Integrand size = 25, antiderivative size = 160 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=-\frac {2 \sqrt {2} a^3 e^{3/2} \text {arctanh}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}+\frac {4 a^3 e \sqrt {e \cot (c+d x)}}{d}-\frac {4 a^3 (e \cot (c+d x))^{3/2}}{3 d}-\frac {32 a^3 (e \cot (c+d x))^{5/2}}{35 d e}-\frac {2 (e \cot (c+d x))^{5/2} \left (a^3+a^3 \cot (c+d x)\right )}{7 d e} \] Output:
-2*2^(1/2)*a^3*e^(3/2)*arctanh(1/2*(e^(1/2)+e^(1/2)*cot(d*x+c))*2^(1/2)/(e *cot(d*x+c))^(1/2))/d+4*a^3*e*(e*cot(d*x+c))^(1/2)/d-4/3*a^3*(e*cot(d*x+c) )^(3/2)/d-32/35*a^3*(e*cot(d*x+c))^(5/2)/d/e-2/7*(e*cot(d*x+c))^(5/2)*(a^3 +a^3*cot(d*x+c))/d/e
Leaf count is larger than twice the leaf count of optimal. \(380\) vs. \(2(160)=320\).
Time = 2.93 (sec) , antiderivative size = 380, normalized size of antiderivative = 2.38 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=-\frac {a^3 (e \cot (c+d x))^{3/2} (1+\cot (c+d x))^3 \left (-420 \arctan \left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot (c+d x)} \sin ^3(c+d x)+420 \text {arctanh}\left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot (c+d x)} \sin ^3(c+d x)+\sqrt [4]{\cot (c+d x)} \sin (c+d x) \left (60 \cos ^2(c+d x) \cot ^{\frac {3}{2}}(c+d x)-210 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \sin ^2(c+d x)+210 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \sin ^2(c+d x)-840 \sqrt {\cot (c+d x)} \sin ^2(c+d x)+280 \cot ^{\frac {3}{2}}(c+d x) \sin ^2(c+d x)-105 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^2(c+d x)+105 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^2(c+d x)+126 \cot ^{\frac {3}{2}}(c+d x) \sin (2 (c+d x))\right )\right )}{210 d \cot ^{\frac {7}{4}}(c+d x) (\cos (c+d x)+\sin (c+d x))^3} \] Input:
Integrate[(e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])^3,x]
Output:
-1/210*(a^3*(e*Cot[c + d*x])^(3/2)*(1 + Cot[c + d*x])^3*(-420*ArcTan[(-Cot [c + d*x]^2)^(1/4)]*(-Cot[c + d*x])^(1/4)*Sin[c + d*x]^3 + 420*ArcTanh[(-C ot[c + d*x]^2)^(1/4)]*(-Cot[c + d*x])^(1/4)*Sin[c + d*x]^3 + Cot[c + d*x]^ (1/4)*Sin[c + d*x]*(60*Cos[c + d*x]^2*Cot[c + d*x]^(3/2) - 210*Sqrt[2]*Arc Tan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sin[c + d*x]^2 + 210*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sin[c + d*x]^2 - 840*Sqrt[Cot[c + d*x]]*Sin[ c + d*x]^2 + 280*Cot[c + d*x]^(3/2)*Sin[c + d*x]^2 - 105*Sqrt[2]*Log[1 - S qrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 + 105*Sqrt[2]*Log [1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 + 126*Cot[c + d*x]^(3/2)*Sin[2*(c + d*x)])))/(d*Cot[c + d*x]^(7/4)*(Cos[c + d*x] + Si n[c + d*x])^3)
Time = 0.87 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.12, number of steps used = 13, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3042, 4049, 25, 3042, 4113, 3042, 4011, 3042, 4011, 3042, 4015, 221}
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.
