3.18 \(\int \frac{(a+a \cot (c+d x))^3}{\sqrt{e \cot (c+d x)}} \, dx\)

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

[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)

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Rubi [A]  time = 0.1691, antiderivative size = 117, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.16, Rules used = {3566, 3630, 3532, 208} \[ -\frac{16 a^3 \sqrt{e \cot (c+d x)}}{3 d e}-\frac{2 \left (a^3 \cot (c+d x)+a^3\right ) \sqrt{e \cot (c+d x)}}{3 d e}+\frac{2 \sqrt{2} a^3 \tanh ^{-1}\left (\frac{\sqrt{e} \cot (c+d x)+\sqrt{e}}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d \sqrt{e}} \]

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 3566

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 3630

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]

Rule 3532

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 208

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

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 ) \operatorname{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*}

Mathematica [C]  time = 5.03914, size = 292, normalized size = 2.5 \[ -\frac{a^3 \sin (c+d x) (\cot (c+d x)+1)^3 \left (8 \cos ^2(c+d x) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )+18 \sin (2 (c+d x))+4 \cos ^2(c+d x)+3 \sqrt{2} \sin ^2(c+d x) \sqrt{\cot (c+d x)} \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-3 \sqrt{2} \sin ^2(c+d x) \sqrt{\cot (c+d x)} \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+6 \sqrt{2} \sin ^2(c+d x) \sqrt{\cot (c+d x)} \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-6 \sqrt{2} \sin ^2(c+d x) \sqrt{\cot (c+d x)} \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )}{6 d \sqrt{e \cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(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, -C
ot[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 + 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]*S
qrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 + 18*Sin[2*(c + d*x)]))/(6*d*Sqrt[e*Cot[c + d*x]]*(Cos[c + d*
x] + Sin[c + d*x])^3)

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Maple [B]  time = 0.026, size = 379, normalized size = 3.2 \begin{align*} -{\frac{2\,{a}^{3}}{3\,d{e}^{2}} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}-6\,{\frac{{a}^{3}\sqrt{e\cot \left ( dx+c \right ) }}{de}}+{\frac{{a}^{3}\sqrt{2}}{2\,de}\sqrt [4]{{e}^{2}}\ln \left ({ \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ) }+{\frac{{a}^{3}\sqrt{2}}{de}\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{{a}^{3}\sqrt{2}}{de}\sqrt [4]{{e}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{{a}^{3}\sqrt{2}}{2\,d}\ln \left ({ \left ( e\cot \left ( dx+c \right ) -\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) \left ( e\cot \left ( dx+c \right ) +\sqrt [4]{{e}^{2}}\sqrt{e\cot \left ( dx+c \right ) }\sqrt{2}+\sqrt{{e}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}-{\frac{{a}^{3}\sqrt{2}}{d}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}}+{\frac{{a}^{3}\sqrt{2}}{d}\arctan \left ( -{\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{e}^{2}}}}} \end{align*}

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)

[Out]

-2/3/d*a^3/e^2*(e*cot(d*x+c))^(3/2)-6*a^3*(e*cot(d*x+c))^(1/2)/d/e+1/2/d*a^3/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)))+1/d*a^3/e*(e^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-1/d*a^3/e*
(e^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)-1/2/d*a^3/(e^2)^(1/4)*2^(1/2)*ln((e*co
t(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)))-1/d*a^3/(e^2)^(1/4)*2^(1/2)*arctan(2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)+1/d*a^3/(
e^2)^(1/4)*2^(1/2)*arctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [A]  time = 1.68323, size = 863, normalized size = 7.38 \begin{align*} \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 ] \end{align*}

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*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) - s
in(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*sin(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))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} 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{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a \cot \left (d x + c\right ) + a\right )}^{3}}{\sqrt{e \cot \left (d x + c\right )}}\,{d x} \end{align*}

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)