3.12.92 \(\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx\) [1192]

Optimal. Leaf size=32 \[ \text {Int}\left (\cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p,x\right ) \]

[Out]

Unintegrable(cos(d*x+c)^4*sin(d*x+c)^n*(a+b*sin(d*x+c))^p,x)

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Rubi [A]
time = 0.06, 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 \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^p,x]

[Out]

Defer[Int][Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^p, x]

Rubi steps

\begin {align*} \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx &=\int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx\\ \end {align*}

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Mathematica [A]
time = 4.88, size = 0, normalized size = 0.00 \begin {gather*} \int \cos ^4(c+d x) \sin ^n(c+d x) (a+b \sin (c+d x))^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^p,x]

[Out]

Integrate[Cos[c + d*x]^4*Sin[c + d*x]^n*(a + b*Sin[c + d*x])^p, x]

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Maple [A]
time = 0.25, size = 0, normalized size = 0.00 \[\int \left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{n}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*sin(d*x+c)^n*(a+b*sin(d*x+c))^p,x)

[Out]

int(cos(d*x+c)^4*sin(d*x+c)^n*(a+b*sin(d*x+c))^p,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(cos(d*x+c)^4*sin(d*x+c)^n*(a+b*sin(d*x+c))^p,x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c) + a)^p*sin(d*x + c)^n*cos(d*x + c)^4, 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(cos(d*x+c)^4*sin(d*x+c)^n*(a+b*sin(d*x+c))^p,x, algorithm="fricas")

[Out]

integral((b*sin(d*x + c) + a)^p*sin(d*x + c)^n*cos(d*x + c)^4, 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(cos(d*x+c)**4*sin(d*x+c)**n*(a+b*sin(d*x+c))**p,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(cos(d*x+c)^4*sin(d*x+c)^n*(a+b*sin(d*x+c))^p,x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)^p*sin(d*x + c)^n*cos(d*x + c)^4, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.03 \begin {gather*} \int {\cos \left (c+d\,x\right )}^4\,{\sin \left (c+d\,x\right )}^n\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(c + d*x)^4*sin(c + d*x)^n*(a + b*sin(c + d*x))^p,x)

[Out]

int(cos(c + d*x)^4*sin(c + d*x)^n*(a + b*sin(c + d*x))^p, x)

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