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=40 \[ \text {Int}\left ((A+B \sin (e+f x)) (g \cos (e+f x))^{-m-1} (a+b \sin (e+f x))^m,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.10, 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 = {} \[ \int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx \]

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*}

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Mathematica [A]  time = 5.17, size = 0, normalized size = 0.00 \[ \int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx \]

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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fricas [A]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{-m - 1} {\left (b \sin \left (f x + e\right ) + a\right )}^{m}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{-m - 1} {\left (b \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]

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]

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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maple [A]  time = 2.56, size = 0, normalized size = 0.00 \[ \int \left (g \cos \left (f x +e \right )\right )^{-1-m} \left (a +b \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right )\, dx \]

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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{-m - 1} {\left (b \sin \left (f x + e\right ) + a\right )}^{m}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^m}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{m+1}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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