3.218 \(\int \frac{(e+f x)^m \csc (c+d x)}{a+a \sin (c+d x)} \, dx\)

Optimal. Leaf size=28 \[ \text{Unintegrable}\left (\frac{\csc (c+d x) (e+f x)^m}{a \sin (c+d x)+a},x\right ) \]

[Out]

Unintegrable[((e + f*x)^m*Csc[c + d*x])/(a + a*Sin[c + d*x]), x]

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Rubi [A]  time = 0.0413263, 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 = {} \[ \int \frac{(e+f x)^m \csc (c+d x)}{a+a \sin (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

Int[((e + f*x)^m*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

Defer[Int][((e + f*x)^m*Csc[c + d*x])/(a + a*Sin[c + d*x]), x]

Rubi steps

\begin{align*} \int \frac{(e+f x)^m \csc (c+d x)}{a+a \sin (c+d x)} \, dx &=\int \frac{(e+f x)^m \csc (c+d x)}{a+a \sin (c+d x)} \, dx\\ \end{align*}

Mathematica [A]  time = 5.39632, size = 0, normalized size = 0. \[ \int \frac{(e+f x)^m \csc (c+d x)}{a+a \sin (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[((e + f*x)^m*Csc[c + d*x])/(a + a*Sin[c + d*x]),x]

[Out]

Integrate[((e + f*x)^m*Csc[c + d*x])/(a + a*Sin[c + d*x]), x]

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Maple [A]  time = 0.083, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( fx+e \right ) ^{m}\csc \left ( dx+c \right ) }{a+a\sin \left ( dx+c \right ) }}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)^m*csc(d*x+c)/(a+a*sin(d*x+c)),x)

[Out]

int((f*x+e)^m*csc(d*x+c)/(a+a*sin(d*x+c)),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (f x + e\right )}^{m} \csc \left (d x + c\right )}{a \sin \left (d x + c\right ) + a}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*csc(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

integrate((f*x + e)^m*csc(d*x + c)/(a*sin(d*x + c) + a), x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*csc(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

integral((f*x + e)^m*csc(d*x + c)/(a*sin(d*x + c) + a), x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{\int \frac{\left (e + f x\right )^{m} \csc{\left (c + d x \right )}}{\sin{\left (c + d x \right )} + 1}\, dx}{a} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)**m*csc(d*x+c)/(a+a*sin(d*x+c)),x)

[Out]

Integral((e + f*x)**m*csc(c + d*x)/(sin(c + d*x) + 1), x)/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)^m*csc(d*x+c)/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate((f*x + e)^m*csc(d*x + c)/(a*sin(d*x + c) + a), x)