Integrand size = 25, antiderivative size = 189 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3} \, dx=\frac {31 \arctan \left (\frac {\sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{8 a^3 d e^{3/2}}+\frac {\text {arctanh}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{2 \sqrt {2} a^3 d e^{3/2}}+\frac {27}{8 a^3 d e \sqrt {e \cot (c+d x)}}-\frac {9}{8 a^3 d e \sqrt {e \cot (c+d x)} (1+\cot (c+d x))}-\frac {1}{4 a d e \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^2} \]
31/8*arctan((e*cot(d*x+c))^(1/2)/e^(1/2))/a^3/d/e^(3/2)+1/4*arctanh(1/2*(e ^(1/2)+cot(d*x+c)*e^(1/2))*2^(1/2)/(e*cot(d*x+c))^(1/2))/a^3/d/e^(3/2)*2^( 1/2)+27/8/a^3/d/e/(e*cot(d*x+c))^(1/2)-9/8/a^3/d/e/(1+cot(d*x+c))/(e*cot(d *x+c))^(1/2)-1/4/a/d/e/(a+a*cot(d*x+c))^2/(e*cot(d*x+c))^(1/2)
Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
Time = 1.28 (sec) , antiderivative size = 325, normalized size of antiderivative = 1.72 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3} \, dx=\frac {-2 \sqrt {2} \arctan \left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right ) \sqrt {e \cot (c+d x)}+2 \sqrt {2} \arctan \left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right ) \sqrt {e \cot (c+d x)}+8 \sqrt {e} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},1,\frac {1}{2},-\cot (c+d x)\right )+16 \sqrt {e} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},2,\frac {1}{2},-\cot (c+d x)\right )+16 \sqrt {e} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},3,\frac {1}{2},-\cot (c+d x)\right )-8 \sqrt {e} \operatorname {Hypergeometric2F1}\left (-\frac {1}{4},1,\frac {3}{4},-\cot ^2(c+d x)\right )-\sqrt {2} \sqrt {e \cot (c+d x)} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )+\sqrt {2} \sqrt {e \cot (c+d x)} \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{16 a^3 d e^{3/2} \sqrt {e \cot (c+d x)}} \]
(-2*Sqrt[2]*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]*Sqrt[e*Cot[ c + d*x]] + 2*Sqrt[2]*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]]*S qrt[e*Cot[c + d*x]] + 8*Sqrt[e]*Hypergeometric2F1[-1/2, 1, 1/2, -Cot[c + d *x]] + 16*Sqrt[e]*Hypergeometric2F1[-1/2, 2, 1/2, -Cot[c + d*x]] + 16*Sqrt [e]*Hypergeometric2F1[-1/2, 3, 1/2, -Cot[c + d*x]] - 8*Sqrt[e]*Hypergeomet ric2F1[-1/4, 1, 3/4, -Cot[c + d*x]^2] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]*Log[S qrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]] + Sqrt[2]*Sq rt[e*Cot[c + d*x]]*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot [c + d*x]]])/(16*a^3*d*e^(3/2)*Sqrt[e*Cot[c + d*x]])
Time = 1.70 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.11, number of steps used = 20, number of rules used = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.760, Rules used = {3042, 4052, 27, 3042, 4132, 25, 3042, 4132, 27, 3042, 4137, 27, 3042, 4015, 221, 4117, 27, 73, 216}
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.
