3.6 \(\int \frac{a+b \csc (c+d x^2)}{x} \, dx\)

Optimal. Leaf size=21 \[ b \text{Unintegrable}\left (\frac{\csc \left (c+d x^2\right )}{x},x\right )+a \log (x) \]

[Out]

a*Log[x] + b*Unintegrable[Csc[c + d*x^2]/x, x]

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Rubi [A]  time = 0.0173375, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \frac{a+b \csc \left (c+d x^2\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Csc[c + d*x^2])/x,x]

[Out]

a*Log[x] + b*Defer[Int][Csc[c + d*x^2]/x, x]

Rubi steps

\begin{align*} \int \frac{a+b \csc \left (c+d x^2\right )}{x} \, dx &=\int \left (\frac{a}{x}+\frac{b \csc \left (c+d x^2\right )}{x}\right ) \, dx\\ &=a \log (x)+b \int \frac{\csc \left (c+d x^2\right )}{x} \, dx\\ \end{align*}

Mathematica [A]  time = 2.31812, size = 0, normalized size = 0. \[ \int \frac{a+b \csc \left (c+d x^2\right )}{x} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Csc[c + d*x^2])/x,x]

[Out]

Integrate[(a + b*Csc[c + d*x^2])/x, x]

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Maple [A]  time = 0.087, size = 0, normalized size = 0. \begin{align*} \int{\frac{a+b\csc \left ( d{x}^{2}+c \right ) }{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} b{\left (\int \frac{\sin \left (d x^{2} + c\right )}{x \cos \left (d x^{2} + c\right )^{2} + x \sin \left (d x^{2} + c\right )^{2} + 2 \, x \cos \left (d x^{2} + c\right ) + x}\,{d x} + \int \frac{\sin \left (d x^{2} + c\right )}{x \cos \left (d x^{2} + c\right )^{2} + x \sin \left (d x^{2} + c\right )^{2} - 2 \, x \cos \left (d x^{2} + c\right ) + x}\,{d x}\right )} + a \log \left (x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{b \csc \left (d x^{2} + c\right ) + a}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*csc(d*x^2 + c) + a)/x, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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