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

Optimal. Leaf size=24 \[ \text {Int}\left (\sin ^m(c+d x) (a+b \sin (c+d x))^n,x\right ) \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx \end {gather*}

Verification is not applicable to the result.

[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*} \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\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 1.53, size = 0, normalized size = 0.00 \begin {gather*} \int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx \end {gather*}

Verification is not applicable to the result.

[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 [A]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (\sin ^{m}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Maxima [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Fricas [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a + b \sin {\left (c + d x \right )}\right )^{n} \sin ^{m}{\left (c + d x \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________

Giac [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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 [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int {\sin \left (c+d\,x\right )}^m\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^n \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

________________________________________________________________________________________