3.1193 \(\int \cos ^4(c+d x) \sin ^{-3-p}(c+d x) (a+b \sin (c+d x))^p \, dx\)

Optimal. Leaf size=35 \[ \text{Unintegrable}\left (\cos ^4(c+d x) \sin ^{-p-3}(c+d x) (a+b \sin (c+d x))^p,x\right ) \]

[Out]

Unintegrable[Cos[c + d*x]^4*Sin[c + d*x]^(-3 - p)*(a + b*Sin[c + d*x])^p, x]

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Rubi [A]  time = 0.113725, 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 = {} \[ \int \cos ^4(c+d x) \sin ^{-3-p}(c+d x) (a+b \sin (c+d x))^p \, dx \]

Verification is Not applicable to the result.

[In]

Int[Cos[c + d*x]^4*Sin[c + d*x]^(-3 - p)*(a + b*Sin[c + d*x])^p,x]

[Out]

Defer[Int][Cos[c + d*x]^4*Sin[c + d*x]^(-3 - p)*(a + b*Sin[c + d*x])^p, x]

Rubi steps

\begin{align*} \int \cos ^4(c+d x) \sin ^{-3-p}(c+d x) (a+b \sin (c+d x))^p \, dx &=\int \cos ^4(c+d x) \sin ^{-3-p}(c+d x) (a+b \sin (c+d x))^p \, dx\\ \end{align*}

Mathematica [A]  time = 4.57067, size = 0, normalized size = 0. \[ \int \cos ^4(c+d x) \sin ^{-3-p}(c+d x) (a+b \sin (c+d x))^p \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[Cos[c + d*x]^4*Sin[c + d*x]^(-3 - p)*(a + b*Sin[c + d*x])^p,x]

[Out]

Integrate[Cos[c + d*x]^4*Sin[c + d*x]^(-3 - p)*(a + b*Sin[c + d*x])^p, x]

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Maple [A]  time = 0.411, size = 0, normalized size = 0. \begin{align*} \int \left ( \cos \left ( dx+c \right ) \right ) ^{4} \left ( \sin \left ( dx+c \right ) \right ) ^{-3-p} \left ( a+b\sin \left ( dx+c \right ) \right ) ^{p}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*sin(d*x+c)^(-3-p)*(a+b*sin(d*x+c))^p,x)

[Out]

int(cos(d*x+c)^4*sin(d*x+c)^(-3-p)*(a+b*sin(d*x+c))^p,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 )}^{p} \sin \left (d x + c\right )^{-p - 3} \cos \left (d x + c\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^(-3-p)*(a+b*sin(d*x+c))^p,x, algorithm="maxima")

[Out]

integrate((b*sin(d*x + c) + a)^p*sin(d*x + c)^(-p - 3)*cos(d*x + c)^4, 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 )}^{p} \sin \left (d x + c\right )^{-p - 3} \cos \left (d x + c\right )^{4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^(-3-p)*(a+b*sin(d*x+c))^p,x, algorithm="fricas")

[Out]

integral((b*sin(d*x + c) + a)^p*sin(d*x + c)^(-p - 3)*cos(d*x + c)^4, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)**4*sin(d*x+c)**(-3-p)*(a+b*sin(d*x+c))**p,x)

[Out]

Timed out

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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 )}^{p} \sin \left (d x + c\right )^{-p - 3} \cos \left (d x + c\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(cos(d*x+c)^4*sin(d*x+c)^(-3-p)*(a+b*sin(d*x+c))^p,x, algorithm="giac")

[Out]

integrate((b*sin(d*x + c) + a)^p*sin(d*x + c)^(-p - 3)*cos(d*x + c)^4, x)