Integrand size = 25, antiderivative size = 117 \[ \int \frac {(a+a \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx=\frac {2 \sqrt {2} a^3 \text {arctanh}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d \sqrt {e}}-\frac {16 a^3 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 \sqrt {e \cot (c+d x)} \left (a^3+a^3 \cot (c+d x)\right )}{3 d e} \] Output:
2*2^(1/2)*a^3*arctanh(1/2*(e^(1/2)+e^(1/2)*cot(d*x+c))*2^(1/2)/(e*cot(d*x+ c))^(1/2))/d/e^(1/2)-16/3*a^3*(e*cot(d*x+c))^(1/2)/d/e-2/3*(e*cot(d*x+c))^ (1/2)*(a^3+a^3*cot(d*x+c))/d/e
Leaf count is larger than twice the leaf count of optimal. \(342\) vs. \(2(117)=234\).
Time = 0.89 (sec) , antiderivative size = 342, normalized size of antiderivative = 2.92 \[ \int \frac {(a+a \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx=-\frac {a^3 (1+\cot (c+d x))^3 \sin (c+d x) \left (4 \cos ^2(c+d x)+6 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \sqrt {\cot (c+d x)} \sin ^2(c+d x)-6 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \sqrt {\cot (c+d x)} \sin ^2(c+d x)+12 \arctan \left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot ^2(c+d x)} \sin ^2(c+d x)-12 \text {arctanh}\left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot ^2(c+d x)} \sin ^2(c+d x)+3 \sqrt {2} \sqrt {\cot (c+d x)} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^2(c+d x)-3 \sqrt {2} \sqrt {\cot (c+d x)} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^2(c+d x)+18 \sin (2 (c+d x))\right )}{6 d \sqrt {e \cot (c+d x)} (\cos (c+d x)+\sin (c+d x))^3} \] Input:
Integrate[(a + a*Cot[c + d*x])^3/Sqrt[e*Cot[c + d*x]],x]
Output:
-1/6*(a^3*(1 + Cot[c + d*x])^3*Sin[c + d*x]*(4*Cos[c + d*x]^2 + 6*Sqrt[2]* ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sin[c + d*x]^2 - 6*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sqrt[Cot[c + d*x]]*Sin[c + d*x]^2 + 12*ArcTan[(-Cot[c + d*x]^2)^(1/4)]*(-Cot[c + d*x]^2)^(1/4)*Sin [c + d*x]^2 - 12*ArcTanh[(-Cot[c + d*x]^2)^(1/4)]*(-Cot[c + d*x]^2)^(1/4)* Sin[c + d*x]^2 + 3*Sqrt[2]*Sqrt[Cot[c + d*x]]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 - 3*Sqrt[2]*Sqrt[Cot[c + d*x]]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 + 18*Sin[2*(c + d*x)]))/(d*Sqrt[e*Cot[c + d*x]]*(Cos[c + d*x] + Sin[c + d*x])^3)
Time = 0.55 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.09, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3042, 4049, 25, 3042, 4113, 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.
\(\displaystyle \int \frac {(a \cot (c+d x)+a)^3}{\sqrt {e \cot (c+d x)}} \, dx\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \int \frac {\left (a-a \tan \left (c+d x+\frac {\pi }{2}\right )\right )^3}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx\) |
\(\Big \downarrow \) 4049 |
\(\displaystyle -\frac {2 \int -\frac {4 e \cot ^2(c+d x) a^3+e a^3+3 e \cot (c+d x) a^3}{\sqrt {e \cot (c+d x)}}dx}{3 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) \sqrt {e \cot (c+d x)}}{3 d e}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {2 \int \frac {4 e \cot ^2(c+d x) a^3+e a^3+3 e \cot (c+d x) a^3}{\sqrt {e \cot (c+d x)}}dx}{3 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) \sqrt {e \cot (c+d x)}}{3 d e}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \int \frac {4 e \tan \left (c+d x+\frac {\pi }{2}\right )^2 a^3+e a^3-3 e \tan \left (c+d x+\frac {\pi }{2}\right ) a^3}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx}{3 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) \sqrt {e \cot (c+d x)}}{3 d e}\) |
\(\Big \downarrow \) 4113 |
