3.207 \(\int \frac {\csc ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\)

Optimal. Leaf size=31 \[ \text {Int}\left (\frac {\csc ^2(c+d x)}{(e+f x) (a \sin (c+d x)+a)},x\right ) \]

[Out]

Unintegrable(csc(d*x+c)^2/(f*x+e)/(a+a*sin(d*x+c)),x)

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Rubi [A]  time = 0.07, 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 = {} \[ \int \frac {\csc ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

Int[Csc[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])),x]

[Out]

Defer[Int][Csc[c + d*x]^2/((e + f*x)*(a + a*Sin[c + d*x])), x]

Rubi steps

\begin {align*} \int \frac {\csc ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx &=\int \frac {\csc ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\\ \end {align*}

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Mathematica [A]  time = 25.27, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^2(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\csc \left (d x + c\right )^{2}}{a f x + a e + {\left (a f x + a e\right )} \sin \left (d x + c\right )}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral(csc(d*x + c)^2/(a*f*x + a*e + (a*f*x + a*e)*sin(d*x + c)), x)

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [A]  time = 4.75, size = 0, normalized size = 0.00 \[ \int \frac {\csc ^{2}\left (d x +c \right )}{\left (f x +e \right ) \left (a +a \sin \left (d x +c \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(csc(d*x+c)^2/(f*x+e)/(a+a*sin(d*x+c)),x)

[Out]

int(csc(d*x+c)^2/(f*x+e)/(a+a*sin(d*x+c)),x)

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maxima [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {1}{{\sin \left (c+d\,x\right )}^2\,\left (e+f\,x\right )\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(sin(c + d*x)^2*(e + f*x)*(a + a*sin(c + d*x))),x)

[Out]

int(1/(sin(c + d*x)^2*(e + f*x)*(a + a*sin(c + d*x))), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\csc ^{2}{\left (c + d x \right )}}{e \sin {\left (c + d x \right )} + e + f x \sin {\left (c + d x \right )} + f x}\, dx}{a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(csc(d*x+c)**2/(f*x+e)/(a+a*sin(d*x+c)),x)

[Out]

Integral(csc(c + d*x)**2/(e*sin(c + d*x) + e + f*x*sin(c + d*x) + f*x), x)/a

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