3.1.64 \(\int \frac {(a+b \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx\) [64]

Optimal. Leaf size=313 \[ \frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d \sqrt {e}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d \sqrt {e}} \]

[Out]

1/2*(a-b)*(a^2+4*a*b+b^2)*arctan(1-2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/d*2^(1/2)/e^(1/2)-1/2*(a-b)*(a^2+4*a*
b+b^2)*arctan(1+2^(1/2)*(e*cot(d*x+c))^(1/2)/e^(1/2))/d*2^(1/2)/e^(1/2)+1/4*(a+b)*(a^2-4*a*b+b^2)*ln(e^(1/2)+c
ot(d*x+c)*e^(1/2)-2^(1/2)*(e*cot(d*x+c))^(1/2))/d*2^(1/2)/e^(1/2)-1/4*(a+b)*(a^2-4*a*b+b^2)*ln(e^(1/2)+cot(d*x
+c)*e^(1/2)+2^(1/2)*(e*cot(d*x+c))^(1/2))/d*2^(1/2)/e^(1/2)-16/3*a*b^2*(e*cot(d*x+c))^(1/2)/d/e-2/3*b^2*(a+b*c
ot(d*x+c))*(e*cot(d*x+c))^(1/2)/d/e

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Rubi [A]
time = 0.29, antiderivative size = 313, normalized size of antiderivative = 1.00, number of steps used = 12, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3647, 3711, 3615, 1182, 1176, 631, 210, 1179, 642} \begin {gather*} \frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}+1\right )}{\sqrt {2} d \sqrt {e}}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \sqrt {e}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}+\sqrt {e}\right )}{2 \sqrt {2} d \sqrt {e}}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{3 d e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((a - b)*(a^2 + 4*a*b + b^2)*ArcTan[1 - (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - ((a - b
)*(a^2 + 4*a*b + b^2)*ArcTan[1 + (Sqrt[2]*Sqrt[e*Cot[c + d*x]])/Sqrt[e]])/(Sqrt[2]*d*Sqrt[e]) - (16*a*b^2*Sqrt
[e*Cot[c + d*x]])/(3*d*e) - (2*b^2*Sqrt[e*Cot[c + d*x]]*(a + b*Cot[c + d*x]))/(3*d*e) + ((a + b)*(a^2 - 4*a*b
+ b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] - Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e]) - ((a + b)*(a
^2 - 4*a*b + b^2)*Log[Sqrt[e] + Sqrt[e]*Cot[c + d*x] + Sqrt[2]*Sqrt[e*Cot[c + d*x]]])/(2*Sqrt[2]*d*Sqrt[e])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 631

Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> With[{q = 1 - 4*Simplify[a*(c/b^2)]}, Dist[-2/b, Sub
st[Int[1/(q - x^2), x], x, 1 + 2*c*(x/b)], x] /; RationalQ[q] && (EqQ[q^2, 1] ||  !RationalQ[b^2 - 4*a*c])] /;
FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 642

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[d*(Log[RemoveContent[a + b*x +
c*x^2, x]]/b), x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rule 1176

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[2*(d/e), 2]}, Dist[e/(2*c), Int[1/S
imp[d/e + q*x + x^2, x], x], x] + Dist[e/(2*c), Int[1/Simp[d/e - q*x + x^2, x], x], x]] /; FreeQ[{a, c, d, e},
x] && EqQ[c*d^2 - a*e^2, 0] && PosQ[d*e]

Rule 1179

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[-2*(d/e), 2]}, Dist[e/(2*c*q), Int[
(q - 2*x)/Simp[d/e + q*x - x^2, x], x], x] + Dist[e/(2*c*q), Int[(q + 2*x)/Simp[d/e - q*x - x^2, x], x], x]] /
; FreeQ[{a, c, d, e}, x] && EqQ[c*d^2 - a*e^2, 0] && NegQ[d*e]

Rule 1182

Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[a*c, 2]}, Dist[(d*q + a*e)/(2*a*c),
Int[(q + c*x^2)/(a + c*x^4), x], x] + Dist[(d*q - a*e)/(2*a*c), Int[(q - c*x^2)/(a + c*x^4), x], x]] /; FreeQ
[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && NeQ[c*d^2 - a*e^2, 0] && NegQ[(-a)*c]

