3.2 \(\int x^4 (a+b \csc (c+d x^2)) \, dx\)

Optimal. Leaf size=25 \[ b \text{Unintegrable}\left (x^4 \csc \left (c+d x^2\right ),x\right )+\frac{a x^5}{5} \]

[Out]

(a*x^5)/5 + b*Unintegrable[x^4*Csc[c + d*x^2], x]

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Rubi [A]  time = 0.0160538, 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 = {} \[ \int x^4 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Int[x^4*(a + b*Csc[c + d*x^2]),x]

[Out]

(a*x^5)/5 + b*Defer[Int][x^4*Csc[c + d*x^2], x]

Rubi steps

\begin{align*} \int x^4 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx &=\int \left (a x^4+b x^4 \csc \left (c+d x^2\right )\right ) \, dx\\ &=\frac{a x^5}{5}+b \int x^4 \csc \left (c+d x^2\right ) \, dx\\ \end{align*}

Mathematica [A]  time = 3.24103, size = 0, normalized size = 0. \[ \int x^4 \left (a+b \csc \left (c+d x^2\right )\right ) \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[x^4*(a + b*Csc[c + d*x^2]),x]

[Out]

Integrate[x^4*(a + b*Csc[c + d*x^2]), x]

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Maple [A]  time = 0.124, size = 0, normalized size = 0. \begin{align*} \int{x}^{4} \left ( a+b\csc \left ( d{x}^{2}+c \right ) \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a+b*csc(d*x^2+c)),x)

[Out]

int(x^4*(a+b*csc(d*x^2+c)),x)

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Maxima [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \frac{1}{5} \, a x^{5} + b{\left (\int \frac{x^{4} \sin \left (d x^{2} + c\right )}{\cos \left (d x^{2} + c\right )^{2} + \sin \left (d x^{2} + c\right )^{2} + 2 \, \cos \left (d x^{2} + c\right ) + 1}\,{d x} + \int \frac{x^{4} \sin \left (d x^{2} + c\right )}{\cos \left (d x^{2} + c\right )^{2} + \sin \left (d x^{2} + c\right )^{2} - 2 \, \cos \left (d x^{2} + c\right ) + 1}\,{d x}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*csc(d*x^2+c)),x, algorithm="maxima")

[Out]

1/5*a*x^5 + b*(integrate(x^4*sin(d*x^2 + c)/(cos(d*x^2 + c)^2 + sin(d*x^2 + c)^2 + 2*cos(d*x^2 + c) + 1), x) +
integrate(x^4*sin(d*x^2 + c)/(cos(d*x^2 + c)^2 + sin(d*x^2 + c)^2 - 2*cos(d*x^2 + c) + 1), x))

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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (b x^{4} \csc \left (d x^{2} + c\right ) + a x^{4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*csc(d*x^2+c)),x, algorithm="fricas")

[Out]

integral(b*x^4*csc(d*x^2 + c) + a*x^4, x)

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Sympy [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int x^{4} \left (a + b \csc{\left (c + d x^{2} \right )}\right )\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(a+b*csc(d*x**2+c)),x)

[Out]

Integral(x**4*(a + b*csc(c + d*x**2)), x)

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \csc \left (d x^{2} + c\right ) + a\right )} x^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(a+b*csc(d*x^2+c)),x, algorithm="giac")

[Out]

integrate((b*csc(d*x^2 + c) + a)*x^4, x)