| \(\displaystyle \int \frac {1}{(a \cot (c+d x)+a)^3 (e \cot (c+d x))^{3/2}} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\) |
| \(\Big \downarrow \) 4052 |
| \(\displaystyle -\frac {\int -\frac {5 e \cot ^2(c+d x) a^2+9 e a^2-4 e \cot (c+d x) a^2}{2 (e \cot (c+d x))^{3/2} (\cot (c+d x) a+a)^2}dx}{4 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\int \frac {5 e \cot ^2(c+d x) a^2+9 e a^2-4 e \cot (c+d x) a^2}{(e \cot (c+d x))^{3/2} (\cot (c+d x) a+a)^2}dx}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {5 e \tan \left (c+d x+\frac {\pi }{2}\right )^2 a^2+9 e a^2+4 e \tan \left (c+d x+\frac {\pi }{2}\right ) a^2}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^2}dx}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {-\frac {\int -\frac {27 e^2 a^4+27 e^2 \cot ^2(c+d x) a^4-8 e^2 \cot (c+d x) a^4}{(e \cot (c+d x))^{3/2} (\cot (c+d x) a+a)}dx}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {27 e^2 a^4+27 e^2 \cot ^2(c+d x) a^4-8 e^2 \cot (c+d x) a^4}{(e \cot (c+d x))^{3/2} (\cot (c+d x) a+a)}dx}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {27 e^2 a^4+27 e^2 \tan \left (c+d x+\frac {\pi }{2}\right )^2 a^4+8 e^2 \tan \left (c+d x+\frac {\pi }{2}\right ) a^4}{\left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 4132 |
| \(\displaystyle \frac {\frac {\frac {2 \int -\frac {35 e^4 a^5+27 e^4 \cot ^2(c+d x) a^5}{2 \sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{a e^3}+\frac {54 a^3 e}{d \sqrt {e \cot (c+d x)}}}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {54 a^3 e}{d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {35 e^4 a^5+27 e^4 \cot ^2(c+d x) a^5}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx}{a e^3}}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {54 a^3 e}{d \sqrt {e \cot (c+d x)}}-\frac {\int \frac {35 e^4 a^5+27 e^4 \tan \left (c+d x+\frac {\pi }{2}\right )^2 a^5}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx}{a e^3}}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 4137 |
| \(\displaystyle \frac {\frac {\frac {54 a^3 e}{d \sqrt {e \cot (c+d x)}}-\frac {31 a^5 e^4 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx+\frac {\int \frac {8 \left (a^6 e^4-a^6 e^4 \cot (c+d x)\right )}{\sqrt {e \cot (c+d x)}}dx}{2 a^2}}{a e^3}}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {54 a^3 e}{d \sqrt {e \cot (c+d x)}}-\frac {31 a^5 e^4 \int \frac {\cot ^2(c+d x)+1}{\sqrt {e \cot (c+d x)} (\cot (c+d x) a+a)}dx+\frac {4 \int \frac {a^6 e^4-a^6 e^4 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}dx}{a^2}}{a e^3}}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {54 a^3 e}{d \sqrt {e \cot (c+d x)}}-\frac {31 a^5 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx+\frac {4 \int \frac {e^4 a^6+e^4 \tan \left (c+d x+\frac {\pi }{2}\right ) a^6}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{a^2}}{a e^3}}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 4015 |
| \(\displaystyle \frac {\frac {\frac {54 a^3 e}{d \sqrt {e \cot (c+d x)}}-\frac {31 a^5 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {8 a^{10} e^8 \int \frac {1}{2 a^{12} e^8-\left (e^4 a^6+e^4 \cot (c+d x) a^6\right )^2 \tan (c+d x)}d\frac {e^4 a^6+e^4 \cot (c+d x) a^6}{\sqrt {e \cot (c+d x)}}}{d}}{a e^3}}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {\frac {\frac {54 a^3 e}{d \sqrt {e \cot (c+d x)}}-\frac {31 a^5 e^4 \int \frac {\tan \left (c+d x+\frac {\pi }{2}\right )^2+1}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )}dx-\frac {4 \sqrt {2} a^4 e^{7/2} \text {arctanh}\left (\frac {a^6 e^4 \cot (c+d x)+a^6 e^4}{\sqrt {2} a^6 e^{7/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{a e^3}}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 4117 |