| \(\displaystyle \int (a \cot (c+d x)+a)^3 (e \cot (c+d x))^{3/2} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3 \left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}dx\) |
| \(\Big \downarrow \) 4049 |
| \(\displaystyle -\frac {2 \int -(e \cot (c+d x))^{3/2} \left (8 e \cot ^2(c+d x) a^3+e a^3+7 e \cot (c+d x) a^3\right )dx}{7 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {2 \int (e \cot (c+d x))^{3/2} \left (8 e \cot ^2(c+d x) a^3+e a^3+7 e \cot (c+d x) a^3\right )dx}{7 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \int \left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (8 e \tan \left (c+d x+\frac {\pi }{2}\right )^2 a^3+e a^3-7 e \tan \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx}{7 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e}\) |
| \(\Big \downarrow \) 4113 |
| \(\displaystyle \frac {2 \left (\int (e \cot (c+d x))^{3/2} \left (7 a^3 e \cot (c+d x)-7 a^3 e\right )dx-\frac {16 a^3 (e \cot (c+d x))^{5/2}}{5 d}\right )}{7 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\int \left (-e \tan \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2} \left (-7 e a^3-7 e \tan \left (c+d x+\frac {\pi }{2}\right ) a^3\right )dx-\frac {16 a^3 (e \cot (c+d x))^{5/2}}{5 d}\right )}{7 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {2 \left (\int \sqrt {e \cot (c+d x)} \left (-7 e^2 a^3-7 e^2 \cot (c+d x) a^3\right )dx-\frac {16 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac {14 a^3 e (e \cot (c+d x))^{3/2}}{3 d}\right )}{7 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\int \sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )} \left (7 a^3 e^2 \tan \left (c+d x+\frac {\pi }{2}\right )-7 a^3 e^2\right )dx-\frac {16 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac {14 a^3 e (e \cot (c+d x))^{3/2}}{3 d}\right )}{7 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e}\) |
| \(\Big \downarrow \) 4011 |
| \(\displaystyle \frac {2 \left (\int \frac {7 a^3 e^3-7 a^3 e^3 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}dx+\frac {14 a^3 e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {16 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac {14 a^3 e (e \cot (c+d x))^{3/2}}{3 d}\right )}{7 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {2 \left (\int \frac {7 a^3 e^3+7 a^3 \tan \left (c+d x+\frac {\pi }{2}\right ) e^3}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx+\frac {14 a^3 e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {16 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac {14 a^3 e (e \cot (c+d x))^{3/2}}{3 d}\right )}{7 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e}\) |
| \(\Big \downarrow \) 4015 |
| \(\displaystyle \frac {2 \left (-\frac {98 a^6 e^6 \int \frac {1}{98 a^6 e^6-49 \left (a^3 e^3+a^3 \cot (c+d x) e^3\right )^2 \tan (c+d x)}d\frac {7 \left (a^3 e^3+a^3 \cot (c+d x) e^3\right )}{\sqrt {e \cot (c+d x)}}}{d}+\frac {14 a^3 e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {14 a^3 e (e \cot (c+d x))^{3/2}}{3 d}-\frac {16 a^3 (e \cot (c+d x))^{5/2}}{5 d}\right )}{7 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e}\) |
| \(\Big \downarrow \) 221 |
| \(\displaystyle \frac {2 \left (-\frac {7 \sqrt {2} a^3 e^{5/2} \text {arctanh}\left (\frac {a^3 e^3 \cot (c+d x)+a^3 e^3}{\sqrt {2} a^3 e^{5/2} \sqrt {e \cot (c+d x)}}\right )}{d}+\frac {14 a^3 e^2 \sqrt {e \cot (c+d x)}}{d}-\frac {16 a^3 (e \cot (c+d x))^{5/2}}{5 d}-\frac {14 a^3 e (e \cot (c+d x))^{3/2}}{3 d}\right )}{7 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{5/2}}{7 d e}\) |
Input:
Int[(e*Cot[c + d*x])^(3/2)*(a + a*Cot[c + d*x])^3,x]
Output:
(-2*(e*Cot[c + d*x])^(5/2)*(a^3 + a^3*Cot[c + d*x]))/(7*d*e) + (2*((-7*Sqr t[2]*a^3*e^(5/2)*ArcTanh[(a^3*e^3 + a^3*e^3*Cot[c + d*x])/(Sqrt[2]*a^3*e^( 5/2)*Sqrt[e*Cot[c + d*x]])])/d + (14*a^3*e^2*Sqrt[e*Cot[c + d*x]])/d - (14 *a^3*e*(e*Cot[c + d*x])^(3/2))/(3*d) - (16*a^3*(e*Cot[c + d*x])^(5/2))/(5* d)))/(7*e)
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[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[d*((a + b*Tan[e + f*x])^m/(f*m)), x] + Int [(a + b*Tan[e + f*x])^(m - 1)*Simp[a*c - b*d + (b*c + a*d)*Tan[e + f*x], x] , x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && GtQ[m, 0]
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 - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Simp[1/(d*(m + n - 1)) Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^n*Simp[a^3*d*(m + n - 1) - b^2*(b*c*(m - 2) + a*d*(1 + n)) + b*d*(m + n - 1)*(3*a^2 - b^2)*Tan[ e + f*x] - b^2*(b*c*(m - 2) - a*d*(3*m + 2*n - 4))*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] && GtQ[m, 2] && (GeQ[n, -1] || I ntegerQ[m]) && !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0] && NeQ[a, 0])) )
Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_.)*((A_.) + (B_.)*tan[(e_.) + (f_.)*(x_)] + (C_.)*tan[(e_.) + (f_.)*(x_)]^2), x_Symbol] :> Simp[C*((a + b*Tan[e + f*x])^(m + 1)/(b*f*(m + 1))), x] + Int[(a + b*Tan[e + f*x])^m*Si mp[A - C + B*Tan[e + f*x], x], x] /; FreeQ[{a, b, e, f, A, B, C, m}, x] && NeQ[A*b^2 - a*b*B + a^2*C, 0] && !LeQ[m, -1]
Leaf count of result is larger than twice the leaf count of optimal. \(338\) vs. \(2(135)=270\).
