3.1191 \(\int \frac{\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt{a+b \sin (c+d x)}} \, dx\)

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

[Out]

Unintegrable[(Cos[c + d*x]^4*Sin[c + d*x]^(1/3))/Sqrt[a + b*Sin[c + d*x]], x]

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Rubi [A]  time = 0.148847, 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 = {} \[ \int \frac{\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt{a+b \sin (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(Cos[c + d*x]^4*Sin[c + d*x]^(1/3))/Sqrt[a + b*Sin[c + d*x]],x]

[Out]

Defer[Int][(Cos[c + d*x]^4*Sin[c + d*x]^(1/3))/Sqrt[a + b*Sin[c + d*x]], x]

Rubi steps

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

Mathematica [A]  time = 23.2607, size = 0, normalized size = 0. \[ \int \frac{\cos ^4(c+d x) \sqrt [3]{\sin (c+d x)}}{\sqrt{a+b \sin (c+d x)}} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^(1/3))/Sqrt[a + b*Sin[c + d*x]],x]

[Out]

Integrate[(Cos[c + d*x]^4*Sin[c + d*x]^(1/3))/Sqrt[a + b*Sin[c + d*x]], x]

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Maple [A]  time = 0.5, size = 0, normalized size = 0. \begin{align*} \int{ \left ( \cos \left ( dx+c \right ) \right ) ^{4}\sqrt [3]{\sin \left ( dx+c \right ) }{\frac{1}{\sqrt{a+b\sin \left ( dx+c \right ) }}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(cos(d*x+c)^4*sin(d*x+c)^(1/3)/(a+b*sin(d*x+c))^(1/2),x)

[Out]

int(cos(d*x+c)^4*sin(d*x+c)^(1/3)/(a+b*sin(d*x+c))^(1/2),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{\frac{1}{3}}}{\sqrt{b \sin \left (d x + c\right ) + a}}\,{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)^(1/3)/(a+b*sin(d*x+c))^(1/2),x, algorithm="maxima")

[Out]

integrate(cos(d*x + c)^4*sin(d*x + c)^(1/3)/sqrt(b*sin(d*x + c) + a), x)

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{\frac{1}{3}}}{\sqrt{b \sin \left (d x + c\right ) + a}}, 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)^(1/3)/(a+b*sin(d*x+c))^(1/2),x, algorithm="fricas")

[Out]

integral(cos(d*x + c)^4*sin(d*x + c)^(1/3)/sqrt(b*sin(d*x + c) + a), 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)**(1/3)/(a+b*sin(d*x+c))**(1/2),x)

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\cos \left (d x + c\right )^{4} \sin \left (d x + c\right )^{\frac{1}{3}}}{\sqrt{b \sin \left (d x + c\right ) + a}}\,{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)^(1/3)/(a+b*sin(d*x+c))^(1/2),x, algorithm="giac")

[Out]

integrate(cos(d*x + c)^4*sin(d*x + c)^(1/3)/sqrt(b*sin(d*x + c) + a), x)