\(\int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx\) [17]

Optimal result

Integrand size = 25, antiderivative size = 138 \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=-\frac {2 \sqrt {2} a^3 \sqrt {e} \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e} \]

[Out]

-8/5*a^3*(e*cot(d*x+c))^(3/2)/d/e-2/5*(e*cot(d*x+c))^(3/2)*(a^3+a^3*cot(d*x+c))/d/e-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)*e^(1/2)/d-4*a^3*(e*cot(d*x+c))^(1/2)/d

Rubi [A] (verified)

Time = 0.23 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3647, 3711, 3609, 3613, 211} \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=-\frac {2 \sqrt {2} a^3 \sqrt {e} \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{3/2}}{5 d e} \]

[In]

Int[Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^3,x]

[Out]

(-2*Sqrt[2]*a^3*Sqrt[e]*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d - (4*a^3*Sq
rt[e*Cot[c + d*x]])/d - (8*a^3*(e*Cot[c + d*x])^(3/2))/(5*d*e) - (2*(e*Cot[c + d*x])^(3/2)*(a^3 + a^3*Cot[c +
d*x]))/(5*d*e)

Rule 211

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 3609

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]

Rule 3613

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[-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]

Rule 3647

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[b^2*(a + b*Tan[e + f*x])^(m - 2)*((c + d*Tan[e + f*x])^(n + 1)/(d*f*(m + n - 1))), x] + Dist[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] || IntegerQ[m]) &&  !(IGtQ[n, 2] && ( !IntegerQ[m] || (EqQ[c, 0]
&& NeQ[a, 0])))

Rule 3711

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*Simp[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]

Rubi steps \begin{align*} \text {integral}& = -\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}-\frac {2 \int \sqrt {e \cot (c+d x)} \left (-a^3 e-5 a^3 e \cot (c+d x)-6 a^3 e \cot ^2(c+d x)\right ) \, dx}{5 e} \\ & = -\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}-\frac {2 \int \sqrt {e \cot (c+d x)} \left (5 a^3 e-5 a^3 e \cot (c+d x)\right ) \, dx}{5 e} \\ & = -\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}-\frac {2 \int \frac {5 a^3 e^2+5 a^3 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{5 e} \\ & = -\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}+\frac {\left (20 a^6 e^3\right ) \text {Subst}\left (\int \frac {1}{-50 a^6 e^4-e x^2} \, dx,x,\frac {5 a^3 e^2-5 a^3 e^2 \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d} \\ & = -\frac {2 \sqrt {2} a^3 \sqrt {e} \arctan \left (\frac {\sqrt {e}-\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d}-\frac {4 a^3 \sqrt {e \cot (c+d x)}}{d}-\frac {8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac {2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e} \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(360\) vs. \(2(138)=276\).

Time = 3.43 (sec) , antiderivative size = 360, normalized size of antiderivative = 2.61 \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=-\frac {a^3 \sqrt {e \cot (c+d x)} (1+\cot (c+d x))^3 \left (-20 \arctan \left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot (c+d x)} \sin ^3(c+d x)+20 \text {arctanh}\left (\sqrt [4]{-\cot ^2(c+d x)}\right ) \sqrt [4]{-\cot (c+d x)} \sin ^3(c+d x)+\sqrt [4]{\cot (c+d x)} \sin (c+d x) \left (4 \cos ^2(c+d x) \sqrt {\cot (c+d x)}+10 \sqrt {2} \arctan \left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \sin ^2(c+d x)-10 \sqrt {2} \arctan \left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \sin ^2(c+d x)+40 \sqrt {\cot (c+d x)} \sin ^2(c+d x)+5 \sqrt {2} \log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^2(c+d x)-5 \sqrt {2} \log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right ) \sin ^2(c+d x)+10 \sqrt {\cot (c+d x)} \sin (2 (c+d x))\right )\right )}{10 d \cot ^{\frac {3}{4}}(c+d x) (\cos (c+d x)+\sin (c+d x))^3} \]

[In]

Integrate[Sqrt[e*Cot[c + d*x]]*(a + a*Cot[c + d*x])^3,x]

[Out]

-1/10*(a^3*Sqrt[e*Cot[c + d*x]]*(1 + Cot[c + d*x])^3*(-20*ArcTan[(-Cot[c + d*x]^2)^(1/4)]*(-Cot[c + d*x])^(1/4
)*Sin[c + d*x]^3 + 20*ArcTanh[(-Cot[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]*(4*Cos[c + d*x]^2*Sqrt[Cot[c + d*x]] + 10*Sqrt[2]*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sin[c
+ d*x]^2 - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sin[c + d*x]^2 + 40*Sqrt[Cot[c + d*x]]*Sin[c + d
*x]^2 + 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 - 5*Sqrt[2]*Log[1 + Sqrt[2
]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 + 10*Sqrt[Cot[c + d*x]]*Sin[2*(c + d*x)])))/(d*Cot[c + d*x
]^(3/4)*(Cos[c + d*x] + Sin[c + d*x])^3)

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(322\) vs. \(2(117)=234\).

