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

Optimal. Leaf size=23 \[ \text {Int}\left (\frac {(e+f x)^m}{a+b \sin (c+d x)},x\right ) \]

[Out]

Unintegrable((f*x+e)^m/(a+b*sin(d*x+c)),x)

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Rubi [A]  time = 0.05, 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 = {} \[ \int \frac {(e+f x)^m}{a+b \sin (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.31, size = 0, normalized size = 0.00 \[ \int \frac {(e+f x)^m}{a+b \sin (c+d x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.44, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (f x + e\right )}^{m}}{b \sin \left (d x + c\right ) + a}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.08, size = 0, normalized size = 0.00 \[ \int \frac {\left (f x +e \right )^{m}}{a +b \sin \left (d x +c \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int \frac {{\left (e+f\,x\right )}^m}{a+b\,\sin \left (c+d\,x\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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