\(\int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx\) [1560]

Optimal result

Integrand size = 37, antiderivative size = 37 \[ \int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\text {Int}\left ((g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)),x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.07 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx \]

[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*} \text {integral}& = \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 [N/A]

Not integrable

Time = 7.99 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05 \[ \int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx \]

[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]

Maple [N/A] (verified)

Not integrable

Time = 0.99 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.00

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

[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)

Fricas [N/A]

Not integrable

Time = 0.29 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05 \[ \int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{-m - 1} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[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)

Sympy [N/A]

Not integrable

Time = 77.17 (sec) , antiderivative size = 36, normalized size of antiderivative = 0.97 \[ \int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int \left (g \cos {\left (e + f x \right )}\right )^{- m - 1} \left (A + B \sin {\left (e + f x \right )}\right ) \left (a + b \sin {\left (e + f x \right )}\right )^{m}\, dx \]

[In]

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

[Out]

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

Maxima [N/A]

Not integrable

Time = 5.89 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05 \[ \int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{-m - 1} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[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)

Giac [N/A]

Not integrable

Time = 1.17 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05 \[ \int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} \left (g \cos \left (f x + e\right )\right )^{-m - 1} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \,d x } \]

[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)

Mupad [N/A]

Not integrable

Time = 14.32 (sec) , antiderivative size = 39, normalized size of antiderivative = 1.05 \[ \int (g \cos (e+f x))^{-1-m} (a+b \sin (e+f x))^m (A+B \sin (e+f x)) \, dx=\int \frac {\left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^m}{{\left (g\,\cos \left (e+f\,x\right )\right )}^{m+1}} \,d x \]

[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)