\(\int \frac {(e+f x)^m \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx\) [215]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {(e+f x)^m \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\text {Int}\left (\frac {(e+f x)^m \sin ^2(c+d x)}{a+a \sin (c+d x)},x\right ) \]

[Out]

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

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 28, 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 = {} \[ \int \frac {(e+f x)^m \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {(e+f x)^m \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 5.88 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(e+f x)^m \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {(e+f x)^m \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.18 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

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

[In]

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

[Out]

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

Fricas [N/A]

Not integrable

Time = 0.28 (sec) , antiderivative size = 33, normalized size of antiderivative = 1.18 \[ \int \frac {(e+f x)^m \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m} \sin \left (d x + c\right )^{2}}{a \sin \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

integral(-(cos(d*x + c)^2 - 1)*(f*x + e)^m/(a*sin(d*x + c) + a), x)

Sympy [N/A]

Not integrable

Time = 5.69 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.93 \[ \int \frac {(e+f x)^m \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\frac {\int \frac {\left (e + f x\right )^{m} \sin ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \]

[In]

integrate((f*x+e)**m*sin(d*x+c)**2/(a+a*sin(d*x+c)),x)

[Out]

Integral((e + f*x)**m*sin(c + d*x)**2/(sin(c + d*x) + 1), x)/a

Maxima [N/A]

Not integrable

Time = 0.77 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(e+f x)^m \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m} \sin \left (d x + c\right )^{2}}{a \sin \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Giac [N/A]

Not integrable

Time = 0.42 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(e+f x)^m \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\int { \frac {{\left (f x + e\right )}^{m} \sin \left (d x + c\right )^{2}}{a \sin \left (d x + c\right ) + a} \,d x } \]

[In]

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

[Out]

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

Mupad [N/A]

Not integrable

Time = 1.55 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {(e+f x)^m \sin ^2(c+d x)}{a+a \sin (c+d x)} \, dx=\int \frac {{\sin \left (c+d\,x\right )}^2\,{\left (e+f\,x\right )}^m}{a+a\,\sin \left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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