3.1.18 \(\int \frac {(a+a \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx\) [18]

Optimal. Leaf size=117 \[ \frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {e}+\sqrt {e} \cot (c+d x)}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d \sqrt {e}}-\frac {16 a^3 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 \sqrt {e \cot (c+d x)} \left (a^3+a^3 \cot (c+d x)\right )}{3 d e} \]

[Out]

2*a^3*arctanh(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)/d/e^(1/2)-16/3*a^3*(e*cot
(d*x+c))^(1/2)/d/e-2/3*(a^3+a^3*cot(d*x+c))*(e*cot(d*x+c))^(1/2)/d/e

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Rubi [A]
time = 0.11, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3647, 3711, 3613, 214} \begin {gather*} -\frac {16 a^3 \sqrt {e \cot (c+d x)}}{3 d e}-\frac {2 \left (a^3 \cot (c+d x)+a^3\right ) \sqrt {e \cot (c+d x)}}{3 d e}+\frac {2 \sqrt {2} a^3 \tanh ^{-1}\left (\frac {\sqrt {e} \cot (c+d x)+\sqrt {e}}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d \sqrt {e}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[2]*a^3*ArcTanh[(Sqrt[e] + Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*Sqrt[e]) - (16*a^3
*Sqrt[e*Cot[c + d*x]])/(3*d*e) - (2*Sqrt[e*Cot[c + d*x]]*(a^3 + a^3*Cot[c + d*x]))/(3*d*e)

Rule 214

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]

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*} \int \frac {(a+a \cot (c+d x))^3}{\sqrt {e \cot (c+d x)}} \, dx &=-\frac {2 \sqrt {e \cot (c+d x)} \left (a^3+a^3 \cot (c+d x)\right )}{3 d e}-\frac {2 \int \frac {-a^3 e-3 a^3 e \cot (c+d x)-4 a^3 e \cot ^2(c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{3 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}-\frac {2 \int \frac {3 a^3 e-3 a^3 e \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx}{3 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}+\frac {\left (12 a^6 e\right ) \text {Subst}\left (\int \frac {1}{18 a^6 e^2-e x^2} \, dx,x,\frac {3 a^3 e+3 a^3 e \cot (c+d x)}{\sqrt {e \cot (c+d x)}}\right )}{d}\\ &=\frac {2 \sqrt {2} a^3 \tanh ^{-1}\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}\\ \end {align*}

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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 3 in optimal.
time = 5.46, size = 292, normalized size = 2.50 \begin {gather*} -\frac {a^3 (1+\cot (c+d x))^3 \sin (c+d x) \left (4 \cos ^2(c+d x)+8 \cos ^2(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\cot ^2(c+d x)\right )+6 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\cot (c+d x)}\right ) \sqrt {\cot (c+d x)} \sin ^2(c+d x)-6 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\cot (c+d x)}\right ) \sqrt {\cot (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} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/6*(a^3*(1 + Cot[c + d*x])^3*Sin[c + d*x]*(4*Cos[c + d*x]^2 + 8*Cos[c + d*x]^2*Hypergeometric2F1[3/4, 1, 7/4
, -Cot[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*Sq
rt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sqrt[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 - 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)

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(308\) vs. \(2(98)=196\).
time = 0.45, size = 309, normalized size = 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	\(-\frac {2 a^{3} \left (\frac {\left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3}+3 e \sqrt {e \cot \left (d x +c \right )}-2 e^{2} \left (\frac {\left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 e}-\frac {\sqrt {2}\, \left (\ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (e^{2}\right )^{\frac {1}{4}}}\right )\right )}{d \,e^{2}}\)	\(309\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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*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))))

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Maxima [A]
time = 0.51, size = 96, normalized size = 0.82 \begin {gather*} \frac {{\left (3 \, {\left (\sqrt {2} \log \left (\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right ) - \sqrt {2} \log \left (-\frac {\sqrt {2}}{\sqrt {\tan \left (d x + c\right )}} + \frac {1}{\tan \left (d x + c\right )} + 1\right )\right )} a^{3} - \frac {18 \, a^{3}}{\sqrt {\tan \left (d x + c\right )}} - \frac {2 \, a^{3}}{\tan \left (d x + c\right )^{\frac {3}{2}}}\right )} e^{\left (-\frac {1}{2}\right )}}{3 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*(sqrt(2)*log(sqrt(2)/sqrt(tan(d*x + c)) + 1/tan(d*x + c) + 1) - sqrt(2)*log(-sqrt(2)/sqrt(tan(d*x + c))
+ 1/tan(d*x + c) + 1))*a^3 - 18*a^3/sqrt(tan(d*x + c)) - 2*a^3/tan(d*x + c)^(3/2))*e^(-1/2)/d

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Fricas [A]
time = 2.58, size = 168, normalized size = 1.44 \begin {gather*} \frac {{\left (3 \, \sqrt {2} a^{3} \log \left (-{\left (\sqrt {2} \cos \left (2 \, d x + 2 \, c\right ) - \sqrt {2} \sin \left (2 \, d x + 2 \, c\right ) - \sqrt {2}\right )} \sqrt {\frac {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}} + 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 {\cos \left (2 \, d x + 2 \, c\right ) + 1}{\sin \left (2 \, d x + 2 \, c\right )}}\right )} e^{\left (-\frac {1}{2}\right )}}{3 \, d \sin \left (2 \, d x + 2 \, c\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/3*(3*sqrt(2)*a^3*log(-(sqrt(2)*cos(2*d*x + 2*c) - sqrt(2)*sin(2*d*x + 2*c) - sqrt(2))*sqrt((cos(2*d*x + 2*c)
+ 1)/sin(2*d*x + 2*c)) + 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((cos(2*d*x + 2*c) + 1)/sin(2*d*x + 2*c)))*e^(-1/2)/(d*sin(2*d*x + 2*c))

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} 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 ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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))

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

[Out]

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

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Mupad [B]
time = 0.62, size = 100, normalized size = 0.85 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(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)

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