3.325 \(\int \cot ^2(c+d x) \sqrt{a+b \sec (c+d x)} \, dx\)

Optimal. Leaf size=246 \[ \frac{\sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right ),\frac{a+b}{a-b}\right )}{d}-\frac{\cot (c+d x) \sqrt{a+b \sec (c+d x)}}{d}+\frac{2 \cot (c+d x) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (\sec (c+d x)+1)}{a+b \sec (c+d x)}} (a+b \sec (c+d x)) \Pi \left (\frac{a}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right )|\frac{a-b}{a+b}\right )}{d \sqrt{a+b}} \]

[Out]

(Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1
- Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (Cot[c + d*x]*Sqrt[a + b*Sec[c + d*x]])
/d + (2*Cot[c + d*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sec[c + d*x]]], (a - b)/(a + b)]*Sqrt
[-((b*(1 - Sec[c + d*x]))/(a + b*Sec[c + d*x]))]*Sqrt[(b*(1 + Sec[c + d*x]))/(a + b*Sec[c + d*x])]*(a + b*Sec[
c + d*x]))/(Sqrt[a + b]*d)

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Rubi [A]  time = 0.211957, antiderivative size = 246, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174, Rules used = {3896, 3780, 3875, 3832} \[ -\frac{\cot (c+d x) \sqrt{a+b \sec (c+d x)}}{d}+\frac{\sqrt{a+b} \cot (c+d x) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (\sec (c+d x)+1)}{a-b}} F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right )}{d}+\frac{2 \cot (c+d x) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (\sec (c+d x)+1)}{a+b \sec (c+d x)}} (a+b \sec (c+d x)) \Pi \left (\frac{a}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right )|\frac{a-b}{a+b}\right )}{d \sqrt{a+b}} \]

Antiderivative was successfully verified.

[In]

Int[Cot[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]],x]

[Out]

(Sqrt[a + b]*Cot[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1
- Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/d - (Cot[c + d*x]*Sqrt[a + b*Sec[c + d*x]])
/d + (2*Cot[c + d*x]*EllipticPi[a/(a + b), ArcSin[Sqrt[a + b]/Sqrt[a + b*Sec[c + d*x]]], (a - b)/(a + b)]*Sqrt
[-((b*(1 - Sec[c + d*x]))/(a + b*Sec[c + d*x]))]*Sqrt[(b*(1 + Sec[c + d*x]))/(a + b*Sec[c + d*x])]*(a + b*Sec[
c + d*x]))/(Sqrt[a + b]*d)

Rule 3896

Int[cot[(c_.) + (d_.)*(x_)]^(m_)*(csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_))^(n_), x_Symbol] :> Int[ExpandIntegrand
[(a + b*Csc[c + d*x])^n, (-1 + Sec[c + d*x]^2)^(-(m/2)), x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[a^2 - b^2,
0] && ILtQ[m/2, 0] && IntegerQ[n - 1/2] && EqQ[m, -2]

Rule 3780

Int[Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(2*(a + b*Csc[c + d*x])*Sqrt[(b*(1 + Csc[c +
d*x]))/(a + b*Csc[c + d*x])]*Sqrt[-((b*(1 - Csc[c + d*x]))/(a + b*Csc[c + d*x]))]*EllipticPi[a/(a + b), ArcSi
n[Rt[a + b, 2]/Sqrt[a + b*Csc[c + d*x]]], (a - b)/(a + b)])/(d*Rt[a + b, 2]*Cot[c + d*x]), x] /; FreeQ[{a, b,
c, d}, x] && NeQ[a^2 - b^2, 0]

Rule 3875

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_)/cos[(e_.) + (f_.)*(x_)]^2, x_Symbol] :> Simp[(Tan[e + f*x]*(a
+ b*Csc[e + f*x])^m)/f, x] + Dist[b*m, Int[Csc[e + f*x]*(a + b*Csc[e + f*x])^(m - 1), x], x] /; FreeQ[{a, b, e
, f, m}, x]

Rule 3832

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[(-2*Rt[a + b, 2]*Sqr
t[(b*(1 - Csc[e + f*x]))/(a + b)]*Sqrt[-((b*(1 + Csc[e + f*x]))/(a - b))]*EllipticF[ArcSin[Sqrt[a + b*Csc[e +
f*x]]/Rt[a + b, 2]], (a + b)/(a - b)])/(b*f*Cot[e + f*x]), x] /; FreeQ[{a, b, e, f}, x] && NeQ[a^2 - b^2, 0]

