3.53 \(\int \frac{a+b \csc (c+d \sqrt{x})}{\sqrt{x}} \, dx\)

Optimal. Leaf size=26 \[ 2 a \sqrt{x}-\frac{2 b \tanh ^{-1}\left (\cos \left (c+d \sqrt{x}\right )\right )}{d} \]

[Out]

2*a*Sqrt[x] - (2*b*ArcTanh[Cos[c + d*Sqrt[x]]])/d

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Rubi [A]  time = 0.0217672, antiderivative size = 26, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15, Rules used = {14, 4205, 3770} \[ 2 a \sqrt{x}-\frac{2 b \tanh ^{-1}\left (\cos \left (c+d \sqrt{x}\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*Csc[c + d*Sqrt[x]])/Sqrt[x],x]

[Out]

2*a*Sqrt[x] - (2*b*ArcTanh[Cos[c + d*Sqrt[x]]])/d

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 4205

Int[((a_.) + Csc[(c_.) + (d_.)*(x_)^(n_)]*(b_.))^(p_.)*(x_)^(m_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*Csc[c + d*x])^p, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p}, x] && IGtQ[Simplify[
(m + 1)/n], 0] && IntegerQ[p]

Rule 3770

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> -Simp[ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rubi steps

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

Mathematica [A]  time = 0.0925451, size = 52, normalized size = 2. \[ \frac{2 \left (a \left (c+d \sqrt{x}\right )+b \log \left (\sin \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )\right )-b \log \left (\cos \left (\frac{1}{2} \left (c+d \sqrt{x}\right )\right )\right )\right )}{d} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*Csc[c + d*Sqrt[x]])/Sqrt[x],x]

[Out]

(2*(a*(c + d*Sqrt[x]) - b*Log[Cos[(c + d*Sqrt[x])/2]] + b*Log[Sin[(c + d*Sqrt[x])/2]]))/d

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Maple [A]  time = 0.024, size = 32, normalized size = 1.2 \begin{align*} 2\,a\sqrt{x}-2\,{\frac{b\ln \left ( \csc \left ( c+d\sqrt{x} \right ) +\cot \left ( c+d\sqrt{x} \right ) \right ) }{d}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*x^(1/2)-2*b/d*ln(csc(c+d*x^(1/2))+cot(c+d*x^(1/2)))

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Maxima [A]  time = 0.98613, size = 42, normalized size = 1.62 \begin{align*} 2 \, a \sqrt{x} - \frac{2 \, b \log \left (\cot \left (d \sqrt{x} + c\right ) + \csc \left (d \sqrt{x} + c\right )\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a*sqrt(x) - 2*b*log(cot(d*sqrt(x) + c) + csc(d*sqrt(x) + c))/d

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Fricas [A]  time = 0.514915, size = 130, normalized size = 5. \begin{align*} \frac{2 \, a d \sqrt{x} - b \log \left (\frac{1}{2} \, \cos \left (d \sqrt{x} + c\right ) + \frac{1}{2}\right ) + b \log \left (-\frac{1}{2} \, \cos \left (d \sqrt{x} + c\right ) + \frac{1}{2}\right )}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*a*d*sqrt(x) - b*log(1/2*cos(d*sqrt(x) + c) + 1/2) + b*log(-1/2*cos(d*sqrt(x) + c) + 1/2))/d

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Sympy [A]  time = 3.8478, size = 58, normalized size = 2.23 \begin{align*} - \begin{cases} - \sqrt{x} \left (2 a + 2 b \csc{\left (c \right )}\right ) & \text{for}\: d = 0 \\- \frac{2 a \left (c + d \sqrt{x}\right ) - 2 b \log{\left (\cot{\left (c + d \sqrt{x} \right )} + \csc{\left (c + d \sqrt{x} \right )} \right )}}{d} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-Piecewise((-sqrt(x)*(2*a + 2*b*csc(c)), Eq(d, 0)), (-(2*a*(c + d*sqrt(x)) - 2*b*log(cot(c + d*sqrt(x)) + csc(
c + d*sqrt(x))))/d, True))

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Giac [A]  time = 1.49557, size = 41, normalized size = 1.58 \begin{align*} \frac{2 \,{\left ({\left (d \sqrt{x} + c\right )} a + b \log \left ({\left | \tan \left (\frac{1}{2} \, d \sqrt{x} + \frac{1}{2} \, c\right ) \right |}\right )\right )}}{d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*((d*sqrt(x) + c)*a + b*log(abs(tan(1/2*d*sqrt(x) + 1/2*c))))/d