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

Optimal. Leaf size=25 \[ \text{Unintegrable}\left ((g \tan (e+f x))^p (a+b \sin (e+f x))^m,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0403311, 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 = {} \[ \int (a+b \sin (e+f x))^m (g \tan (e+f x))^p \, dx \]

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*}

Mathematica [A]  time = 2.69509, size = 0, normalized size = 0. \[ \int (a+b \sin (e+f x))^m (g \tan (e+f x))^p \, dx \]

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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Maple [A]  time = 0.862, size = 0, normalized size = 0. \begin{align*} \int \left ( a+b\sin \left ( fx+e \right ) \right ) ^{m} \left ( g\tan \left ( fx+e \right ) \right ) ^{p}\, dx \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right ) + a\right )}^{m} \left (g \tan \left (f x + e\right )\right )^{p}\,{d x} \end{align*}

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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Fricas [A]  time = 0., size = 0, normalized size = 0. \begin{align*}{\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 ) \end{align*}

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

[Out]

Timed out

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Giac [A]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sin \left (f x + e\right ) + a\right )}^{m} \left (g \tan \left (f x + e\right )\right )^{p}\,{d x} \end{align*}

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)