Time = 0.30 (sec) , antiderivative size = 339, normalized size of antiderivative = 2.12
| method | result | size |
| derivativedivides | \(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {3 e \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}} e^{2}}{3}-2 e^{3} \sqrt {e \cot \left (d x +c \right )}+2 e^{4} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(339\) |
| default | \(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}{7}+\frac {3 e \left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}} e^{2}}{3}-2 e^{3} \sqrt {e \cot \left (d x +c \right )}+2 e^{4} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(339\) |
| parts | \(-\frac {2 a^{3} e \left (\sqrt {e \cot \left (d x +c \right )}-\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}\right )}{d}-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {7}{2}}}{7}-\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}} e^{2}}{3}+\frac {e^{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 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d \,e^{2}}+\frac {3 a^{3} \left (-\frac {2 \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+\frac {e^{2} \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}}}\right )}{d}-\frac {6 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-e^{2} \sqrt {e \cot \left (d x +c \right )}+\frac {e^{2} \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}\right )}{d e}\) | \(654\) |
Input:
int((e*cot(d*x+c))^(3/2)*(a+a*cot(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
-2/d*a^3/e^2*(1/7*(e*cot(d*x+c))^(7/2)+3/5*e*(e*cot(d*x+c))^(5/2)+2/3*(e*c ot(d*x+c))^(3/2)*e^2-2*e^3*(e*cot(d*x+c))^(1/2)+2*e^4*(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))))
Time = 0.10 (sec) , antiderivative size = 487, normalized size of antiderivative = 3.04 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=\left [\frac {105 \, \sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - a^{3} e\right )} \sqrt {e} \log \left (\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} + 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) \sin \left (2 \, d x + 2 \, c\right ) - 2 \, {\left (55 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right )^{2} - 30 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 85 \, a^{3} e - 21 \, {\left (13 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 7 \, a^{3} e\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{105 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )} \sin \left (2 \, d x + 2 \, c\right )}, \frac {2 \, {\left (105 \, \sqrt {2} {\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - a^{3} e\right )} \sqrt {-e} \arctan \left (\frac {\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) \sin \left (2 \, d x + 2 \, c\right ) - {\left (55 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right )^{2} - 30 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 85 \, a^{3} e - 21 \, {\left (13 \, a^{3} e \cos \left (2 \, d x + 2 \, c\right ) - 7 \, a^{3} e\right )} \sin \left (2 \, d x + 2 \, c\right )\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right )}}{105 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )} \sin \left (2 \, d x + 2 \, c\right )}\right ] \] Input:
integrate((e*cot(d*x+c))^(3/2)*(a+a*cot(d*x+c))^3,x, algorithm="fricas")
Output:
