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=36 \[ \text {Int}\left (\cos ^4(c+d x) \sin ^{-p-3}(c+d x) (a+b \sin (c+d x))^p,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.11, 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 = {} \[ \int \cos ^4(c+d x) \sin ^{-3-p}(c+d x) (a+b \sin (c+d x))^p \, dx \]

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*}

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Mathematica [A]  time = 4.99, size = 0, normalized size = 0.00 \[ \int \cos ^4(c+d x) \sin ^{-3-p}(c+d x) (a+b \sin (c+d x))^p \, dx \]

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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fricas [A]  time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sin \left (d x + c\right ) + a\right )}^{p} \sin \left (d x + c\right )^{-p - 3} \cos \left (d x + c\right )^{4}, x\right ) \]

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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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{p} \sin \left (d x + c\right )^{-p - 3} \cos \left (d x + c\right )^{4}\,{d x} \]

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)

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maple [A]  time = 1.60, size = 0, normalized size = 0.00 \[ \int \left (\cos ^{4}\left (d x +c \right )\right ) \left (\sin ^{-3-p}\left (d x +c \right )\right ) \left (a +b \sin \left (d x +c \right )\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*sin(d*x+c)^(-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 = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{p} \sin \left (d x + c\right )^{-p - 3} \cos \left (d x + c\right )^{4}\,{d x} \]

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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mupad [A]  time = 0.00, size = -1, normalized size = -0.03 \[ \int \frac {{\cos \left (c+d\,x\right )}^4\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^p}{{\sin \left (c+d\,x\right )}^{p+3}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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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