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

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

[Out]

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

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Rubi [A]  time = 0.06, 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 = {} \[ \int \frac {(a+b \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 2.65, size = 0, normalized size = 0.00 \[ \int \frac {(a+b \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 1.42, size = 0, normalized size = 0.00 \[ \int \frac {\left (a +b \sin \left (f x +e \right )\right )^{m}}{c +d \sin \left (f x +e \right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

Timed out

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