| \(\displaystyle \frac {\frac {\frac {54 a^3 e}{d \sqrt {e \cot (c+d x)}}-\frac {\frac {31 a^5 e^4 \int \frac {1}{a \sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{d}-\frac {4 \sqrt {2} a^4 e^{7/2} \text {arctanh}\left (\frac {a^6 e^4 \cot (c+d x)+a^6 e^4}{\sqrt {2} a^6 e^{7/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{a e^3}}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {54 a^3 e}{d \sqrt {e \cot (c+d x)}}-\frac {\frac {31 a^4 e^4 \int \frac {1}{\sqrt {e \cot (c+d x)} (\cot (c+d x)+1)}d(-\cot (c+d x))}{d}-\frac {4 \sqrt {2} a^4 e^{7/2} \text {arctanh}\left (\frac {a^6 e^4 \cot (c+d x)+a^6 e^4}{\sqrt {2} a^6 e^{7/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{a e^3}}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle \frac {\frac {\frac {54 a^3 e}{d \sqrt {e \cot (c+d x)}}-\frac {-\frac {62 a^4 e^3 \int \frac {1}{\frac {\cot ^2(c+d x)}{e}+1}d\sqrt {e \cot (c+d x)}}{d}-\frac {4 \sqrt {2} a^4 e^{7/2} \text {arctanh}\left (\frac {a^6 e^4 \cot (c+d x)+a^6 e^4}{\sqrt {2} a^6 e^{7/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{a e^3}}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle \frac {\frac {\frac {54 a^3 e}{d \sqrt {e \cot (c+d x)}}-\frac {\frac {62 a^4 e^{7/2} \arctan \left (\frac {\cot (c+d x)}{\sqrt {e}}\right )}{d}-\frac {4 \sqrt {2} a^4 e^{7/2} \text {arctanh}\left (\frac {a^6 e^4 \cot (c+d x)+a^6 e^4}{\sqrt {2} a^6 e^{7/2} \sqrt {e \cot (c+d x)}}\right )}{d}}{a e^3}}{2 a^3 e}-\frac {9}{d (\cot (c+d x)+1) \sqrt {e \cot (c+d x)}}}{8 a^3 e}-\frac {1}{4 a d e (a \cot (c+d x)+a)^2 \sqrt {e \cot (c+d x)}}\) |
-1/4*1/(a*d*e*Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^2) + (-9/(d*Sqrt[e *Cot[c + d*x]]*(1 + Cot[c + d*x])) + (-(((62*a^4*e^(7/2)*ArcTan[Cot[c + d* x]/Sqrt[e]])/d - (4*Sqrt[2]*a^4*e^(7/2)*ArcTanh[(a^6*e^4 + a^6*e^4*Cot[c + d*x])/(Sqrt[2]*a^6*e^(7/2)*Sqrt[e*Cot[c + d*x]])])/d)/(a*e^3)) + (54*a^3* e)/(d*Sqrt[e*Cot[c + d*x]]))/(2*a^3*e))/(8*a^3*e)
3.1.39.3.1 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_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
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 + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(a^2 + b^2)*(b*c - a*d))), x] + Simp[1 /((m + 1)*(a^2 + b^2)*(b*c - a*d)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[a*(b*c - a*d)*(m + 1) - b^2*d*(m + n + 2) - b*(b*c - a*d)*(m + 1)*Tan[e + f*x] - b^2*d*(m + n + 2)*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] && LtQ[m, -1] && (LtQ[n, 0] || Integ erQ[m]) && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_.)*((A_) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[A/f Subst[Int[(a + b*x)^m*(c + d*x)^n, x], x, Tan[e + f*x]], x] /; FreeQ[{a, b, c, d, e, f, A, C, m, n}, x] && EqQ[A, C]
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[(A*b^2 - a*(b*B - a*C))*(a + b*Tan[e + f*x])^(m + 1)*((c + d*Tan[e + f*x])^(n + 1)/(f*(m + 1)*(b*c - a*d)*(a^2 + b^2))), x] + Simp[1/((m + 1)*(b*c - a*d)*(a^2 + b^2)) Int[(a + b*Tan[e + f*x])^(m + 1)*(c + d*Tan[e + f*x])^n*Simp[A*(a*(b*c - a*d)*(m + 1) - b^2*d* (m + n + 2)) + (b*B - a*C)*(b*c*(m + 1) + a*d*(n + 1)) - (m + 1)*(b*c - a*d )*(A*b - a*B - b*C)*Tan[e + f*x] - d*(A*b^2 - a*(b*B - a*C))*(m + n + 2)*Ta n[e + f*x]^2, x], x], x] /; FreeQ[{a, b, c, d, e, f, A, B, C, n}, x] && NeQ [b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && LtQ[m, -1] && !(ILtQ[n, -1] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])))
Int[(((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_)*((A_.) + (C_.)*tan[(e_.) + (f_.)*(x_)]^2))/((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Sim p[1/(a^2 + b^2) Int[(c + d*Tan[e + f*x])^n*Simp[a*(A - C) - (A*b - b*C)*T an[e + f*x], x], x], x] + Simp[(A*b^2 + a^2*C)/(a^2 + b^2) Int[(c + d*Tan [e + f*x])^n*((1 + Tan[e + f*x]^2)/(a + b*Tan[e + f*x])), x], x] /; FreeQ[{ a, b, c, d, e, f, A, C, n}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && !GtQ[n, 0] && !LeQ[n, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(363\) vs. \(2(156)=312\).