[1/105*(105*sqrt(2)*(a^3*e*cos(2*d*x + 2*c) - a^3*e)*sqrt(e)*log(sqrt(2)*s qrt(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)*sin(2*d*x + 2*c) - 2*(5 5*a^3*e*cos(2*d*x + 2*c)^2 - 30*a^3*e*cos(2*d*x + 2*c) - 85*a^3*e - 21*(13 *a^3*e*cos(2*d*x + 2*c) - 7*a^3*e)*sin(2*d*x + 2*c))*sqrt((e*cos(2*d*x + 2 *c) + e)/sin(2*d*x + 2*c)))/((d*cos(2*d*x + 2*c) - d)*sin(2*d*x + 2*c)), 2 /105*(105*sqrt(2)*(a^3*e*cos(2*d*x + 2*c) - a^3*e)*sqrt(-e)*arctan(1/2*sqr t(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))*sin(2*d*x + 2*c) - (55*a^3*e*cos(2*d*x + 2*c)^2 - 30*a^3*e*cos(2*d*x + 2*c) - 85*a^3*e - 21* (13*a^3*e*cos(2*d*x + 2*c) - 7*a^3*e)*sin(2*d*x + 2*c))*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/((d*cos(2*d*x + 2*c) - d)*sin(2*d*x + 2*c)) ]
\[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=a^{3} \left (\int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int 3 \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot {\left (c + d x \right )}\, dx + \int 3 \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot ^{2}{\left (c + d x \right )}\, dx + \int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot ^{3}{\left (c + d x \right )}\, dx\right ) \] Input:
integrate((e*cot(d*x+c))**(3/2)*(a+a*cot(d*x+c))**3,x)
Output:
a**3*(Integral((e*cot(c + d*x))**(3/2), x) + Integral(3*(e*cot(c + d*x))** (3/2)*cot(c + d*x), x) + Integral(3*(e*cot(c + d*x))**(3/2)*cot(c + d*x)** 2, x) + Integral((e*cot(c + d*x))**(3/2)*cot(c + d*x)**3, x))
Exception generated. \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=\text {Exception raised: ValueError} \] Input:
integrate((e*cot(d*x+c))^(3/2)*(a+a*cot(d*x+c))^3,x, algorithm="maxima")
Output:
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
\[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=\int { {\left (a \cot \left (d x + c\right ) + a\right )}^{3} \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}} \,d x } \] Input:
integrate((e*cot(d*x+c))^(3/2)*(a+a*cot(d*x+c))^3,x, algorithm="giac")
Output:
integrate((a*cot(d*x + c) + a)^3*(e*cot(d*x + c))^(3/2), x)
Time = 10.40 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.89 \[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=\frac {4\,a^3\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}-\frac {6\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}{5\,d\,e}-\frac {2\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{7/2}}{7\,d\,e^2}-\frac {4\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}+\frac {\sqrt {2}\,a^3\,e^{3/2}\,\mathrm {atan}\left (\frac {\sqrt {2}\,a^6\,e^{9/2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,32{}\mathrm {i}}{32\,a^6\,e^5+32\,a^6\,e^5\,\mathrm {cot}\left (c+d\,x\right )}\right )\,2{}\mathrm {i}}{d} \] Input:
int((e*cot(c + d*x))^(3/2)*(a + a*cot(c + d*x))^3,x)
Output:
(4*a^3*e*(e*cot(c + d*x))^(1/2))/d - (6*a^3*(e*cot(c + d*x))^(5/2))/(5*d*e ) - (2*a^3*(e*cot(c + d*x))^(7/2))/(7*d*e^2) - (4*a^3*(e*cot(c + d*x))^(3/ 2))/(3*d) + (2^(1/2)*a^3*e^(3/2)*atan((2^(1/2)*a^6*e^(9/2)*(e*cot(c + d*x) )^(1/2)*32i)/(32*a^6*e^5 + 32*a^6*e^5*cot(c + d*x)))*2i)/d
\[ \int (e \cot (c+d x))^{3/2} (a+a \cot (c+d x))^3 \, dx=\frac {\sqrt {e}\, a^{3} e \left (-6 \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2}+20 \sqrt {\cot \left (d x +c \right )}+10 \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) d +5 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{4}d x \right ) d +15 \left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2}d x \right ) d \right )}{5 d} \] Input:
int((e*cot(d*x+c))^(3/2)*(a+a*cot(d*x+c))^3,x)
Output:
(sqrt(e)*a**3*e*( - 6*sqrt(cot(c + d*x))*cot(c + d*x)**2 + 20*sqrt(cot(c + d*x)) + 10*int(sqrt(cot(c + d*x))/cot(c + d*x),x)*d + 5*int(sqrt(cot(c + d*x))*cot(c + d*x)**4,x)*d + 15*int(sqrt(cot(c + d*x))*cot(c + d*x)**2,x)* d))/(5*d)