3.1.34 \(\int \frac {a+b \csc (c+d \sqrt {x})}{x} \, dx\) [34]

Optimal. Leaf size=24 \[ a \log (x)+b \text {Int}\left (\frac {\csc \left (c+d \sqrt {x}\right )}{x},x\right ) \]

[Out]

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

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Rubi [A]
time = 0.01, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(a + b*Csc[c + d*Sqrt[x]])/x,x]

[Out]

a*Log[x] + b*Defer[Int][Csc[c + d*Sqrt[x]]/x, x]

Rubi steps

\begin {align*} \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x} \, dx &=\int \left (\frac {a}{x}+\frac {b \csc \left (c+d \sqrt {x}\right )}{x}\right ) \, dx\\ &=a \log (x)+b \int \frac {\csc \left (c+d \sqrt {x}\right )}{x} \, dx\\ \end {align*}

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Mathematica [A]
time = 4.16, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a+b \csc \left (c+d \sqrt {x}\right )}{x} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.09, size = 0, normalized size = 0.00 \[\int \frac {a +b \csc \left (c +d \sqrt {x}\right )}{x}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*csc(d*sqrt(x) + c) + a)/x, x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {a + b \csc {\left (c + d \sqrt {x} \right )}}{x}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((a + b*csc(c + d*sqrt(x)))/x, x)

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Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {a+\frac {b}{\sin \left (c+d\,\sqrt {x}\right )}}{x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/sin(c + d*x^(1/2)))/x,x)

[Out]

int((a + b/sin(c + d*x^(1/2)))/x, x)

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