3.5 \(\int \frac {a+a \cot (c+d x)}{\sqrt {e \cot (c+d x)}} \, dx\)

Optimal. Leaf size=49 \[ \frac {\sqrt {2} a \tan ^{-1}\left (\frac {\sqrt {e} (1-\cot (c+d x))}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d \sqrt {e}} \]

[Out]

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

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Rubi [A]  time = 0.04, antiderivative size = 49, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3532, 205} \[ \frac {\sqrt {2} a \tan ^{-1}\left (\frac {\sqrt {e} (1-\cot (c+d x))}{\sqrt {2} \sqrt {e \cot (c+d x)}}\right )}{d \sqrt {e}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[2]*a*ArcTan[(Sqrt[e]*(1 - Cot[c + d*x]))/(Sqrt[2]*Sqrt[e*Cot[c + d*x]])])/(d*Sqrt[e])

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

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Mathematica [C]  time = 0.22, size = 165, normalized size = 3.37 \[ \frac {a \left (8 \tan ^{\frac {3}{2}}(c+d x) \, _2F_1\left (\frac {3}{4},1;\frac {7}{4};-\tan ^2(c+d x)\right )+3 \sqrt {2} \left (-2 \tan ^{-1}\left (1-\sqrt {2} \sqrt {\tan (c+d x)}\right )+2 \tan ^{-1}\left (\sqrt {2} \sqrt {\tan (c+d x)}+1\right )-\log \left (\tan (c+d x)-\sqrt {2} \sqrt {\tan (c+d x)}+1\right )+\log \left (\tan (c+d x)+\sqrt {2} \sqrt {\tan (c+d x)}+1\right )\right )\right )}{12 d \sqrt {\tan (c+d x)} \sqrt {e \cot (c+d x)}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(a*(3*Sqrt[2]*(-2*ArcTan[1 - Sqrt[2]*Sqrt[Tan[c + d*x]]] + 2*ArcTan[1 + Sqrt[2]*Sqrt[Tan[c + d*x]]] - Log[1 -
Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]] + Log[1 + Sqrt[2]*Sqrt[Tan[c + d*x]] + Tan[c + d*x]]) + 8*Hypergeom
etric2F1[3/4, 1, 7/4, -Tan[c + d*x]^2]*Tan[c + d*x]^(3/2)))/(12*d*Sqrt[e*Cot[c + d*x]]*Sqrt[Tan[c + d*x]])

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fricas [B]  time = 0.60, size = 172, normalized size = 3.51 \[ \left [\frac {\sqrt {2} a \sqrt {-\frac {1}{e}} \log \left (-\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} \sqrt {-\frac {1}{e}} {\left (\cos \left (2 \, d x + 2 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right ) - 1\right )} - 2 \, \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}{2 \, d}, \frac {\sqrt {2} a \arctan \left (-\frac {\sqrt {2} \sqrt {\frac {e \cos \left (2 \, d x + 2 \, c\right ) + e}{\sin \left (2 \, d x + 2 \, c\right )}} {\left (\cos \left (2 \, d x + 2 \, c\right ) - \sin \left (2 \, d x + 2 \, c\right ) + 1\right )}}{2 \, \sqrt {e} {\left (\cos \left (2 \, d x + 2 \, c\right ) + 1\right )}}\right )}{d \sqrt {e}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/2*sqrt(2)*a*sqrt(-1/e)*log(-sqrt(2)*sqrt((e*cos(2*d*x + 2*c) + e)/sin(2*d*x + 2*c))*sqrt(-1/e)*(cos(2*d*x +
2*c) + sin(2*d*x + 2*c) - 1) - 2*sin(2*d*x + 2*c) + 1)/d, sqrt(2)*a*arctan(-1/2*sqrt(2)*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)/(sqrt(e)*(cos(2*d*x + 2*c) + 1)))/(d*sqrt
(e))]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.43, size = 327, normalized size = 6.67 \[ -\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{4 d e}-\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d e}+\frac {a \left (e^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d e}-\frac {a \sqrt {2}\, \ln \left (\frac {e \cot \left (d x +c \right )-\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}{e \cot \left (d x +c \right )+\left (e^{2}\right )^{\frac {1}{4}} \sqrt {e \cot \left (d x +c \right )}\, \sqrt {2}+\sqrt {e^{2}}}\right )}{4 d \left (e^{2}\right )^{\frac {1}{4}}}-\frac {a \sqrt {2}\, \arctan \left (\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d \left (e^{2}\right )^{\frac {1}{4}}}+\frac {a \sqrt {2}\, \arctan \left (-\frac {\sqrt {2}\, \sqrt {e \cot \left (d x +c \right )}}{\left (e^{2}\right )^{\frac {1}{4}}}+1\right )}{2 d \left (e^{2}\right )^{\frac {1}{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-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*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*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*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 [B]  time = 0.69, size = 83, normalized size = 1.69 \[ -\frac {a {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} + 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {e} - 2 \, \sqrt {\frac {e}{\tan \left (d x + c\right )}}\right )}}{2 \, \sqrt {e}}\right )}{\sqrt {e}}\right )}}{d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.73, size = 65, normalized size = 1.33 \[ \frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atan}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (-1+1{}\mathrm {i}\right )}{d\,\sqrt {e}}+\frac {{\left (-1\right )}^{1/4}\,a\,\mathrm {atanh}\left (\frac {{\left (-1\right )}^{1/4}\,\sqrt {e\,\mathrm {cot}\left (c+d\,x\right )}}{\sqrt {e}}\right )\,\left (1+1{}\mathrm {i}\right )}{d\,\sqrt {e}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ a \left (\int \frac {1}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx + \int \frac {\cot {\left (c + d x \right )}}{\sqrt {e \cot {\left (c + d x \right )}}}\, dx\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

a*(Integral(1/sqrt(e*cot(c + d*x)), x) + Integral(cot(c + d*x)/sqrt(e*cot(c + d*x)), x))

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