Time = 0.05 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.34

method	result	size
derivativedivides	\(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}+2 \sqrt {e \cot \left (d x +c \right )}\, e^{2}-2 e^{3} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\)	\(323\)
default	\(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}+e \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}+2 \sqrt {e \cot \left (d x +c \right )}\, e^{2}-2 e^{3} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}+\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\)	\(323\)
parts	\(-\frac {a^{3} e \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 {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {5}{2}}}{5}-\sqrt {e \cot \left (d x +c \right )}\, e^{2}+\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^{2}}+\frac {3 a^{3} \left (-2 \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 )}{4}\right )}{d}-\frac {6 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}\)	\(623\)

[In]

int((e*cot(d*x+c))^(1/2)*(a+a*cot(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

-2/d*a^3/e^2*(1/5*(e*cot(d*x+c))^(5/2)+e*(e*cot(d*x+c))^(3/2)+2*(e*cot(d*x+c))^(1/2)*e^2-2*e^3*(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))))

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 366, normalized size of antiderivative = 2.65 \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=\left [\frac {5 \, \sqrt {2} {\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) - a^{3}\right )} \sqrt {-e} \log \left (\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) - 2 \, {\left (9 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 5 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) - 11 \, a^{3}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{5 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )}}, -\frac {2 \, {\left (5 \, \sqrt {2} {\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) - a^{3}\right )} \sqrt {e} \arctan \left (-\frac {\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) + {\left (9 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 5 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) - 11 \, a^{3}\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right )}}{5 \, {\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )}}\right ] \]

[In]

integrate((e*cot(d*x+c))^(1/2)*(a+a*cot(d*x+c))^3,x, algorithm="fricas")

[Out]

[1/5*(5*sqrt(2)*(a^3*cos(2*d*x + 2*c) - a^3)*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) - 2*(9*a^3*cos(2*d*x + 2*c)
- 5*a^3*sin(2*d*x + 2*c) - 11*a^3)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*cos(2*d*x + 2*c) - d),
-2/5*(5*sqrt(2)*(a^3*cos(2*d*x + 2*c) - a^3)*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)) + (9*a^3*cos(2*d*x + 2
*c) - 5*a^3*sin(2*d*x + 2*c) - 11*a^3)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*cos(2*d*x + 2*c) -
d)]

Sympy [F]

\[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=a^{3} \left (\int \sqrt {e \cot {\left (c + d x \right )}}\, dx + \int 3 \sqrt {e \cot {\left (c + d x \right )}} \cot {\left (c + d x \right )}\, dx + \int 3 \sqrt {e \cot {\left (c + d x \right )}} \cot ^{2}{\left (c + d x \right )}\, dx + \int \sqrt {e \cot {\left (c + d x \right )}} \cot ^{3}{\left (c + d x \right )}\, dx\right ) \]

[In]

integrate((e*cot(d*x+c))**(1/2)*(a+a*cot(d*x+c))**3,x)

[Out]

a**3*(Integral(sqrt(e*cot(c + d*x)), x) + Integral(3*sqrt(e*cot(c + d*x))*cot(c + d*x), x) + Integral(3*sqrt(e
*cot(c + d*x))*cot(c + d*x)**2, x) + Integral(sqrt(e*cot(c + d*x))*cot(c + d*x)**3, x))

Maxima [F(-2)]

Exception generated. \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=\text {Exception raised: ValueError} \]

[In]

integrate((e*cot(d*x+c))^(1/2)*(a+a*cot(d*x+c))^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(e>0)', see `assume?` for more
details)Is e

Giac [F]

\[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=\int { {\left (a \cot \left (d x + c\right ) + a\right )}^{3} \sqrt {e \cot \left (d x + c\right )} \,d x } \]

[In]

integrate((e*cot(d*x+c))^(1/2)*(a+a*cot(d*x+c))^3,x, algorithm="giac")

[Out]

integrate((a*cot(d*x + c) + a)^3*sqrt(e*cot(d*x + c)), x)

Mupad [B] (verification not implemented)

Time = 13.52 (sec) , antiderivative size = 136, normalized size of antiderivative = 0.99 \[ \int \sqrt {e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx=\frac {\sqrt {2}\,a^3\,\sqrt {e}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{2\,\sqrt {e}}+\frac {\sqrt {2}\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{2\,e^{3/2}}\right )\right )}{d}-\frac {2\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{d\,e}-\frac {2\,a^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{5/2}}{5\,d\,e^2}-\frac {4\,a^3\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d} \]

[In]

int((e*cot(c + d*x))^(1/2)*(a + a*cot(c + d*x))^3,x)

[Out]

(2^(1/2)*a^3*e^(1/2)*(2*atan((2^(1/2)*(e*cot(c + d*x))^(1/2))/(2*e^(1/2))) + 2*atan((2^(1/2)*(e*cot(c + d*x))^
(1/2))/(2*e^(1/2)) + (2^(1/2)*(e*cot(c + d*x))^(3/2))/(2*e^(3/2)))))/d - (2*a^3*(e*cot(c + d*x))^(3/2))/(d*e)
- (2*a^3*(e*cot(c + d*x))^(5/2))/(5*d*e^2) - (4*a^3*(e*cot(c + d*x))^(1/2))/d