\(\int \frac {\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx\) [195]

Optimal result

Integrand size = 28, antiderivative size = 28 \[ \int \frac {\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\text {Int}\left (\frac {\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))},x\right ) \]

[Out]

Unintegrable(sin(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x)

Rubi [N/A]

Not integrable

Time = 0.05 (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 = {} \[ \int \frac {\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int \frac {\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]

[In]

Int[Sin[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])),x]

[Out]

Defer[Int][Sin[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \\ \end{align*}

Mathematica [N/A]

Not integrable

Time = 3.53 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int \frac {\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx \]

[In]

Integrate[Sin[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])),x]

[Out]

Integrate[Sin[c + d*x]^3/((e + f*x)*(a + a*Sin[c + d*x])), x]

Maple [N/A] (verified)

Not integrable

Time = 0.63 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.00

\[\int \frac {\sin ^{3}\left (d x +c \right )}{\left (f x +e \right ) \left (a +a \sin \left (d x +c \right )\right )}d x\]

[In]

int(sin(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x)

[Out]

int(sin(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 45, normalized size of antiderivative = 1.61 \[ \int \frac {\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int { \frac {\sin \left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(sin(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

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

Sympy [F(-2)]

Exception generated. \[ \int \frac {\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\text {Exception raised: HeuristicGCDFailed} \]

[In]

integrate(sin(d*x+c)**3/(f*x+e)/(a+a*sin(d*x+c)),x)

[Out]

Exception raised: HeuristicGCDFailed >> no luck

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\text {Exception raised: RuntimeError} \]

[In]

integrate(sin(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: THROW: The catch RAT-ERR is undefined.

Giac [N/A]

Not integrable

Time = 0.39 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int { \frac {\sin \left (d x + c\right )^{3}}{{\left (f x + e\right )} {\left (a \sin \left (d x + c\right ) + a\right )}} \,d x } \]

[In]

integrate(sin(d*x+c)^3/(f*x+e)/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

integrate(sin(d*x + c)^3/((f*x + e)*(a*sin(d*x + c) + a)), x)

Mupad [N/A]

Not integrable

Time = 1.17 (sec) , antiderivative size = 30, normalized size of antiderivative = 1.07 \[ \int \frac {\sin ^3(c+d x)}{(e+f x) (a+a \sin (c+d x))} \, dx=\int \frac {{\sin \left (c+d\,x\right )}^3}{\left (e+f\,x\right )\,\left (a+a\,\sin \left (c+d\,x\right )\right )} \,d x \]

[In]

int(sin(c + d*x)^3/((e + f*x)*(a + a*sin(c + d*x))),x)

[Out]

int(sin(c + d*x)^3/((e + f*x)*(a + a*sin(c + d*x))), x)