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

Optimal. Leaf size=138 \[ -\frac{8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac{4 a^3 \sqrt{e \cot (c+d x)}}{d}-\frac{2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{3/2}}{5 d e}-\frac{2 \sqrt{2} a^3 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d} \]

[Out]

(-2*Sqrt[2]*a^3*Sqrt[e]*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d - (4*a^3*Sq
rt[e*Cot[c + d*x]])/d - (8*a^3*(e*Cot[c + d*x])^(3/2))/(5*d*e) - (2*(e*Cot[c + d*x])^(3/2)*(a^3 + a^3*Cot[c +
d*x]))/(5*d*e)

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Rubi [A]  time = 0.203041, antiderivative size = 138, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {3566, 3630, 3528, 3532, 205} \[ -\frac{8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac{4 a^3 \sqrt{e \cot (c+d x)}}{d}-\frac{2 \left (a^3 \cot (c+d x)+a^3\right ) (e \cot (c+d x))^{3/2}}{5 d e}-\frac{2 \sqrt{2} a^3 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*Sqrt[2]*a^3*Sqrt[e]*ArcTan[(Sqrt[e] - Sqrt[e]*Cot[c + d*x])/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/d - (4*a^3*Sq
rt[e*Cot[c + d*x]])/d - (8*a^3*(e*Cot[c + d*x])^(3/2))/(5*d*e) - (2*(e*Cot[c + d*x])^(3/2)*(a^3 + a^3*Cot[c +
d*x]))/(5*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 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 \sqrt{e \cot (c+d x)} (a+a \cot (c+d x))^3 \, dx &=-\frac{2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}-\frac{2 \int \sqrt{e \cot (c+d x)} \left (-a^3 e-5 a^3 e \cot (c+d x)-6 a^3 e \cot ^2(c+d x)\right ) \, dx}{5 e}\\ &=-\frac{8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac{2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}-\frac{2 \int \sqrt{e \cot (c+d x)} \left (5 a^3 e-5 a^3 e \cot (c+d x)\right ) \, dx}{5 e}\\ &=-\frac{4 a^3 \sqrt{e \cot (c+d x)}}{d}-\frac{8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac{2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}-\frac{2 \int \frac{5 a^3 e^2+5 a^3 e^2 \cot (c+d x)}{\sqrt{e \cot (c+d x)}} \, dx}{5 e}\\ &=-\frac{4 a^3 \sqrt{e \cot (c+d x)}}{d}-\frac{8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac{2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}+\frac{\left (20 a^6 e^3\right ) \operatorname{Subst}\left (\int \frac{1}{-50 a^6 e^4-e x^2} \, dx,x,\frac{5 a^3 e^2-5 a^3 e^2 \cot (c+d x)}{\sqrt{e \cot (c+d x)}}\right )}{d}\\ &=-\frac{2 \sqrt{2} a^3 \sqrt{e} \tan ^{-1}\left (\frac{\sqrt{e}-\sqrt{e} \cot (c+d x)}{\sqrt{2} \sqrt{e \cot (c+d x)}}\right )}{d}-\frac{4 a^3 \sqrt{e \cot (c+d x)}}{d}-\frac{8 a^3 (e \cot (c+d x))^{3/2}}{5 d e}-\frac{2 (e \cot (c+d x))^{3/2} \left (a^3+a^3 \cot (c+d x)\right )}{5 d e}\\ \end{align*}