\(\displaystyle \frac {2 \left (\int \frac {3 a^3 e \cot (c+d x)-3 a^3 e}{\sqrt {e \cot (c+d x)}}dx-\frac {8 a^3 \sqrt {e \cot (c+d x)}}{d}\right )}{3 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) \sqrt {e \cot (c+d x)}}{3 d e}\) |
\(\Big \downarrow \) 3042 |
\(\displaystyle \frac {2 \left (\int \frac {-3 e a^3-3 e \tan \left (c+d x+\frac {\pi }{2}\right ) a^3}{\sqrt {-e \tan \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {8 a^3 \sqrt {e \cot (c+d x)}}{d}\right )}{3 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) \sqrt {e \cot (c+d x)}}{3 d e}\) |
\(\Big \downarrow \) 4015 |
\(\displaystyle \frac {2 \left (-\frac {18 a^6 e^2 \int \frac {1}{18 a^6 e^2-9 \left (e a^3+e \cot (c+d x) a^3\right )^2 \tan (c+d x)}d\left (-\frac {3 \left (e a^3+e \cot (c+d x) a^3\right )}{\sqrt {e \cot (c+d x)}}\right )}{d}-\frac {8 a^3 \sqrt {e \cot (c+d x)}}{d}\right )}{3 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) \sqrt {e \cot (c+d x)}}{3 d e}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle \frac {2 \left (\frac {3 \sqrt {2} a^3 \sqrt {e} \text {arctanh}\left (\frac {a^3 e \cot (c+d x)+a^3 e}{\sqrt {2} a^3 \sqrt {e} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {8 a^3 \sqrt {e \cot (c+d x)}}{d}\right )}{3 e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) \sqrt {e \cot (c+d x)}}{3 d e}\) |
Input:
Int[(a + a*Cot[c + d*x])^3/Sqrt[e*Cot[c + d*x]],x]
Output:
(-2*Sqrt[e*Cot[c + d*x]]*(a^3 + a^3*Cot[c + d*x]))/(3*d*e) + (2*((3*Sqrt[2 ]*a^3*Sqrt[e]*ArcTanh[(a^3*e + a^3*e*Cot[c + d*x])/(Sqrt[2]*a^3*Sqrt[e]*Sq rt[e*Cot[c + d*x]])])/d - (8*a^3*Sqrt[e*Cot[c + d*x]])/d))/(3*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[((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. \(308\) vs. \(2(98)=196\).
Time = 0.33 (sec) , antiderivative size = 309, normalized size of antiderivative = 2.64
method | result | size |
derivativedivides | \(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+3 e \sqrt {e \cot \left (d x +c \right )}-2 e^{2} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(309\) |
default | \(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+3 e \sqrt {e \cot \left (d x +c \right )}-2 e^{2} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\) | \(309\) |
parts | \(-\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}-\frac {2 a^{3} \left (\frac {\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 )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )}{d \,e^{2}}-\frac {3 a^{3} \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 \left (e^{2}\right )^{\frac {1}{4}}}-\frac {6 a^{3} \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 e}\) | \(590\) |
Input:
int((a+a*cot(d*x+c))^3/(e*cot(d*x+c))^(1/2),x,method=_RETURNVERBOSE)
Output:
-2/d*a^3/e^2*(1/3*(e*cot(d*x+c))^(3/2)+3*e*(e*cot(d*x+c))^(1/2)-2*e^2*(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*arc tan(-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*arcta n(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 = 349, normalized size of antiderivative = 2.98 \[ \int \frac {(a+a \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx=\left [\frac {3 \, \sqrt {2} a^{3} \sqrt {e} \log \left (-\frac {\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) - 1\right )}}{\sqrt {e}} + 2 \, \sin \left (2 \, d x + 2 \, c\right ) + 1\right ) \sin \left (2 \, d x + 2 \, c\right ) - 2 \, {\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) + 9 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) + a^{3}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{3 \, d e \sin \left (2 \, d x + 2 \, c\right )}, -\frac {2 \, {\left (3 \, \sqrt {2} a^{3} e \sqrt {-\frac {1}{e}} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sqrt {-\frac {1}{e}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )}}\right ) \sin \left (2 \, d x + 2 \, c\right ) + {\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) + 9 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) + a^{3}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right )}}{3 \, d e \sin \left (2 \, d x + 2 \, c\right )}\right ] \] Input:
