3.208 \(\int (a+b \sin (e+f x))^m (g \tan (e+f x))^p \, dx\)

Optimal. Leaf size=26 \[ \text {Int}\left ((g \tan (e+f x))^p (a+b \sin (e+f x))^m,x\right ) \]

[Out]

Unintegrable((a+b*sin(f*x+e))^m*(g*tan(f*x+e))^p,x)

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Rubi [A]  time = 0.04, 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 = {} \[ \int (a+b \sin (e+f x))^m (g \tan (e+f x))^p \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sin[e + f*x])^m*(g*Tan[e + f*x])^p,x]

[Out]

Defer[Int][(a + b*Sin[e + f*x])^m*(g*Tan[e + f*x])^p, x]

Rubi steps

\begin {align*} \int (a+b \sin (e+f x))^m (g \tan (e+f x))^p \, dx &=\int (a+b \sin (e+f x))^m (g \tan (e+f x))^p \, dx\\ \end {align*}

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Mathematica [A]  time = 3.13, size = 0, normalized size = 0.00 \[ \int (a+b \sin (e+f x))^m (g \tan (e+f x))^p \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Sin[e + f*x])^m*(g*Tan[e + f*x])^p,x]

[Out]

Integrate[(a + b*Sin[e + f*x])^m*(g*Tan[e + f*x])^p, x]

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fricas [A]  time = 0.71, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sin \left (f x + e\right ) + a\right )}^{m} \left (g \tan \left (f x + e\right )\right )^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))^m*(g*tan(f*x+e))^p,x, algorithm="fricas")

[Out]

integral((b*sin(f*x + e) + a)^m*(g*tan(f*x + e))^p, x)

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giac [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \left (g \tan \left (f x + e\right )\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))^m*(g*tan(f*x+e))^p,x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e) + a)^m*(g*tan(f*x + e))^p, x)

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maple [A]  time = 1.52, size = 0, normalized size = 0.00 \[ \int \left (a +b \sin \left (f x +e \right )\right )^{m} \left (g \tan \left (f x +e \right )\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*sin(f*x+e))^m*(g*tan(f*x+e))^p,x)

[Out]

int((a+b*sin(f*x+e))^m*(g*tan(f*x+e))^p,x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (f x + e\right ) + a\right )}^{m} \left (g \tan \left (f x + e\right )\right )^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e))^m*(g*tan(f*x+e))^p,x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e) + a)^m*(g*tan(f*x + e))^p, x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.04 \[ \int {\left (g\,\mathrm {tan}\left (e+f\,x\right )\right )}^p\,{\left (a+b\,\sin \left (e+f\,x\right )\right )}^m \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((g*tan(e + f*x))^p*(a + b*sin(e + f*x))^m,x)

[Out]

int((g*tan(e + f*x))^p*(a + b*sin(e + f*x))^m, 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((a+b*sin(f*x+e))**m*(g*tan(f*x+e))**p,x)

[Out]

Timed out

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