Rule 3615

Int[((c_) + (d_.)*tan[(e_.) + (f_.)*(x_)])/Sqrt[(b_.)*tan[(e_.) + (f_.)*(x_)]], x_Symbol] :> Dist[2/f, Subst[I
nt[(b*c + d*x^2)/(b^2 + x^4), x], x, Sqrt[b*Tan[e + f*x]]], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[c^2 - d^2,
0] && NeQ[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*} \int \frac {(a+b \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx &=-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}-\frac {2 \int \frac {-\frac {1}{2} a \left (3 a^2-b^2\right ) e-\frac {3}{2} b \left (3 a^2-b^2\right ) e \cot (c+d x)-4 a b^2 e \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{3 e}\\ &=-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}-\frac {2 \int \frac {-\frac {3}{2} a \left (a^2-3 b^2\right ) e-\frac {3}{2} b \left (3 a^2-b^2\right ) e \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{3 e}\\ &=-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}-\frac {4 \text {Subst}\left (\int \frac {\frac {3}{2} a \left (a^2-3 b^2\right ) e^2+\frac {3}{2} b \left (3 a^2-b^2\right ) e x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{3 d e}\\ &=-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}-\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {e-x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {e+x^2}{e^2+x^4} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{d}\\ &=-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{e-\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 d}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{e+\sqrt {2} \sqrt {e} x+x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 d}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}+2 x}{-e-\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d \sqrt {e}}+\frac {\left ((a+b) \left (a^2-4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {\sqrt {2} \sqrt {e}-2 x}{-e+\sqrt {2} \sqrt {e} x-x^2} \, dx,x,\sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d \sqrt {e}}\\ &=-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d \sqrt {e}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d \sqrt {e}}-\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}+\frac {\left ((a-b) \left (a^2+4 a b+b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}\\ &=\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}-\frac {(a-b) \left (a^2+4 a b+b^2\right ) \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {e \cot (c+d x)}}{\sqrt {e}}\right )}{\sqrt {2} d \sqrt {e}}-\frac {16 a b^2 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 b^2 \sqrt {e \cot (c+d x)} (a+b \cot (c+d x))}{3 d e}+\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)-\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d \sqrt {e}}-\frac {(a+b) \left (a^2-4 a b+b^2\right ) \log \left (\sqrt {e}+\sqrt {e} \cot (c+d x)+\sqrt {2} \sqrt {e \cot (c+d x)}\right )}{2 \sqrt {2} d \sqrt {e}}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 1.08, size = 216, normalized size = 0.69 \begin {gather*} -\frac {2 \sqrt {\cot (c+d x)} \left (9 a b^2 \sqrt {\cot (c+d x)}+b^3 \cot ^{\frac {3}{2}}(c+d x)-b \left (-3 a^2+b^2\right ) \cot ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )-\frac {3 a \left (a^2-3 b^2\right ) \left (2 \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right )-2 \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right )+\log \left (1-\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )-\log \left (1+\sqrt {2} \sqrt {\cot (c+d x)}+\cot (c+d x)\right )\right )}{4 \sqrt {2}}\right )}{3 d \sqrt {e \cot (c+d x)}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[Cot[c + d*x]]*(9*a*b^2*Sqrt[Cot[c + d*x]] + b^3*Cot[c + d*x]^(3/2) - b*(-3*a^2 + b^2)*Cot[c + d*x]^(3
/2)*Hypergeometric2F1[3/4, 1, 7/4, -Cot[c + d*x]^2] - (3*a*(a^2 - 3*b^2)*(2*ArcTan[1 - Sqrt[2]*Sqrt[Cot[c + d*
x]]] - 2*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]] + Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]] - Log[1 +
Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]))/(4*Sqrt[2])))/(3*d*Sqrt[e*Cot[c + d*x]])

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Maple [A]
time = 0.51, size = 337, normalized size = 1.08

method	result	size
derivativedivides	\(-\frac {2 \left (\frac {b^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+3 a \,b^{2} e \sqrt {e \cot \left (d x +c \right )}+e^{2} \left (\frac {\left (a^{3} e -3 a \,b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\)	\(337\)
default	\(-\frac {2 \left (\frac {b^{3} \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+3 a \,b^{2} e \sqrt {e \cot \left (d x +c \right )}+e^{2} \left (\frac {\left (a^{3} e -3 a \,b^{2} e \right ) \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e^{2}}+\frac {\left (3 a^{2} b -b^{3}\right ) \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\)	\(337\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.50, size = 222, normalized size = 0.71 \begin {gather*} -\frac {{\left (6 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} + \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 6 \, \sqrt {2} {\left (a^{3} + 3 \, a^{2} b - 3 \, a b^{2} - b^{3}\right )} \arctan \left (-\frac {1}{2} \, \sqrt {2} {\left (\sqrt {2} - \frac {2}{\sqrt {\tan \left (d x + c\right )}}\right )}\right ) + 3 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - 3 \, \sqrt {2} {\left (a^{3} - 3 \, a^{2} b - 3 \, a b^{2} + b^{3}\right )} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) + \frac {72 \, a b^{2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {8 \, b^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}}\right )} e^{\left (-\frac {1}{2}\right )}}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/12*(6*sqrt(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(1/2*sqrt(2)*(sqrt(2) + 2/sqrt(tan(d*x + c)))) + 6*sqrt
(2)*(a^3 + 3*a^2*b - 3*a*b^2 - b^3)*arctan(-1/2*sqrt(2)*(sqrt(2) - 2/sqrt(tan(d*x + c)))) + 3*sqrt(2)*(a^3 - 3
*a^2*b - 3*a*b^2 + b^3)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - 3*sqrt(2)*(a^3 - 3*a^2*b - 3*a*
b^2 + b^3)*log(-sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) + 72*a*b^2/sqrt(tan(d*x + c)) + 8*b^3/tan(d*x
+ c)^(3/2))*e^(-1/2)/d

