3.19 \(\int \frac{(a+a \cot (c+d x))^3}{(e \cot (c+d x))^{3/2}} \, dx\)

Optimal. Leaf size=114 \[ -\frac{4 a^3 \sqrt{e \cot (c+d x)}}{d e^2}+\frac{2 \sqrt{2} a^3 \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d e^{3/2}}+\frac{2 \left (a^3 \cot (c+d x)+a^3\right )}{d e \sqrt{e \cot (c+d x)}} \]

[Out]

(2*Sqrt[2]*a^3*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*e^(3/2)) - (4*a^3*S
qrt[e*Cot[c + d*x]])/(d*e^2) + (2*(a^3 + a^3*Cot[c + d*x]))/(d*e*Sqrt[e*Cot[c + d*x]])

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Rubi [A]  time = 0.175359, antiderivative size = 114, 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 = {3565, 3630, 3532, 205} \[ -\frac{4 a^3 \sqrt{e \cot (c+d x)}}{d e^2}+\frac{2 \sqrt{2} a^3 \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d e^{3/2}}+\frac{2 \left (a^3 \cot (c+d x)+a^3\right )}{d e \sqrt{e \cot (c+d x)}} \]

Antiderivative was successfully verified.

[In]

Int[(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(3/2),x]

[Out]

(2*Sqrt[2]*a^3*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*e^(3/2)) - (4*a^3*S
qrt[e*Cot[c + d*x]])/(d*e^2) + (2*(a^3 + a^3*Cot[c + d*x]))/(d*e*Sqrt[e*Cot[c + d*x]])

Rule 3565

Int[((a_.) + (b_.)*tan[(e_.) + (f_.)*(x_)])^(m_)*((c_.) + (d_.)*tan[(e_.) + (f_.)*(x_)])^(n_), x_Symbol] :> Si
mp[((b*c - a*d)^2*(a + b*Tan[e + f*x])^(m - 2)*(c + d*Tan[e + f*x])^(n + 1))/(d*f*(n + 1)*(c^2 + d^2)), x] - D
ist[1/(d*(n + 1)*(c^2 + d^2)), Int[(a + b*Tan[e + f*x])^(m - 3)*(c + d*Tan[e + f*x])^(n + 1)*Simp[a^2*d*(b*d*(
m - 2) - a*c*(n + 1)) + b*(b*c - 2*a*d)*(b*c*(m - 2) + a*d*(n + 1)) - d*(n + 1)*(3*a^2*b*c - b^3*c - a^3*d + 3
*a*b^2*d)*Tan[e + f*x] - b*(a*d*(2*b*c - a*d)*(m + n - 1) - b^2*(c^2*(m - 2) - d^2*(n + 1)))*Tan[e + f*x]^2, x
], x], x] /; FreeQ[{a, b, c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 + b^2, 0] && NeQ[c^2 + d^2, 0] && Gt
Q[m, 2] && LtQ[n, -1] && IntegerQ[2*m]

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 205

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

Rubi steps

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

Mathematica [C]  time = 2.91324, size = 311, normalized size = 2.73 \[ \frac{a^3 (\cot (c+d x)+1)^3 \left (\sin (c+d x) \left (2 \sin (2 (c+d x)) \text{Hypergeometric2F1}\left (-\frac{1}{4},1,\frac{3}{4},-\cot ^2(c+d x)\right )-4 \cos ^2(c+d x)+\sqrt{2} \sin ^2(c+d x) \cot ^{\frac{3}{2}}(c+d x) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-\sqrt{2} \sin ^2(c+d x) \cot ^{\frac{3}{2}}(c+d x) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+2 \sqrt{2} \sin ^2(c+d x) \cot ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-2 \sqrt{2} \sin ^2(c+d x) \cot ^{\frac{3}{2}}(c+d x) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )-4 \cos ^3(c+d x) \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )\right )}{2 d (e \cot (c+d x))^{3/2} (\sin (c+d x)+\cos (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + a*Cot[c + d*x])^3/(e*Cot[c + d*x])^(3/2),x]

[Out]

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

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Maple [B]  time = 0.023, size = 388, normalized size = 3.4 \begin{align*} -2\,{\frac{{a}^{3}\sqrt{e\cot \left ( dx+c \right ) }}{d{e}^{2}}}-{\frac{{a}^{3}\sqrt{2}}{2\,d{e}^{2}}\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}}{d{e}^{2}}\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}}{d{e}^{2}}\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\,de}\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}}{de}\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}}{de}\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}}}}}+2\,{\frac{{a}^{3}}{de\sqrt{e\cot \left ( dx+c \right ) }}} \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))^(3/2),x)

[Out]

-2*a^3*(e*cot(d*x+c))^(1/2)/d/e^2-1/2/d*a^3/e^2*(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^2*
(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*(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/(e^2)^(1/4)*2^(1/2)*ln((e*cot(d*x+c)-(e^2)^(1/4)*(e*c
ot(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)*ar
ctan(-2^(1/2)/(e^2)^(1/4)*(e*cot(d*x+c))^(1/2)+1)+2/d*a^3/e/(e*cot(d*x+c))^(1/2)

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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))^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.68629, size = 903, normalized size = 7.92 \begin{align*} \left [\frac{\sqrt{2}{\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) + a^{3} e\right )} \sqrt{-\frac{1}{e}} \log \left (-\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 \, \sin \left (2 \, d x + 2 \, c\right ) + 1\right ) - 2 \,{\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 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 )}}}{d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + d e^{2}}, \frac{2 \,{\left (\frac{\sqrt{2}{\left (a^{3} e \cos \left (2 \, d x + 2 \, c\right ) + a^{3} e\right )} \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 )}}{\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, \sqrt{e}{\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )}}\right )}{\sqrt{e}} -{\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 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 )}}{d e^{2} \cos \left (2 \, d x + 2 \, c\right ) + d e^{2}}\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))^(3/2),x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} a^{3} \left (\int \frac{1}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{3 \cot{\left (c + d x \right )}}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{3 \cot ^{2}{\left (c + d x \right )}}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, dx + \int \frac{\cot ^{3}{\left (c + d x \right )}}{\left (e \cot{\left (c + d x \right )}\right )^{\frac{3}{2}}}\, 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))**(3/2),x)

[Out]

a**3*(Integral((e*cot(c + d*x))**(-3/2), x) + Integral(3*cot(c + d*x)/(e*cot(c + d*x))**(3/2), x) + Integral(3
*cot(c + d*x)**2/(e*cot(c + d*x))**(3/2), x) + Integral(cot(c + d*x)**3/(e*cot(c + d*x))**(3/2), 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}}{\left (e \cot \left (d x + c\right )\right )^{\frac{3}{2}}}\,{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))^(3/2),x, algorithm="giac")

[Out]

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