
| \(\displaystyle \frac {2 \left (\frac {\int \frac {5 \left (a^3 e^3-a^3 e^3 \cot (c+d x)\right )}{(e \cot (c+d x))^{3/2}}dx}{e^2}+\frac {4 a^3 e}{d (e \cot (c+d x))^{3/2}}\right )}{5 e^3}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {2 \left (\frac {5 \int \frac {a^3 e^3-a^3 e^3 \cot (c+d x)}{(e \cot (c+d x))^{3/2}}dx}{e^2}+\frac {4 a^3 e}{d (e \cot (c+d x))^{3/2}}\right )}{5 e^3}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {5 \int \frac {a^3 e^3+a^3 \tan \left (c+d x+\frac {\pi }{2}\right ) e^3}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{e^2}+\frac {4 a^3 e}{d (e \cot (c+d x))^{3/2}}\right )}{5 e^3}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4012 |
| \(\displaystyle \frac {2 \left (\frac {5 \left (\frac {\int -\frac {a^3 e^4+a^3 \cot (c+d x) e^4}{\sqrt {e \cot (c+d x)}}dx}{e^2}+\frac {2 a^3 e^2}{d \sqrt {e \cot (c+d x)}}\right )}{e^2}+\frac {4 a^3 e}{d (e \cot (c+d x))^{3/2}}\right )}{5 e^3}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \left (\frac {5 \left (\frac {2 a^3 e^2}{d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {a^3 e^4+a^3 \cot (c+d x) e^4}{\sqrt {e \cot (c+d x)}}dx}{e^2}\right )}{e^2}+\frac {4 a^3 e}{d (e \cot (c+d x))^{3/2}}\right )}{5 e^3}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\frac {5 \left (\frac {2 a^3 e^2}{d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {a^3 e^4-a^3 e^4 \tan \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{e^2}\right )}{e^2}+\frac {4 a^3 e}{d (e \cot (c+d x))^{3/2}}\right )}{5 e^3}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 4015 |
| \(\displaystyle \frac {2 \left (\frac {5 \left (\frac {2 a^6 e^6 \int \frac {1}{-2 a^6 e^8-\left (a^3 e^4-a^3 e^4 \cot (c+d x)\right )^2 \tan (c+d x)}d\frac {a^3 e^4-a^3 e^4 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}}{d}+\frac {2 a^3 e^2}{d \sqrt {e \cot (c+d x)}}\right )}{e^2}+\frac {4 a^3 e}{d (e \cot (c+d x))^{3/2}}\right )}{5 e^3}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{5 d e (e \cot (c+d x))^{5/2}}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 \left (\frac {5 \left (\frac {2 a^3 e^2}{d \sqrt {e \cot (c+d x)}}-\frac {\sqrt {2} a^3 e^{3/2} \arctan \left (\frac {a^3 e^4-a^3 e^4 \cot (c+d x)}{\sqrt {2} a^3 e^{7/2} \sqrt {e \cot (c+d x)}}\right )}{d}\right )}{e^2}+\frac {4 a^3 e}{d (e \cot (c+d x))^{3/2}}\right )}{5 e^3}+\frac {2 \left (a^3 \cot (c+d x)+a^3\right )}{5 d e (e \cot (c+d x))^{5/2}}\) |
(2*(a^3 + a^3*Cot[c + d*x]))/(5*d*e*(e*Cot[c + d*x])^(5/2)) + (2*((4*a^3*e )/(d*(e*Cot[c + d*x])^(3/2)) + (5*(-((Sqrt[2]*a^3*e^(3/2)*ArcTan[(a^3*e^4 - a^3*e^4*Cot[c + d*x])/(Sqrt[2]*a^3*e^(7/2)*Sqrt[e*Cot[c + d*x]])])/d) + (2*a^3*e^2)/(d*Sqrt[e*Cot[c + d*x]])))/e^2))/(5*e^3)
3.1.21.3.1 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/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[(b*c - a*d)*((a + b*Tan[e + f*x])^(m + 1)/ (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 + b*d - (b*c - a*d)*Tan[e + f*x], x], x], x] /; FreeQ[{a , b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && LtQ[m, -1 ]
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*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(n + 1)*(c^2 + d^2))), x] - Simp[1 /(d*(n + 1)*(c^2 + d^2)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c *(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3*a*b^2*d) *Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*( n + 1)))*Tan[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[ b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && GtQ[m, 2] && LtQ [n, -1] && IntegerQ[2*m]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*b*B + a^2*C)*((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[b*B + a*(A - C) - (A*b - a*B - b*C)*Tan[e + f*x], x], x], x] /; FreeQ[{a, b, e, f, A, B , C}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && LtQ[m, -1] && NeQ[a^2 + b^2, 0 ]
Leaf count of result is larger than twice the leaf count of optimal. \(322\) vs. \(2(120)=240\).
