3.670 \(\int \cos (x) \sqrt{1+\csc (x)} \, dx\)

Optimal. Leaf size=21 \[ \sin (x) \sqrt{\csc (x)+1}+\tanh ^{-1}\left (\sqrt{\csc (x)+1}\right ) \]

[Out]

ArcTanh[Sqrt[1 + Csc[x]]] + Sqrt[1 + Csc[x]]*Sin[x]

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Rubi [A]  time = 0.0239342, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364, Rules used = {3873, 47, 63, 207} \[ \sin (x) \sqrt{\csc (x)+1}+\tanh ^{-1}\left (\sqrt{\csc (x)+1}\right ) \]

Antiderivative was successfully verified.

[In]

Int[Cos[x]*Sqrt[1 + Csc[x]],x]

[Out]

ArcTanh[Sqrt[1 + Csc[x]]] + Sqrt[1 + Csc[x]]*Sin[x]

Rule 3873

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> -Dist[(f*b^(p - 1)
)^(-1), Subst[Int[((-a + b*x)^((p - 1)/2)*(a + b*x)^(m + (p - 1)/2))/x^(p + 1), x], x, Csc[e + f*x]], x] /; Fr
eeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]

Rule 47

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((a + b*x)^(m + 1)*(c + d*x)^n)/(b*
(m + 1)), x] - Dist[(d*n)/(b*(m + 1)), Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d},
x] && NeQ[b*c - a*d, 0] && GtQ[n, 0] && LtQ[m, -1] &&  !(IntegerQ[n] &&  !IntegerQ[m]) &&  !(ILeQ[m + n + 2, 0
] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c, d, m, n, x]

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 207

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

Rubi steps

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

Mathematica [A]  time = 0.0151509, size = 21, normalized size = 1. \[ \sin (x) \sqrt{\csc (x)+1}+\tanh ^{-1}\left (\sqrt{\csc (x)+1}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Cos[x]*Sqrt[1 + Csc[x]],x]

[Out]

ArcTanh[Sqrt[1 + Csc[x]]] + Sqrt[1 + Csc[x]]*Sin[x]

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Maple [B]  time = 0.034, size = 48, normalized size = 2.3 \begin{align*}{\frac{1}{2} \left ( 1+\sqrt{1+\csc \left ( x \right ) } \right ) ^{-1}}+{\frac{1}{2}\ln \left ( 1+\sqrt{1+\csc \left ( x \right ) } \right ) }+{\frac{1}{2} \left ( \sqrt{1+\csc \left ( x \right ) }-1 \right ) ^{-1}}-{\frac{1}{2}\ln \left ( \sqrt{1+\csc \left ( x \right ) }-1 \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(x)*(1+csc(x))^(1/2),x)

[Out]

1/2/(1+(1+csc(x))^(1/2))+1/2*ln(1+(1+csc(x))^(1/2))+1/2/((1+csc(x))^(1/2)-1)-1/2*ln((1+csc(x))^(1/2)-1)

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Maxima [B]  time = 0.94912, size = 51, normalized size = 2.43 \begin{align*} \sqrt{\frac{1}{\sin \left (x\right )} + 1} \sin \left (x\right ) + \frac{1}{2} \, \log \left (\sqrt{\frac{1}{\sin \left (x\right )} + 1} + 1\right ) - \frac{1}{2} \, \log \left (\sqrt{\frac{1}{\sin \left (x\right )} + 1} - 1\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(1+csc(x))^(1/2),x, algorithm="maxima")

[Out]

sqrt(1/sin(x) + 1)*sin(x) + 1/2*log(sqrt(1/sin(x) + 1) + 1) - 1/2*log(sqrt(1/sin(x) + 1) - 1)

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Fricas [B]  time = 2.02988, size = 271, normalized size = 12.9 \begin{align*} \sqrt{\frac{\sin \left (x\right ) + 1}{\sin \left (x\right )}} \sin \left (x\right ) + \frac{1}{2} \, \log \left (\frac{2 \,{\left (\sqrt{\frac{\sin \left (x\right ) + 1}{\sin \left (x\right )}} \sin \left (x\right ) + \sin \left (x\right ) + 1\right )}}{\cos \left (x\right ) + \sin \left (x\right ) + 1}\right ) - \frac{1}{2} \, \log \left (-\frac{2 \,{\left (\sqrt{\frac{\sin \left (x\right ) + 1}{\sin \left (x\right )}} \sin \left (x\right ) - \sin \left (x\right ) - 1\right )}}{\cos \left (x\right ) + \sin \left (x\right ) + 1}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(1+csc(x))^(1/2),x, algorithm="fricas")

[Out]

sqrt((sin(x) + 1)/sin(x))*sin(x) + 1/2*log(2*(sqrt((sin(x) + 1)/sin(x))*sin(x) + sin(x) + 1)/(cos(x) + sin(x)
+ 1)) - 1/2*log(-2*(sqrt((sin(x) + 1)/sin(x))*sin(x) - sin(x) - 1)/(cos(x) + sin(x) + 1))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sqrt{\csc{\left (x \right )} + 1} \cos{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(1+csc(x))**(1/2),x)

[Out]

Integral(sqrt(csc(x) + 1)*cos(x), x)

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Giac [B]  time = 1.14278, size = 51, normalized size = 2.43 \begin{align*} \frac{1}{2} \,{\left (2 \, \sqrt{\sin \left (x\right )^{2} + \sin \left (x\right )} - \log \left ({\left | 2 \, \sqrt{\sin \left (x\right )^{2} + \sin \left (x\right )} - 2 \, \sin \left (x\right ) - 1 \right |}\right )\right )} \mathrm{sgn}\left (\sin \left (x\right )\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(x)*(1+csc(x))^(1/2),x, algorithm="giac")

[Out]

1/2*(2*sqrt(sin(x)^2 + sin(x)) - log(abs(2*sqrt(sin(x)^2 + sin(x)) - 2*sin(x) - 1)))*sgn(sin(x))