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

Optimal. Leaf size=94 \[ -\frac{\sqrt{2} a e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d}-\frac{2 a e \sqrt{e \cot (c+d x)}}{d}-\frac{2 a (e \cot (c+d x))^{3/2}}{3 d} \]

[Out]

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

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Rubi [A]  time = 0.116296, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3528, 3532, 205} \[ -\frac{\sqrt{2} a e^{3/2} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d}-\frac{2 a e \sqrt{e \cot (c+d x)}}{d}-\frac{2 a (e \cot (c+d x))^{3/2}}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 3528

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

Mathematica [C]  time = 0.108132, size = 67, normalized size = 0.71 \[ -\frac{2 a e \sqrt{e \cot (c+d x)} \left (3 \text{Hypergeometric2F1}\left (-\frac{1}{4},1,\frac{3}{4},-\tan ^2(c+d x)\right )+\cot (c+d x) \text{Hypergeometric2F1}\left (-\frac{3}{4},1,\frac{1}{4},-\tan ^2(c+d x)\right )\right )}{3 d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*a*e*Sqrt[e*Cot[c + d*x]]*(Cot[c + d*x]*Hypergeometric2F1[-3/4, 1, 1/4, -Tan[c + d*x]^2] + 3*Hypergeometric
2F1[-1/4, 1, 3/4, -Tan[c + d*x]^2]))/(3*d)

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Maple [B]  time = 0.017, size = 363, normalized size = 3.9 \begin{align*} -{\frac{2\,a}{3\,d} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{3}{2}}}}-2\,{\frac{ae\sqrt{e\cot \left ( dx+c \right ) }}{d}}+{\frac{ae\sqrt{2}}{4\,d}\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{ae\sqrt{2}}{2\,d}\sqrt [4]{{e}^{2}}\arctan \left ({\sqrt{2}\sqrt{e\cot \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{e}^{2}}}}}+1 \right ) }-{\frac{ae\sqrt{2}}{2\,d}\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{e}^{2}\sqrt{2}}{4\,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{e}^{2}\sqrt{2}}{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{e}^{2}\sqrt{2}}{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((e*cot(d*x+c))^(3/2)*(a+a*cot(d*x+c)),x)

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.02722, size = 855, normalized size = 9.1 \begin{align*} \left [\frac{3 \, \sqrt{2} a \sqrt{-e} 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 ) \sin \left (2 \, d x + 2 \, c\right ) - 4 \,{\left (a e \cos \left (2 \, d x + 2 \, c\right ) + 3 \, a e \sin \left (2 \, d x + 2 \, c\right ) + a e\right )} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{6 \, d \sin \left (2 \, d x + 2 \, c\right )}, -\frac{3 \, \sqrt{2} a e^{\frac{3}{2}} \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 ) \sin \left (2 \, d x + 2 \, c\right ) + 2 \,{\left (a e \cos \left (2 \, d x + 2 \, c\right ) + 3 \, a e \sin \left (2 \, d x + 2 \, c\right ) + a e\right )} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{3 \, d \sin \left (2 \, d x + 2 \, c\right )}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/6*(3*sqrt(2)*a*sqrt(-e)*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)*sin(2*d*x + 2*c) - 4*(a*e*cos(2*d*x + 2*c) + 3*a*e*s
in(2*d*x + 2*c) + a*e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*sin(2*d*x + 2*c)), -1/3*(3*sqrt(2)*
a*e^(3/2)*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))*sin(2*d*x + 2*c) + 2*(a*e*cos(2*d*x + 2*c) + 3*a*e*sin(2*d*x + 2*c
) + a*e)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*sin(2*d*x + 2*c))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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