\(\int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx\) [221]

Optimal result

Integrand size = 21, antiderivative size = 21 \[ \int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx=\text {Int}\left (\sin ^m(c+d x) (a+b \sin (c+d x))^n,x\right ) \]

[Out]

Unintegrable(sin(d*x+c)^m*(a+b*sin(d*x+c))^n,x)

Rubi [N/A]

Not integrable

Time = 0.03 (sec) , antiderivative size = 21, 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 = {} \[ \int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx=\int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx \]

[In]

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

[Out]

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

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

Mathematica [N/A]

Not integrable

Time = 2.24 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx=\int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx \]

[In]

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

[Out]

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

Maple [N/A] (verified)

Not integrable

Time = 0.30 (sec) , antiderivative size = 21, normalized size of antiderivative = 1.00

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

[In]

int(sin(d*x+c)^m*(a+b*sin(d*x+c))^n,x)

[Out]

int(sin(d*x+c)^m*(a+b*sin(d*x+c))^n,x)

Fricas [N/A]

Not integrable

Time = 0.35 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{m} \,d x } \]

[In]

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

[Out]

integral((b*sin(d*x + c) + a)^n*sin(d*x + c)^m, x)

Sympy [N/A]

Not integrable

Time = 14.96 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.95 \[ \int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx=\int \left (a + b \sin {\left (c + d x \right )}\right )^{n} \sin ^{m}{\left (c + d x \right )}\, dx \]

[In]

integrate(sin(d*x+c)**m*(a+b*sin(d*x+c))**n,x)

[Out]

Integral((a + b*sin(c + d*x))**n*sin(c + d*x)**m, x)

Maxima [N/A]

Not integrable

Time = 1.55 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{m} \,d x } \]

[In]

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

[Out]

integrate((b*sin(d*x + c) + a)^n*sin(d*x + c)^m, x)

Giac [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx=\int { {\left (b \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{m} \,d x } \]

[In]

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

[Out]

integrate((b*sin(d*x + c) + a)^n*sin(d*x + c)^m, x)

Mupad [N/A]

Not integrable

Time = 7.17 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.10 \[ \int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx=\int {\sin \left (c+d\,x\right )}^m\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^n \,d x \]

[In]

int(sin(c + d*x)^m*(a + b*sin(c + d*x))^n,x)

[Out]

int(sin(c + d*x)^m*(a + b*sin(c + d*x))^n, x)