\(\int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx\) [1527]

Optimal result

Integrand size = 33, antiderivative size = 33 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx=\text {Int}\left (\frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2},x\right ) \]

[Out]

Unintegrable(cos(f*x+e)^2*(c+d*sin(f*x+e))^n/(a+b*sin(f*x+e))^2,x)

Rubi [N/A]

Not integrable

Time = 0.08 (sec) , antiderivative size = 33, 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 = {} \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx=\int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx \]

[In]

Int[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + b*Sin[e + f*x])^2,x]

[Out]

Defer[Int][(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + b*Sin[e + f*x])^2, x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 51.98 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx=\int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx \]

[In]

Integrate[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + b*Sin[e + f*x])^2,x]

[Out]

Integrate[(Cos[e + f*x]^2*(c + d*Sin[e + f*x])^n)/(a + b*Sin[e + f*x])^2, x]

Maple [N/A] (verified)

Not integrable

Time = 1.52 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.00

\[\int \frac {\left (\cos ^{2}\left (f x +e \right )\right ) \left (c +d \sin \left (f x +e \right )\right )^{n}}{\left (a +b \sin \left (f x +e \right )\right )^{2}}d x\]

[In]

int(cos(f*x+e)^2*(c+d*sin(f*x+e))^n/(a+b*sin(f*x+e))^2,x)

[Out]

int(cos(f*x+e)^2*(c+d*sin(f*x+e))^n/(a+b*sin(f*x+e))^2,x)

Fricas [N/A]

Not integrable

Time = 0.38 (sec) , antiderivative size = 59, normalized size of antiderivative = 1.79 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(c+d*sin(f*x+e))^n/(a+b*sin(f*x+e))^2,x, algorithm="fricas")

[Out]

integral(-(d*sin(f*x + e) + c)^n*cos(f*x + e)^2/(b^2*cos(f*x + e)^2 - 2*a*b*sin(f*x + e) - a^2 - b^2), x)

Sympy [F(-2)]

Exception generated. \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate(cos(f*x+e)**2*(c+d*sin(f*x+e))**n/(a+b*sin(f*x+e))**2,x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [N/A]

Not integrable

Time = 30.49 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(c+d*sin(f*x+e))^n/(a+b*sin(f*x+e))^2,x, algorithm="maxima")

[Out]

integrate((d*sin(f*x + e) + c)^n*cos(f*x + e)^2/(b*sin(f*x + e) + a)^2, x)

Giac [N/A]

Not integrable

Time = 0.46 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx=\int { \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n} \cos \left (f x + e\right )^{2}}{{\left (b \sin \left (f x + e\right ) + a\right )}^{2}} \,d x } \]

[In]

integrate(cos(f*x+e)^2*(c+d*sin(f*x+e))^n/(a+b*sin(f*x+e))^2,x, algorithm="giac")

[Out]

integrate((d*sin(f*x + e) + c)^n*cos(f*x + e)^2/(b*sin(f*x + e) + a)^2, x)

Mupad [N/A]

Not integrable

Time = 20.03 (sec) , antiderivative size = 35, normalized size of antiderivative = 1.06 \[ \int \frac {\cos ^2(e+f x) (c+d \sin (e+f x))^n}{(a+b \sin (e+f x))^2} \, dx=\int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{{\left (a+b\,\sin \left (e+f\,x\right )\right )}^2} \,d x \]

[In]

int((cos(e + f*x)^2*(c + d*sin(e + f*x))^n)/(a + b*sin(e + f*x))^2,x)

[Out]

int((cos(e + f*x)^2*(c + d*sin(e + f*x))^n)/(a + b*sin(e + f*x))^2, x)