Rubi steps

\begin{align*} \int \cot ^2(c+d x) \sqrt{a+b \sec (c+d x)} \, dx &=\int \left (-\sqrt{a+b \sec (c+d x)}+\csc ^2(c+d x) \sqrt{a+b \sec (c+d x)}\right ) \, dx\\ &=-\int \sqrt{a+b \sec (c+d x)} \, dx+\int \csc ^2(c+d x) \sqrt{a+b \sec (c+d x)} \, dx\\ &=-\frac{\cot (c+d x) \sqrt{a+b \sec (c+d x)}}{d}+\frac{2 \cot (c+d x) \Pi \left (\frac{a}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right )|\frac{a-b}{a+b}\right ) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (1+\sec (c+d x))}{a+b \sec (c+d x)}} (a+b \sec (c+d x))}{\sqrt{a+b} d}+\frac{1}{2} b \int \frac{\sec (c+d x)}{\sqrt{a+b \sec (c+d x)}} \, dx\\ &=\frac{\sqrt{a+b} \cot (c+d x) F\left (\sin ^{-1}\left (\frac{\sqrt{a+b \sec (c+d x)}}{\sqrt{a+b}}\right )|\frac{a+b}{a-b}\right ) \sqrt{\frac{b (1-\sec (c+d x))}{a+b}} \sqrt{-\frac{b (1+\sec (c+d x))}{a-b}}}{d}-\frac{\cot (c+d x) \sqrt{a+b \sec (c+d x)}}{d}+\frac{2 \cot (c+d x) \Pi \left (\frac{a}{a+b};\sin ^{-1}\left (\frac{\sqrt{a+b}}{\sqrt{a+b \sec (c+d x)}}\right )|\frac{a-b}{a+b}\right ) \sqrt{-\frac{b (1-\sec (c+d x))}{a+b \sec (c+d x)}} \sqrt{\frac{b (1+\sec (c+d x))}{a+b \sec (c+d x)}} (a+b \sec (c+d x))}{\sqrt{a+b} d}\\ \end{align*}

Mathematica [A]  time = 3.3656, size = 156, normalized size = 0.63 \[ \frac{\sqrt{a+b \sec (c+d x)} \left (-\frac{2 \cos ^2\left (\frac{1}{2} (c+d x)\right ) \sqrt{\frac{1}{\sec (c+d x)+1}} \sqrt{\frac{a+b \sec (c+d x)}{(a+b) (\sec (c+d x)+1)}} \left ((b-2 a) \text{EllipticF}\left (\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right ),\frac{a-b}{a+b}\right )-4 a \Pi \left (-1;-\sin ^{-1}\left (\tan \left (\frac{1}{2} (c+d x)\right )\right )|\frac{a-b}{a+b}\right )\right )}{a \cos (c+d x)+b}-\cot (c+d x)\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[Cot[c + d*x]^2*Sqrt[a + b*Sec[c + d*x]],x]

[Out]

(Sqrt[a + b*Sec[c + d*x]]*(-Cot[c + d*x] - (2*Cos[(c + d*x)/2]^2*((-2*a + b)*EllipticF[ArcSin[Tan[(c + d*x)/2]
], (a - b)/(a + b)] - 4*a*EllipticPi[-1, -ArcSin[Tan[(c + d*x)/2]], (a - b)/(a + b)])*Sqrt[(1 + Sec[c + d*x])^
(-1)]*Sqrt[(a + b*Sec[c + d*x])/((a + b)*(1 + Sec[c + d*x]))])/(b + a*Cos[c + d*x])))/d

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Maple [B]  time = 0.296, size = 628, normalized size = 2.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cot(d*x+c)^2*(a+b*sec(d*x+c))^(1/2),x)

[Out]

-1/d*(-1+cos(d*x+c))^2*(2*EllipticF((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*cos(d*x+c)*(cos(d*x+c)/(co
s(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*sin(d*x+c)*a-EllipticF((-1+cos(d*x+c))/sin(
d*x+c),((a-b)/(a+b))^(1/2))*cos(d*x+c)*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)
+1))^(1/2)*sin(d*x+c)*b-4*a*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*
EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*sin(d*x+c)*cos(d*x+c)+2*EllipticF((-1+cos(d*x+c)
)/sin(d*x+c),((a-b)/(a+b))^(1/2))*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^
(1/2)*sin(d*x+c)*a-(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticF
((-1+cos(d*x+c))/sin(d*x+c),((a-b)/(a+b))^(1/2))*b*sin(d*x+c)-4*a*(cos(d*x+c)/(cos(d*x+c)+1))^(1/2)*(1/(a+b)*(
b+a*cos(d*x+c))/(cos(d*x+c)+1))^(1/2)*EllipticPi((-1+cos(d*x+c))/sin(d*x+c),-1,((a-b)/(a+b))^(1/2))*sin(d*x+c)
+a*cos(d*x+c)^2+b*cos(d*x+c))*(cos(d*x+c)+1)^2*((b+a*cos(d*x+c))/cos(d*x+c))^(1/2)/(b+a*cos(d*x+c))/sin(d*x+c)
^5

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sec(d*x + c) + a)*cot(d*x + c)^2, x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)**2*(a+b*sec(d*x+c))**(1/2),x)

[Out]

Integral(sqrt(a + b*sec(c + d*x))*cot(c + d*x)**2, x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{b \sec \left (d x + c\right ) + a} \cot \left (d x + c\right )^{2}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cot(d*x+c)^2*(a+b*sec(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(sqrt(b*sec(d*x + c) + a)*cot(d*x + c)^2, x)