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

Optimal. Leaf size=27 \[ \text{Unintegrable}\left (\frac{(a+b \sin (e+f x))^m}{c+d \sin (e+f x)},x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.057435, 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 = {} \[ \int \frac{(a+b \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx \]

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*}

Mathematica [A]  time = 2.58644, size = 0, normalized size = 0. \[ \int \frac{(a+b \sin (e+f x))^m}{c+d \sin (e+f x)} \, dx \]

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]

________________________________________________________________________________________

Maple [A]  time = 0.913, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( a+b\sin \left ( fx+e \right ) \right ) ^{m}}{c+d\sin \left ( fx+e \right ) }}\, dx \end{align*}

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)

________________________________________________________________________________________

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

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)

________________________________________________________________________________________

Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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)

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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

________________________________________________________________________________________

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

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)