3.2.45 \(\int (c+d x)^m (a+a \sin (e+f x))^n \, dx\) [145]

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

[Out]

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

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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 (c+d x)^m (a+a \sin (e+f x))^n \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

\begin {align*} \int (c+d x)^m (a+a \sin (e+f x))^n \, dx &=\int (c+d x)^m (a+a \sin (e+f x))^n \, dx\\ \end {align*}

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Mathematica [A]
time = 0.75, size = 0, normalized size = 0.00 \begin {gather*} \int (c+d x)^m (a+a \sin (e+f x))^n \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.02, size = 0, normalized size = 0.00 \[\int \left (d x +c \right )^{m} \left (a +a \sin \left (f x +e \right )\right )^{n}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

[Out]

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

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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