Time = 0.05 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.29
| method | result | size |
| derivativedivides | \(-\frac {2 a^{3} \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{e}-\frac {e}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {1}{\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2}{e \sqrt {e \cot \left (d x +c \right )}}\right )}{d \,e^{2}}\) | \(323\) |
| default | \(-\frac {2 a^{3} \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{4 \left (e^{2}\right )^{\frac {1}{4}}}}{e}-\frac {e}{5 \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {1}{\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}-\frac {2}{e \sqrt {e \cot \left (d x +c \right )}}\right )}{d \,e^{2}}\) | \(323\) |
| parts | \(-\frac {2 a^{3} 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 {a^{3} \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 d \,e^{4}}+\frac {3 a^{3} \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}-\frac {6 a^{3} \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}\) | \(634\) |
-2/d*a^3/e^2*(1/e*(-1/4/e*(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*c ot(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 /4/(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*arc tan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)))-1/5*e/(e*cot(d*x+c))^(5 /2)-1/(e*cot(d*x+c))^(3/2)-2/e/(e*cot(d*x+c))^(1/2))
Time = 0.28 (sec) , antiderivative size = 485, normalized size of antiderivative = 3.44 \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=\left [\frac {5 \, \sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) + a^{3} e\right )} \sqrt {-\frac {1}{e}} \log \left (\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sqrt {-\frac {1}{e}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, \sin \left (2 \, d x + 2 \, c\right ) + 1\right ) - 2 \, {\left (5 \, a^{3} \cos \left (2 \, d x + 2 \, c\right )^{2} - 5 \, a^{3} - {\left (9 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) + 11 \, a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{5 \, {\left (d e^{4} \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, d e^{4} \cos \left (2 \, d x + 2 \, c\right ) + d e^{4}\right )}}, -\frac {2 \, {\left (\frac {5 \, \sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) + a^{3} e\right )} \arctan \left (-\frac {\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, \sqrt {e} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )}}\right )}{\sqrt {e}} + {\left (5 \, a^{3} \cos \left (2 \, d x + 2 \, c\right )^{2} - 5 \, a^{3} - {\left (9 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) + 11 \, a^{3}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right )}}{5 \, {\left (d e^{4} \cos \left (2 \, d x + 2 \, c\right )^{2} + 2 \, d e^{4} \cos \left (2 \, d x + 2 \, c\right ) + d e^{4}\right )}}\right ] \]
[1/5*(5*sqrt(2)*(a^3*e*cos(2*d*x + 2*c)^2 + 2*a^3*e*cos(2*d*x + 2*c) + a^3 *e)*sqrt(-1/e)*log(sqrt(2)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)) *sqrt(-1/e)*(cos(2*d*x + 2*c) + sin(2*d*x + 2*c) - 1) - 2*sin(2*d*x + 2*c) + 1) - 2*(5*a^3*cos(2*d*x + 2*c)^2 - 5*a^3 - (9*a^3*cos(2*d*x + 2*c) + 11 *a^3)*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), -2/5*(5*sqrt (2)*(a^3*e*cos(2*d*x + 2*c)^2 + 2*a^3*e*cos(2*d*x + 2*c) + a^3*e)*arctan(- 1/2*sqrt(2)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*(cos(2*d*x + 2 *c) - sin(2*d*x + 2*c) + 1)/(sqrt(e)*(cos(2*d*x + 2*c) + 1)))/sqrt(e) + (5 *a^3*cos(2*d*x + 2*c)^2 - 5*a^3 - (9*a^3*cos(2*d*x + 2*c) + 11*a^3)*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)]
\[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=a^{3} \left (\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {3 \cot {\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {3 \cot ^{2}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx + \int \frac {\cot ^{3}{\left (c + d x \right )}}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {7}{2}}}\, dx\right ) \]
a**3*(Integral((e*cot(c + d*x))**(-7/2), x) + Integral(3*cot(c + d*x)/(e*c ot(c + d*x))**(7/2), x) + Integral(3*cot(c + d*x)**2/(e*cot(c + d*x))**(7/ 2), x) + Integral(cot(c + d*x)**3/(e*cot(c + d*x))**(7/2), x))
Exception generated. \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=\text {Exception raised: ValueError} \]
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
Timed out. \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=\text {Timed out} \]
Time = 13.54 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.89 \[ \int \frac {(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{7/2}} \, dx=\frac {4\,e\,a^3\,{\mathrm {cot}\left (c+d\,x\right )}^2+2\,e\,a^3\,\mathrm {cot}\left (c+d\,x\right )+\frac {2\,e\,a^3}{5}}{d\,e^2\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}+\frac {\sqrt {2}\,a^3\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{d\,e^{7/2}} \]