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Fricas [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \cot {\left (c + d x \right )}\right )^{3}}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [B]
time = 1.41, size = 1896, normalized size = 6.06 \begin {gather*} -\frac {2\,b^3\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d\,e^2}-\frac {6\,a\,b^2\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d\,e}+\mathrm {atan}\left (\frac {\left (\frac {16\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (a^6\,e^2-15\,a^4\,b^2\,e^2+15\,a^2\,b^4\,e^2-b^6\,e^2\right )}{d^2}-\frac {8\,\left (4\,a^3\,d^2\,e^3-12\,a\,b^2\,d^2\,e^3\right )\,\sqrt {\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}}{d^3}\right )\,\sqrt {\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}\,1{}\mathrm {i}+\left (\frac {16\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (a^6\,e^2-15\,a^4\,b^2\,e^2+15\,a^2\,b^4\,e^2-b^6\,e^2\right )}{d^2}+\frac {8\,\left (4\,a^3\,d^2\,e^3-12\,a\,b^2\,d^2\,e^3\right )\,\sqrt {\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}}{d^3}\right )\,\sqrt {\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}\,1{}\mathrm {i}}{\frac {16\,\left (3\,a^8\,b\,e^2+8\,a^6\,b^3\,e^2+6\,a^4\,b^5\,e^2-b^9\,e^2\right )}{d^3}+\left (\frac {16\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (a^6\,e^2-15\,a^4\,b^2\,e^2+15\,a^2\,b^4\,e^2-b^6\,e^2\right )}{d^2}+\frac {8\,\left (4\,a^3\,d^2\,e^3-12\,a\,b^2\,d^2\,e^3\right )\,\sqrt {\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}}{d^3}\right )\,\sqrt {\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}-\left (\frac {16\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (a^6\,e^2-15\,a^4\,b^2\,e^2+15\,a^2\,b^4\,e^2-b^6\,e^2\right )}{d^2}-\frac {8\,\left (4\,a^3\,d^2\,e^3-12\,a\,b^2\,d^2\,e^3\right )\,\sqrt {\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}}{d^3}\right )\,\sqrt {\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}}\right )\,\sqrt {\frac {\left (-a^6+a^5\,b\,6{}\mathrm {i}+15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}-15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}+b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}\,2{}\mathrm {i}+\mathrm {atan}\left (\frac {\left (\frac {16\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (a^6\,e^2-15\,a^4\,b^2\,e^2+15\,a^2\,b^4\,e^2-b^6\,e^2\right )}{d^2}-\frac {8\,\left (4\,a^3\,d^2\,e^3-12\,a\,b^2\,d^2\,e^3\right )\,\sqrt {\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}}{d^3}\right )\,\sqrt {\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}\,1{}\mathrm {i}+\left (\frac {16\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (a^6\,e^2-15\,a^4\,b^2\,e^2+15\,a^2\,b^4\,e^2-b^6\,e^2\right )}{d^2}+\frac {8\,\left (4\,a^3\,d^2\,e^3-12\,a\,b^2\,d^2\,e^3\right )\,\sqrt {\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}}{d^3}\right )\,\sqrt {\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}\,1{}\mathrm {i}}{\frac {16\,\left (3\,a^8\,b\,e^2+8\,a^6\,b^3\,e^2+6\,a^4\,b^5\,e^2-b^9\,e^2\right )}{d^3}+\left (\frac {16\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (a^6\,e^2-15\,a^4\,b^2\,e^2+15\,a^2\,b^4\,e^2-b^6\,e^2\right )}{d^2}+\frac {8\,\left (4\,a^3\,d^2\,e^3-12\,a\,b^2\,d^2\,e^3\right )\,\sqrt {\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}}{d^3}\right )\,\sqrt {\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}-\left (\frac {16\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}\,\left (a^6\,e^2-15\,a^4\,b^2\,e^2+15\,a^2\,b^4\,e^2-b^6\,e^2\right )}{d^2}-\frac {8\,\left (4\,a^3\,d^2\,e^3-12\,a\,b^2\,d^2\,e^3\right )\,\sqrt {\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}}{d^3}\right )\,\sqrt {\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}}\right )\,\sqrt {\frac {\left (a^6+a^5\,b\,6{}\mathrm {i}-15\,a^4\,b^2-a^3\,b^3\,20{}\mathrm {i}+15\,a^2\,b^4+a\,b^5\,6{}\mathrm {i}-b^6\right )\,1{}\mathrm {i}}{4\,d^2\,e}}\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

