3.93 \(\int \cot ^2(e+f x) \sqrt{a+a \sin (e+f x)} \, dx\)

Optimal. Leaf size=89 \[ \frac{3 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}-\frac{\cot (e+f x) \sqrt{a \sin (e+f x)+a}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right )}{f} \]

[Out]

-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f) + (3*a*Cos[e + f*x])/(f*Sqrt[a + a*Sin
[e + f*x]]) - (Cot[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/f

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Rubi [A]  time = 0.192413, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {2716, 2981, 2773, 206} \[ \frac{3 a \cos (e+f x)}{f \sqrt{a \sin (e+f x)+a}}-\frac{\cot (e+f x) \sqrt{a \sin (e+f x)+a}}{f}-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a \sin (e+f x)+a}}\right )}{f} \]

Antiderivative was successfully verified.

[In]

Int[Cot[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]],x]

[Out]

-((Sqrt[a]*ArcTanh[(Sqrt[a]*Cos[e + f*x])/Sqrt[a + a*Sin[e + f*x]]])/f) + (3*a*Cos[e + f*x])/(f*Sqrt[a + a*Sin
[e + f*x]]) - (Cot[e + f*x]*Sqrt[a + a*Sin[e + f*x]])/f

Rule 2716

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)/tan[(e_.) + (f_.)*(x_)]^2, x_Symbol] :> -Simp[(a + b*Sin[e +
f*x])^m/(f*Tan[e + f*x]), x] + Dist[1/a, Int[((a + b*Sin[e + f*x])^m*(b*m - a*(m + 1)*Sin[e + f*x]))/Sin[e + f
*x], x], x] /; FreeQ[{a, b, e, f, m}, x] && EqQ[a^2 - b^2, 0] && IntegerQ[m - 1/2] &&  !LtQ[m, -1]

Rule 2981

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]*((A_.) + (B_.)*sin[(e_.) + (f_.)*(x_)])*((c_.) + (d_.)*sin[(e_.
) + (f_.)*(x_)])^(n_), x_Symbol] :> Simp[(-2*b*B*Cos[e + f*x]*(c + d*Sin[e + f*x])^(n + 1))/(d*f*(2*n + 3)*Sqr
t[a + b*Sin[e + f*x]]), x] + Dist[(A*b*d*(2*n + 3) - B*(b*c - 2*a*d*(n + 1)))/(b*d*(2*n + 3)), Int[Sqrt[a + b*
Sin[e + f*x]]*(c + d*Sin[e + f*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, A, B, n}, x] && NeQ[b*c - a*d, 0] &&
EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0] &&  !LtQ[n, -1]

Rule 2773

Int[Sqrt[(a_) + (b_.)*sin[(e_.) + (f_.)*(x_)]]/((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Dist[(-2*
b)/f, Subst[Int[1/(b*c + a*d - d*x^2), x], x, (b*Cos[e + f*x])/Sqrt[a + b*Sin[e + f*x]]], x] /; FreeQ[{a, b, c
, d, e, f}, x] && NeQ[b*c - a*d, 0] && EqQ[a^2 - b^2, 0] && NeQ[c^2 - d^2, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \cot ^2(e+f x) \sqrt{a+a \sin (e+f x)} \, dx &=-\frac{\cot (e+f x) \sqrt{a+a \sin (e+f x)}}{f}+\frac{\int \csc (e+f x) \left (\frac{a}{2}-\frac{3}{2} a \sin (e+f x)\right ) \sqrt{a+a \sin (e+f x)} \, dx}{a}\\ &=\frac{3 a \cos (e+f x)}{f \sqrt{a+a \sin (e+f x)}}-\frac{\cot (e+f x) \sqrt{a+a \sin (e+f x)}}{f}+\frac{1}{2} \int \csc (e+f x) \sqrt{a+a \sin (e+f x)} \, dx\\ &=\frac{3 a \cos (e+f x)}{f \sqrt{a+a \sin (e+f x)}}-\frac{\cot (e+f x) \sqrt{a+a \sin (e+f x)}}{f}-\frac{a \operatorname{Subst}\left (\int \frac{1}{a-x^2} \, dx,x,\frac{a \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{f}\\ &=-\frac{\sqrt{a} \tanh ^{-1}\left (\frac{\sqrt{a} \cos (e+f x)}{\sqrt{a+a \sin (e+f x)}}\right )}{f}+\frac{3 a \cos (e+f x)}{f \sqrt{a+a \sin (e+f x)}}-\frac{\cot (e+f x) \sqrt{a+a \sin (e+f x)}}{f}\\ \end{align*}