Mathematica [C]  time = 1.54702, size = 315, normalized size = 2.28 \[ -\frac{a^3 \sin (c+d x) (\cot (c+d x)+1)^3 \sqrt{e \cot (c+d x)} \left (3 \left (4 \cos ^2(c+d x) \sqrt{\cot (c+d x)}+40 \sin ^2(c+d x) \sqrt{\cot (c+d x)}+10 \sin (2 (c+d x)) \sqrt{\cot (c+d x)}+5 \sqrt{2} \sin ^2(c+d x) \log \left (\cot (c+d x)-\sqrt{2} \sqrt{\cot (c+d x)}+1\right )-5 \sqrt{2} \sin ^2(c+d x) \log \left (\cot (c+d x)+\sqrt{2} \sqrt{\cot (c+d x)}+1\right )+10 \sqrt{2} \sin ^2(c+d x) \tan ^{-1}\left (1-\sqrt{2} \sqrt{\cot (c+d x)}\right )-10 \sqrt{2} \sin ^2(c+d x) \tan ^{-1}\left (\sqrt{2} \sqrt{\cot (c+d x)}+1\right )\right )-20 \sin (2 (c+d x)) \sqrt{\cot (c+d x)} \text{Hypergeometric2F1}\left (\frac{3}{4},1,\frac{7}{4},-\cot ^2(c+d x)\right )\right )}{30 d \sqrt{\cot (c+d x)} (\sin (c+d x)+\cos (c+d x))^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(a^3*Sqrt[e*Cot[c + d*x]]*(1 + Cot[c + d*x])^3*Sin[c + d*x]*(-20*Sqrt[Cot[c + d*x]]*Hypergeometric2F1[3/4, 1,
7/4, -Cot[c + d*x]^2]*Sin[2*(c + d*x)] + 3*(4*Cos[c + d*x]^2*Sqrt[Cot[c + d*x]] + 10*Sqrt[2]*ArcTan[1 - Sqrt[
2]*Sqrt[Cot[c + d*x]]]*Sin[c + d*x]^2 - 10*Sqrt[2]*ArcTan[1 + Sqrt[2]*Sqrt[Cot[c + d*x]]]*Sin[c + d*x]^2 + 40*
Sqrt[Cot[c + d*x]]*Sin[c + d*x]^2 + 5*Sqrt[2]*Log[1 - Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^
2 - 5*Sqrt[2]*Log[1 + Sqrt[2]*Sqrt[Cot[c + d*x]] + Cot[c + d*x]]*Sin[c + d*x]^2 + 10*Sqrt[Cot[c + d*x]]*Sin[2*
(c + d*x)])))/(30*d*Sqrt[Cot[c + d*x]]*(Cos[c + d*x] + Sin[c + d*x])^3)

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Maple [B]  time = 0.026, size = 391, normalized size = 2.8 \begin{align*} -{\frac{2\,{a}^{3}}{5\,d{e}^{2}} \left ( e\cot \left ( dx+c \right ) \right ) ^{{\frac{5}{2}}}}-2\,{\frac{{a}^{3} \left ( e\cot \left ( dx+c \right ) \right ) ^{3/2}}{de}}-4\,{\frac{{a}^{3}\sqrt{e\cot \left ( dx+c \right ) }}{d}}+{\frac{{a}^{3}\sqrt{2}}{2\,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{{a}^{3}\sqrt{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}^{3}\sqrt{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}^{3}e\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}e\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}e\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((e*cot(d*x+c))^(1/2)*(a+a*cot(d*x+c))^3,x)

[Out]

-2/5/d*a^3/e^2*(e*cot(d*x+c))^(5/2)-2*a^3*(e*cot(d*x+c))^(3/2)/d/e-4*a^3*(e*cot(d*x+c))^(1/2)/d+1/2/d*a^3*(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)^(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)+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)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.67005, size = 900, normalized size = 6.52 \begin{align*} \left [\frac{5 \, \sqrt{2}{\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) - a^{3}\right )} \sqrt{-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 ) - 2 \,{\left (9 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 5 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) - 11 \, a^{3}\right )} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}}{5 \,{\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )}}, -\frac{2 \,{\left (5 \, \sqrt{2}{\left (a^{3} \cos \left (2 \, d x + 2 \, c\right ) - a^{3}\right )} \sqrt{e} \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 ) +{\left (9 \, a^{3} \cos \left (2 \, d x + 2 \, c\right ) - 5 \, a^{3} \sin \left (2 \, d x + 2 \, c\right ) - 11 \, a^{3}\right )} \sqrt{\frac{e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}}\right )}}{5 \,{\left (d \cos \left (2 \, d x + 2 \, c\right ) - d\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/5*(5*sqrt(2)*(a^3*cos(2*d*x + 2*c) - a^3)*sqrt(-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) - 2*(9*a^3*cos(2*d*x + 2*c)
- 5*a^3*sin(2*d*x + 2*c) - 11*a^3)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*cos(2*d*x + 2*c) - d),
-2/5*(5*sqrt(2)*(a^3*cos(2*d*x + 2*c) - a^3)*sqrt(e)*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)) + (9*a^3*cos(2*d*x + 2
*c) - 5*a^3*sin(2*d*x + 2*c) - 11*a^3)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c)))/(d*cos(2*d*x + 2*c) -
d)]

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

[Out]

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

[Out]

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