3.1560 \(\int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx\)

Optimal. Leaf size=39 \[ \text{Unintegrable}\left ((A+B \sin (e+f x)) (g \cos (e+f x))^{-m-1} (a+b \sin (e+f x))^m,x\right ) \]

[Out]

Unintegrable[(g*Cos[e + f*x])^(-1 - m)*(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x]

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Rubi [A]  time = 0.0973966, 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 = {} \[ \int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx \]

Verification is Not applicable to the result.

[In]

Int[(g*Cos[e + f*x])^(-1 - m)*(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]

[Out]

Defer[Int][(g*Cos[e + f*x])^(-1 - m)*(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x]

Rubi steps

\begin{align*} \int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx &=\int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx\\ \end{align*}

Mathematica [A]  time = 4.73579, size = 0, normalized size = 0. \[ \int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(g*Cos[e + f*x])^(-1 - m)*(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x]),x]

[Out]

Integrate[(g*Cos[e + f*x])^(-1 - m)*(a + b*Sin[e + f*x])^m*(A + B*Sin[e + f*x]), x]

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Maple [A]  time = 0.451, size = 0, normalized size = 0. \begin{align*} \int \left ( g\cos \left ( fx+e \right ) \right ) ^{-1-m} \left ( a+b\sin \left ( fx+e \right ) \right ) ^{m} \left ( A+B\sin \left ( fx+e \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*cos(f*x+e))^(-1-m)*(a+b*sin(f*x+e))^m*(A+B*sin(f*x+e)),x)

[Out]

int((g*cos(f*x+e))^(-1-m)*(a+b*sin(f*x+e))^m*(A+B*sin(f*x+e)),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{-m - 1}{\left (b \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(-1-m)*(a+b*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="maxima")

[Out]

integrate((B*sin(f*x + e) + A)*(g*cos(f*x + e))^(-m - 1)*(b*sin(f*x + e) + a)^m, x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{-m - 1}{\left (b \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(-1-m)*(a+b*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="fricas")

[Out]

integral((B*sin(f*x + e) + A)*(g*cos(f*x + e))^(-m - 1)*(b*sin(f*x + e) + a)^m, x)

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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((g*cos(f*x+e))**(-1-m)*(a+b*sin(f*x+e))**m*(A+B*sin(f*x+e)),x)

[Out]

Timed out

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: AttributeError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*cos(f*x+e))^(-1-m)*(a+b*sin(f*x+e))^m*(A+B*sin(f*x+e)),x, algorithm="giac")

[Out]

Exception raised: AttributeError