atan((((16*(e*cot(c + d*x))^(1/2)*(a^6*e^2 - b^6*e^2 + 15*a^2*b^4*e^2 - 15*a^4*b^2*e^2))/d^2 - (8*(4*a^3*d^2*e
^3 - 12*a*b^2*d^2*e^3)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*
e))^(1/2))/d^3)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e))^(1/
2)*1i + ((16*(e*cot(c + d*x))^(1/2)*(a^6*e^2 - b^6*e^2 + 15*a^2*b^4*e^2 - 15*a^4*b^2*e^2))/d^2 + (8*(4*a^3*d^2
*e^3 - 12*a*b^2*d^2*e^3)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^
2*e))^(1/2))/d^3)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e))^(
1/2)*1i)/(((16*(e*cot(c + d*x))^(1/2)*(a^6*e^2 - b^6*e^2 + 15*a^2*b^4*e^2 - 15*a^4*b^2*e^2))/d^2 + (8*(4*a^3*d
^2*e^3 - 12*a*b^2*d^2*e^3)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*
d^2*e))^(1/2))/d^3)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e))
^(1/2) - ((16*(e*cot(c + d*x))^(1/2)*(a^6*e^2 - b^6*e^2 + 15*a^2*b^4*e^2 - 15*a^4*b^2*e^2))/d^2 - (8*(4*a^3*d^
2*e^3 - 12*a*b^2*d^2*e^3)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d
^2*e))^(1/2))/d^3)*(((a*b^5*6i + a^5*b*6i - a^6 + b^6 - 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e))^
(1/2) + (16*(3*a^8*b*e^2 - b^9*e^2 + 6*a^4*b^5*e^2 + 8*a^6*b^3*e^2))/d^3))*(((a*b^5*6i + a^5*b*6i - a^6 + b^6
- 15*a^2*b^4 - a^3*b^3*20i + 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2)*2i + atan((((16*(e*cot(c + d*x))^(1/2)*(a^6*e^2
- b^6*e^2 + 15*a^2*b^4*e^2 - 15*a^4*b^2*e^2))/d^2 - (8*(4*a^3*d^2*e^3 - 12*a*b^2*d^2*e^3)*(((a*b^5*6i + a^5*b*
6i + a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2))/d^3)*(((a*b^5*6i + a^5*b*6i + a^
6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2)*1i + ((16*(e*cot(c + d*x))^(1/2)*(a^6*e^
2 - b^6*e^2 + 15*a^2*b^4*e^2 - 15*a^4*b^2*e^2))/d^2 + (8*(4*a^3*d^2*e^3 - 12*a*b^2*d^2*e^3)*(((a*b^5*6i + a^5*
b*6i + a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2))/d^3)*(((a*b^5*6i + a^5*b*6i +
a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2)*1i)/(((16*(e*cot(c + d*x))^(1/2)*(a^6*
e^2 - b^6*e^2 + 15*a^2*b^4*e^2 - 15*a^4*b^2*e^2))/d^2 + (8*(4*a^3*d^2*e^3 - 12*a*b^2*d^2*e^3)*(((a*b^5*6i + a^
5*b*6i + a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2))/d^3)*(((a*b^5*6i + a^5*b*6i
+ a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2) - ((16*(e*cot(c + d*x))^(1/2)*(a^6*e
^2 - b^6*e^2 + 15*a^2*b^4*e^2 - 15*a^4*b^2*e^2))/d^2 - (8*(4*a^3*d^2*e^3 - 12*a*b^2*d^2*e^3)*(((a*b^5*6i + a^5
*b*6i + a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2))/d^3)*(((a*b^5*6i + a^5*b*6i +
a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i)/(4*d^2*e))^(1/2) + (16*(3*a^8*b*e^2 - b^9*e^2 + 6*a^4*
b^5*e^2 + 8*a^6*b^3*e^2))/d^3))*(((a*b^5*6i + a^5*b*6i + a^6 - b^6 + 15*a^2*b^4 - a^3*b^3*20i - 15*a^4*b^2)*1i
)/(4*d^2*e))^(1/2)*2i - (2*b^3*(e*cot(c + d*x))^(3/2))/(3*d*e^2) - (6*a*b^2*(e*cot(c + d*x))^(1/2))/(d*e)

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