Mathematica [B]  time = 0.954969, size = 206, normalized size = 2.31 \[ \frac{\csc ^4\left (\frac{1}{2} (e+f x)\right ) \sqrt{a (\sin (e+f x)+1)} \left (4 \sin \left (\frac{1}{2} (e+f x)\right )+2 \sin \left (\frac{3}{2} (e+f x)\right )-4 \cos \left (\frac{1}{2} (e+f x)\right )+2 \cos \left (\frac{3}{2} (e+f x)\right )-\sin (e+f x) \log \left (-\sin \left (\frac{1}{2} (e+f x)\right )+\cos \left (\frac{1}{2} (e+f x)\right )+1\right )+\sin (e+f x) \log \left (\sin \left (\frac{1}{2} (e+f x)\right )-\cos \left (\frac{1}{2} (e+f x)\right )+1\right )\right )}{f \left (\cot \left (\frac{1}{2} (e+f x)\right )+1\right ) \left (\csc \left (\frac{1}{4} (e+f x)\right )-\sec \left (\frac{1}{4} (e+f x)\right )\right ) \left (\csc \left (\frac{1}{4} (e+f x)\right )+\sec \left (\frac{1}{4} (e+f x)\right )\right )} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[e + f*x]^2*Sqrt[a + a*Sin[e + f*x]],x]

[Out]

(Csc[(e + f*x)/2]^4*Sqrt[a*(1 + Sin[e + f*x])]*(-4*Cos[(e + f*x)/2] + 2*Cos[(3*(e + f*x))/2] + 4*Sin[(e + f*x)
/2] - Log[1 + Cos[(e + f*x)/2] - Sin[(e + f*x)/2]]*Sin[e + f*x] + Log[1 - Cos[(e + f*x)/2] + Sin[(e + f*x)/2]]
*Sin[e + f*x] + 2*Sin[(3*(e + f*x))/2]))/(f*(1 + Cot[(e + f*x)/2])*(Csc[(e + f*x)/4] - Sec[(e + f*x)/4])*(Csc[
(e + f*x)/4] + Sec[(e + f*x)/4]))

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Maple [A]  time = 0.55, size = 125, normalized size = 1.4 \begin{align*}{\frac{1+\sin \left ( fx+e \right ) }{\sin \left ( fx+e \right ) \cos \left ( fx+e \right ) f}\sqrt{-a \left ( -1+\sin \left ( fx+e \right ) \right ) } \left ( \sin \left ( fx+e \right ) \left ( 2\,\sqrt{a-a\sin \left ( fx+e \right ) }{a}^{3/2}-{\it Artanh} \left ({\sqrt{a-a\sin \left ( fx+e \right ) }{\frac{1}{\sqrt{a}}}} \right ){a}^{2} \right ) -\sqrt{a-a\sin \left ( fx+e \right ) }{a}^{{\frac{3}{2}}} \right ){a}^{-{\frac{3}{2}}}{\frac{1}{\sqrt{a+a\sin \left ( fx+e \right ) }}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(f*x+e)^2*(a+a*sin(f*x+e))^(1/2),x)

[Out]

(1+sin(f*x+e))*(-a*(-1+sin(f*x+e)))^(1/2)*(sin(f*x+e)*(2*(a-a*sin(f*x+e))^(1/2)*a^(3/2)-arctanh((a-a*sin(f*x+e
))^(1/2)/a^(1/2))*a^2)-(a-a*sin(f*x+e))^(1/2)*a^(3/2))/sin(f*x+e)/a^(3/2)/cos(f*x+e)/(a+a*sin(f*x+e))^(1/2)/f

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{a \sin \left (f x + e\right ) + a} \cot \left (f x + e\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(a+a*sin(f*x+e))^(1/2),x, algorithm="maxima")

[Out]

integrate(sqrt(a*sin(f*x + e) + a)*cot(f*x + e)^2, x)

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Fricas [B]  time = 1.93572, size = 749, normalized size = 8.42 \begin{align*} \frac{{\left (\cos \left (f x + e\right )^{2} -{\left (\cos \left (f x + e\right ) + 1\right )} \sin \left (f x + e\right ) - 1\right )} \sqrt{a} \log \left (\frac{a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} - 4 \,{\left (\cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt{a \sin \left (f x + e\right ) + a} \sqrt{a} - 9 \, a \cos \left (f x + e\right ) +{\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} +{\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) - 4 \,{\left (2 \, \cos \left (f x + e\right )^{2} +{\left (2 \, \cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 3\right )} \sqrt{a \sin \left (f x + e\right ) + a}}{4 \,{\left (f \cos \left (f x + e\right )^{2} -{\left (f \cos \left (f x + e\right ) + f\right )} \sin \left (f x + e\right ) - f\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(a+a*sin(f*x+e))^(1/2),x, algorithm="fricas")

[Out]

