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]

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

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Rubi [A]  time = 0.12, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, 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 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]

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]

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*}

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Mathematica [C]  time = 0.11, size = 67, normalized size = 0.71 \[ -\frac {2 a e \sqrt {e \cot (c+d x)} \left (3 \, _2F_1\left (-\frac {1}{4},1;\frac {3}{4};-\tan ^2(c+d x)\right )+\cot (c+d x) \, _2F_1\left (-\frac {3}{4},1;\frac {1}{4};-\tan ^2(c+d x)\right )\right )}{3 d} \]

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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fricas [B]  time = 0.75, size = 334, normalized size = 3.55 \[ \left [\frac {3 \, \sqrt {2} a \sqrt {-e} e \log \left (\sqrt {2} \sqrt {-e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, e \sin \left (2 \, d x + 2 \, c\right ) + e\right ) \sin \left (2 \, d x + 2 \, c\right ) - 4 \, {\left (a e \cos \left (2 \, d x + 2 \, c\right ) + 3 \, a e \sin \left (2 \, d x + 2 \, c\right ) + a e\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{6 \, d \sin \left (2 \, d x + 2 \, c\right )}, -\frac {3 \, \sqrt {2} a e^{\frac {3}{2}} \arctan \left (-\frac {\sqrt {2} \sqrt {e} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, {\left (e \cos \left (2 \, d x + 2 \, c\right ) + e\right )}}\right ) \sin \left (2 \, d x + 2 \, c\right ) + 2 \, {\left (a e \cos \left (2 \, d x + 2 \, c\right ) + 3 \, a e \sin \left (2 \, d x + 2 \, c\right ) + a e\right )} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{3 \, d \sin \left (2 \, d x + 2 \, c\right )}\right ] \]

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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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (a \cot \left (d x + c\right ) + a\right )} \left (e \cot \left (d x + c\right )\right )^{\frac {3}{2}}\,{d x} \]

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)

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maple [B]  time = 0.43, size = 363, normalized size = 3.86 \[ -\frac {2 a \left (e \cot \left (d x +c \right )\right )^{\frac {3}{2}}}{3 d}-\frac {2 a e \sqrt {e \cot \left (d x +c \right )}}{d}+\frac {a e \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{4 d}+\frac {a e \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d}-\frac {a e \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d}+\frac {a \,e^{2} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{4 d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {a \,e^{2} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {a \,e^{2} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d \left (e^{2}\right )^{\frac {1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.62, size = 124, normalized size = 1.32 \[ \frac {{\left (3 \, a e {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}}\right )} - \frac {2 \, {\left (3 \, a e \sqrt {\frac {e}{\tan \left (d x + c\right )}} + a \left (\frac {e}{\tan \left (d x + c\right )}\right )^{\frac {3}{2}}\right )}}{e}\right )} e}{3 \, d} \]

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]

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

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mupad [B]  time = 1.16, size = 98, normalized size = 1.04 \[ -\frac {2\,a\,{\left (e\,\mathrm {cot}\left (c+d\,x\right )\right )}^{3/2}}{3\,d}-\frac {2\,a\,e\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{d}+\frac {{\left (-1\right )}^{1/4}\,a\,e^{3/2}\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (1-\mathrm {i}\right )}{d}+\frac {{\left (-1\right )}^{1/4}\,a\,e^{3/2}\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (-1-\mathrm {i}\right )}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}}\, dx + \int \left (e \cot {\left (c + d x \right )}\right )^{\frac {3}{2}} \cot {\left (c + d x \right )}\, dx\right ) \]

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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