\(\int (a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx\) [358]

Optimal result

Integrand size = 35, antiderivative size = 35 \[ \int (a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\text {Int}\left ((a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n,x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.05 (sec) , antiderivative size = 35, 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 = {} \[ \int (a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int (a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx \]

[In]

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

[Out]

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

Rubi steps \begin{align*} \text {integral}& = \int (a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 14.64 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int (a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int (a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 1.00 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.00

\[\int \left (a +b \sin \left (f x +e \right )\right )^{m} \left (A +B \sin \left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{n}d x\]

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.54 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int (a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int (a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate((a+b*sin(f*x+e))**m*(A+B*sin(f*x+e))*(c+d*sin(f*x+e))**n,x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [N/A]

Not integrable

Time = 29.15 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int (a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 3.79 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int (a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int { {\left (B \sin \left (f x + e\right ) + A\right )} {\left (b \sin \left (f x + e\right ) + a\right )}^{m} {\left (d \sin \left (f x + e\right ) + c\right )}^{n} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 18.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 1.06 \[ \int (a+b \sin (e+f x))^m (A+B \sin (e+f x)) (c+d \sin (e+f x))^n \, dx=\int \left (A+B\,\sin \left (e+f\,x\right )\right )\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^m\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n \,d x \]

[In]

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

[Out]

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