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

Optimal. Leaf size=23 \[ \text{Unintegrable}\left (\sin ^m(c+d x) (a+b \sin (c+d x))^n,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0397752, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx \]

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 = 2.27161, size = 0, normalized size = 0. \[ \int \sin ^m(c+d x) (a+b \sin (c+d x))^n \, dx \]

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]

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Maple [A]  time = 0.602, size = 0, normalized size = 0. \begin{align*} \int \left ( \sin \left ( dx+c \right ) \right ) ^{m} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

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)

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

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)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right )^{m}, x\right ) \end{align*}

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)

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

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)

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

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)