Time = 0.05 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.93
| method | result | size |
| derivativedivides | \(-\frac {2 e^{4} \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{4 e^{5}}-\frac {1}{e^{5} \sqrt {e \cot \left (d x +c \right )}}-\frac {\frac {\frac {11 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{4}+\frac {13 e \sqrt {e \cot \left (d x +c \right )}}{4}}{\left (e \cot \left (d x +c \right )+e \right )^{2}}+\frac {31 \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{4 \sqrt {e}}}{4 e^{5}}\right )}{d \,a^{3}}\) | \(364\) |
| default | \(-\frac {2 e^{4} \left (\frac {-\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}}{4 e^{5}}-\frac {1}{e^{5} \sqrt {e \cot \left (d x +c \right )}}-\frac {\frac {\frac {11 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{4}+\frac {13 e \sqrt {e \cot \left (d x +c \right )}}{4}}{\left (e \cot \left (d x +c \right )+e \right )^{2}}+\frac {31 \arctan \left (\frac {\sqrt {e \cot \left (d x +c \right )}}{\sqrt {e}}\right )}{4 \sqrt {e}}}{4 e^{5}}\right )}{d \,a^{3}}\) | \(364\) |
-2/d/a^3*e^4*(1/4/e^5*(-1/8/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*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 ))+1/8/(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/e^5/(e*cot(d*x+c) )^(1/2)-1/4/e^5*((11/4*(e*cot(d*x+c))^(3/2)+13/4*e*(e*cot(d*x+c))^(1/2))/( e*cot(d*x+c)+e)^2+31/4/e^(1/2)*arctan((e*cot(d*x+c))^(1/2)/e^(1/2))))
Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (156) = 312\).
Time = 0.29 (sec) , antiderivative size = 697, normalized size of antiderivative = 3.69 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3} \, dx=\left [-\frac {4 \, \sqrt {2} {\left ({\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (2 \, d x + 2 \, c\right ) + \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} \sqrt {-e} \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 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) + 31 \, {\left ({\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (2 \, d x + 2 \, c\right ) + \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sqrt {-e} \log \left (\frac {e \cos \left (2 \, d x + 2 \, c\right ) - e \sin \left (2 \, d x + 2 \, c\right ) - 2 \, \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sin \left (2 \, d x + 2 \, c\right ) + e}{\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1}\right ) + {\left (45 \, \cos \left (2 \, d x + 2 \, c\right )^{2} - {\left (11 \, \cos \left (2 \, d x + 2 \, c\right ) + 43\right )} \sin \left (2 \, d x + 2 \, c\right ) - 45\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{16 \, {\left (a^{3} d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + a^{3} d e^{2} + {\left (a^{3} d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + a^{3} d e^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )}}, \frac {2 \, \sqrt {2} {\left ({\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (2 \, d x + 2 \, c\right ) + \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sqrt {e} \log \left (-\sqrt {2} \sqrt {e} \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 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) + 62 \, {\left ({\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (2 \, d x + 2 \, c\right ) + \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sqrt {e} \arctan \left (\frac {\sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{\sqrt {e}}\right ) - {\left (45 \, \cos \left (2 \, d x + 2 \, c\right )^{2} - {\left (11 \, \cos \left (2 \, d x + 2 \, c\right ) + 43\right )} \sin \left (2 \, d x + 2 \, c\right ) - 45\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{16 \, {\left (a^{3} d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + a^{3} d e^{2} + {\left (a^{3} d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + a^{3} d e^{2}\right )} \sin \left (2 \, d x + 2 \, c\right )\right )}}\right ] \]
