3.1.72 \(\int (e x)^m (a+b \csc (c+d x^n))^p \, dx\) [72]

Optimal. Leaf size=32 \[ x^{-m} (e x)^m \text {Int}\left (x^m \left (a+b \csc \left (c+d x^n\right )\right )^p,x\right ) \]

[Out]

(e*x)^m*Unintegrable(x^m*(a+b*csc(c+d*x^n))^p,x)/(x^m)

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Rubi [A]
time = 0.03, 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 (e x)^m \left (a+b \csc \left (c+d x^n\right )\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(e*x)^m*(a + b*Csc[c + d*x^n])^p,x]

[Out]

((e*x)^m*Defer[Int][x^m*(a + b*Csc[c + d*x^n])^p, x])/x^m

Rubi steps

\begin {align*} \int (e x)^m \left (a+b \csc \left (c+d x^n\right )\right )^p \, dx &=\left (x^{-m} (e x)^m\right ) \int x^m \left (a+b \csc \left (c+d x^n\right )\right )^p \, dx\\ \end {align*}

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Mathematica [A]
time = 3.00, size = 0, normalized size = 0.00 \begin {gather*} \int (e x)^m \left (a+b \csc \left (c+d x^n\right )\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(e*x)^m*(a + b*Csc[c + d*x^n])^p,x]

[Out]

Integrate[(e*x)^m*(a + b*Csc[c + d*x^n])^p, x]

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Maple [A]
time = 0.12, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (a +b \csc \left (c +d \,x^{n}\right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^m*(a+b*csc(c+d*x^n))^p,x)

[Out]

int((e*x)^m*(a+b*csc(c+d*x^n))^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((e*x)^m*(a+b*csc(c+d*x^n))^p,x, algorithm="maxima")

[Out]

integrate((b*csc(d*x^n + c) + a)^p*(x*e)^m, 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((e*x)^m*(a+b*csc(c+d*x^n))^p,x, algorithm="fricas")

[Out]

integral((b*csc(d*x^n + c) + a)^p*(x*e)^m, x)

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Sympy [A]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (e x\right )^{m} \left (a + b \csc {\left (c + d x^{n} \right )}\right )^{p}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**m*(a+b*csc(c+d*x**n))**p,x)

[Out]

Integral((e*x)**m*(a + b*csc(c + d*x**n))**p, x)

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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((e*x)^m*(a+b*csc(c+d*x^n))^p,x, algorithm="giac")

[Out]

integrate((e*x)^m*(b*csc(d*x^n + c) + a)^p, x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b/sin(c + d*x^n))^p*(e*x)^m,x)

[Out]

int((a + b/sin(c + d*x^n))^p*(e*x)^m, x)

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