1/4*((cos(f*x + e)^2 - (cos(f*x + e) + 1)*sin(f*x + e) - 1)*sqrt(a)*log((a*cos(f*x + e)^3 - 7*a*cos(f*x + e)^2
- 4*(cos(f*x + e)^2 + (cos(f*x + e) + 3)*sin(f*x + e) - 2*cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a)*sqrt(a)
- 9*a*cos(f*x + e) + (a*cos(f*x + e)^2 + 8*a*cos(f*x + e) - a)*sin(f*x + e) - a)/(cos(f*x + e)^3 + cos(f*x + e
)^2 + (cos(f*x + e)^2 - 1)*sin(f*x + e) - cos(f*x + e) - 1)) - 4*(2*cos(f*x + e)^2 + (2*cos(f*x + e) + 3)*sin(
f*x + e) - cos(f*x + e) - 3)*sqrt(a*sin(f*x + e) + a))/(f*cos(f*x + e)^2 - (f*cos(f*x + e) + f)*sin(f*x + e) -
f)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)**2*(a+a*sin(f*x+e))**(1/2),x)

[Out]

Integral(sqrt(a*(sin(e + f*x) + 1))*cot(e + f*x)**2, x)

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Giac [B]  time = 2.0909, size = 570, normalized size = 6.4 \begin{align*} \frac{\frac{2 \, a \arctan \left (-\frac{\sqrt{a} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - \sqrt{a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + a}}{\sqrt{-a}}\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}{\sqrt{-a}} - \sqrt{a} \log \left ({\left | -\sqrt{a} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + \sqrt{a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + a} \right |}\right ) \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right ) + \frac{2 \, a^{\frac{3}{2}} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}{{\left (\sqrt{a} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - \sqrt{a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + a}\right )}^{2} - a} - \frac{{\left (2 \, \sqrt{2} a \arctan \left (\frac{\sqrt{2} \sqrt{a} + \sqrt{a}}{\sqrt{-a}}\right ) - \sqrt{2} \sqrt{-a} \sqrt{a} \log \left (\sqrt{2} \sqrt{a} + \sqrt{a}\right ) + 2 \, a \arctan \left (\frac{\sqrt{2} \sqrt{a} + \sqrt{a}}{\sqrt{-a}}\right ) - \sqrt{-a} \sqrt{a} \log \left (\sqrt{2} \sqrt{a} + \sqrt{a}\right ) + 5 \, \sqrt{2} \sqrt{-a} \sqrt{a} + 11 \, \sqrt{-a} \sqrt{a}\right )} \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )}{\sqrt{2} \sqrt{-a} + \sqrt{-a}} + \frac{5 \, a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right ) +{\left (a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right ) \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) - 4 \, a \mathrm{sgn}\left (\tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right ) + 1\right )\right )} \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )}{\sqrt{a \tan \left (\frac{1}{2} \, f x + \frac{1}{2} \, e\right )^{2} + a}}}{2 \, f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(f*x+e)^2*(a+a*sin(f*x+e))^(1/2),x, algorithm="giac")

[Out]

1/2*(2*a*arctan(-(sqrt(a)*tan(1/2*f*x + 1/2*e) - sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a))/sqrt(-a))*sgn(tan(1/2*f*x
+ 1/2*e) + 1)/sqrt(-a) - sqrt(a)*log(abs(-sqrt(a)*tan(1/2*f*x + 1/2*e) + sqrt(a*tan(1/2*f*x + 1/2*e)^2 + a)))
*sgn(tan(1/2*f*x + 1/2*e) + 1) + 2*a^(3/2)*sgn(tan(1/2*f*x + 1/2*e) + 1)/((sqrt(a)*tan(1/2*f*x + 1/2*e) - sqrt
(a*tan(1/2*f*x + 1/2*e)^2 + a))^2 - a) - (2*sqrt(2)*a*arctan((sqrt(2)*sqrt(a) + sqrt(a))/sqrt(-a)) - sqrt(2)*s
qrt(-a)*sqrt(a)*log(sqrt(2)*sqrt(a) + sqrt(a)) + 2*a*arctan((sqrt(2)*sqrt(a) + sqrt(a))/sqrt(-a)) - sqrt(-a)*s
qrt(a)*log(sqrt(2)*sqrt(a) + sqrt(a)) + 5*sqrt(2)*sqrt(-a)*sqrt(a) + 11*sqrt(-a)*sqrt(a))*sgn(tan(1/2*f*x + 1/
2*e) + 1)/(sqrt(2)*sqrt(-a) + sqrt(-a)) + (5*a*sgn(tan(1/2*f*x + 1/2*e) + 1) + (a*sgn(tan(1/2*f*x + 1/2*e) + 1
)*tan(1/2*f*x + 1/2*e) - 4*a*sgn(tan(1/2*f*x + 1/2*e) + 1))*tan(1/2*f*x + 1/2*e))/sqrt(a*tan(1/2*f*x + 1/2*e)^
2 + a))/f