integrate((a+a*cot(d*x+c))^3/(e*cot(d*x+c))^(1/2),x, algorithm="fricas")
Output:
[1/3*(3*sqrt(2)*a^3*sqrt(e)*log(-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) + 2*sin(2 *d*x + 2*c) + 1)*sin(2*d*x + 2*c) - 2*(a^3*cos(2*d*x + 2*c) + 9*a^3*sin(2* d*x + 2*c) + a^3)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*e*si n(2*d*x + 2*c)), -2/3*(3*sqrt(2)*a^3*e*sqrt(-1/e)*arctan(1/2*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)/(cos(2*d*x + 2*c) + 1))*sin(2*d*x + 2*c) + (a^3*cos( 2*d*x + 2*c) + 9*a^3*sin(2*d*x + 2*c) + a^3)*sqrt((e*cos(2*d*x + 2*c) + e) /sin(2*d*x + 2*c)))/(d*e*sin(2*d*x + 2*c))]
\[ \int \frac {(a+a \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx=a^{3} \left (\int \frac {1}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx + \int \frac {3 \cot {\left (c + d x \right )}}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx + \int \frac {3 \cot ^{2}{\left (c + d x \right )}}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx + \int \frac {\cot ^{3}{\left (c + d x \right )}}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx\right ) \] Input:
integrate((a+a*cot(d*x+c))**3/(e*cot(d*x+c))**(1/2),x)
Output:
a**3*(Integral(1/sqrt(e*cot(c + d*x)), x) + Integral(3*cot(c + d*x)/sqrt(e *cot(c + d*x)), x) + Integral(3*cot(c + d*x)**2/sqrt(e*cot(c + d*x)), x) + Integral(cot(c + d*x)**3/sqrt(e*cot(c + d*x)), x))
Exception generated. \[ \int \frac {(a+a \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+a*cot(d*x+c))^3/(e*cot(d*x+c))^(1/2),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
\[ \int \frac {(a+a \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx=\int { \frac {{\left (a \cot \left (d x + c\right ) + a\right )}^{3}}{\sqrt {e \cot \left (d x + c\right )}} \,d x } \] Input:
integrate((a+a*cot(d*x+c))^3/(e*cot(d*x+c))^(1/2),x, algorithm="giac")
Output:
integrate((a*cot(d*x + c) + a)^3/sqrt(e*cot(d*x + c)), x)
Time = 9.74 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.85 \[ \int \frac {(a+a \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx=\frac {2\,\sqrt {2}\,a^3\,\mathrm {atanh}\left (\frac {32\,\sqrt {2}\,a^6\,\sqrt {e}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{32\,a^6\,e+32\,a^6\,e\,\mathrm {cot}\left (c+d\,x\right )}\right )}{d\,\sqrt {e}}-\frac {2\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d\,e^2}-\frac {6\,a^3\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d\,e} \] Input:
int((a + a*cot(c + d*x))^3/(e*cot(c + d*x))^(1/2),x)
Output:
(2*2^(1/2)*a^3*atanh((32*2^(1/2)*a^6*e^(1/2)*(e*cot(c + d*x))^(1/2))/(32*a ^6*e + 32*a^6*e*cot(c + d*x))))/(d*e^(1/2)) - (2*a^3*(e*cot(c + d*x))^(3/2 ))/(3*d*e^2) - (6*a^3*(e*cot(c + d*x))^(1/2))/(d*e)
\[ \int \frac {(a+a \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx=\frac {\sqrt {e}\, a^{3} \left (-6 \sqrt {\cot \left (d x +c \right )}-2 \left (\int \frac {\sqrt {\cot \left (d x +c \right )}}{\cot \left (d x +c \right )}d x \right ) d +3 \left (\int \sqrt {\cot \left (d x +c \right )}d x \right ) d +\left (\int \sqrt {\cot \left (d x +c \right )}\, \cot \left (d x +c \right )^{2}d x \right ) d \right )}{d e} \] Input:
int((a+a*cot(d*x+c))^3/(e*cot(d*x+c))^(1/2),x)
Output:
(sqrt(e)*a**3*( - 6*sqrt(cot(c + d*x)) - 2*int(sqrt(cot(c + d*x))/cot(c + d*x),x)*d + 3*int(sqrt(cot(c + d*x)),x)*d + int(sqrt(cot(c + d*x))*cot(c + d*x)**2,x)*d))/(d*e)