[-1/16*(4*sqrt(2)*((cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c) + cos(2*d*x + 2 *c) + 1)*sqrt(-e)*arctan(1/2*sqrt(2)*sqrt(-e)*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)/(e*cos(2*d*x + 2*c) + e)) + 31*((cos(2*d*x + 2*c) + 1)*sin(2*d*x + 2*c) + cos(2*d*x + 2*c) + 1)*sqrt(-e)*log((e*cos(2*d*x + 2*c) - e*sin(2*d*x + 2*c) - 2*sqrt(- e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*sin(2*d*x + 2*c) + e)/( cos(2*d*x + 2*c) + sin(2*d*x + 2*c) + 1)) + (45*cos(2*d*x + 2*c)^2 - (11*c os(2*d*x + 2*c) + 43)*sin(2*d*x + 2*c) - 45)*sqrt((e*cos(2*d*x + 2*c) + e) /sin(2*d*x + 2*c)))/(a^3*d*e^2*cos(2*d*x + 2*c) + a^3*d*e^2 + (a^3*d*e^2*c os(2*d*x + 2*c) + a^3*d*e^2)*sin(2*d*x + 2*c)), 1/16*(2*sqrt(2)*((cos(2*d* x + 2*c) + 1)*sin(2*d*x + 2*c) + cos(2*d*x + 2*c) + 1)*sqrt(e)*log(-sqrt(2 )*sqrt(e)*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) + 2*e*sin(2*d*x + 2*c) + e) + 62*((cos(2*d*x + 2 *c) + 1)*sin(2*d*x + 2*c) + cos(2*d*x + 2*c) + 1)*sqrt(e)*arctan(sqrt((e*c os(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))/sqrt(e)) - (45*cos(2*d*x + 2*c)^2 - (11*cos(2*d*x + 2*c) + 43)*sin(2*d*x + 2*c) - 45)*sqrt((e*cos(2*d*x + 2*c ) + e)/sin(2*d*x + 2*c)))/(a^3*d*e^2*cos(2*d*x + 2*c) + a^3*d*e^2 + (a^3*d *e^2*cos(2*d*x + 2*c) + a^3*d*e^2)*sin(2*d*x + 2*c))]
\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3} \, dx=\frac {\int \frac {1}{\left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot ^{3}{\left (c + d x \right )} + 3 \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot ^{2}{\left (c + d x \right )} + 3 \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot {\left (c + d x \right )} + \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx}{a^{3}} \]
Integral(1/((e*cot(c + d*x))**(3/2)*cot(c + d*x)**3 + 3*(e*cot(c + d*x))** (3/2)*cot(c + d*x)**2 + 3*(e*cot(c + d*x))**(3/2)*cot(c + d*x) + (e*cot(c + d*x))**(3/2)), x)/a**3
Exception generated. \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3} \, dx=\text {Exception raised: ValueError} \]
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
\[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3} \, dx=\int { \frac {1}{{\left (a \cot \left (d x + c\right ) + a\right )}^{3} \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
Time = 13.34 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.93 \[ \int \frac {1}{(e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3} \, dx=\frac {\frac {27\,e\,{\mathrm {cot}\left (c+d\,x\right )}^2}{8}+\frac {45\,e\,\mathrm {cot}\left (c+d\,x\right )}{8}+2\,e}{a^3\,d\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}+2\,a^3\,d\,e\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}+a^3\,d\,e^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}+\frac {31\,\mathrm {atan}\left (\frac {\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )}{8\,a^3\,d\,e^{3/2}}+\frac {\sqrt {2}\,\mathrm {atanh}\left (\frac {63504384\,\sqrt {2}\,a^9\,d^3\,e^{15/2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{63504384\,a^9\,d^3\,e^8+63504384\,a^9\,d^3\,e^8\,\mathrm {cot}\left (c+d\,x\right )}\right )}{4\,a^3\,d\,e^{3/2}} \]
(2*e + (45*e*cot(c + d*x))/8 + (27*e*cot(c + d*x)^2)/8)/(a^3*d*(e*cot(c + d*x))^(5/2) + 2*a^3*d*e*(e*cot(c + d*x))^(3/2) + a^3*d*e^2*(e*cot(c + d*x) )^(1/2)) + (31*atan((e*cot(c + d*x))^(1/2)/e^(1/2)))/(8*a^3*d*e^(3/2)) + ( 2^(1/2)*atanh((63504384*2^(1/2)*a^9*d^3*e^(15/2)*(e*cot(c + d*x))^(1/2))/( 63504384*a^9*d^3*e^8 + 63504384*a^9*d^3*e^8*cot(c + d*x))))/(4*a^3*d*e^(3/ 2))