3.3.8 \(\int (a+b \sin (e+f x))^m (g \tan (e+f x))^p \, dx\) [208]

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

[Out]

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

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Rubi [A]
time = 0.03, 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 (a+b \sin (e+f x))^m (g \tan (e+f x))^p \, dx \end {gather*}

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 = 1.66, size = 0, normalized size = 0.00 \begin {gather*} \int (a+b \sin (e+f x))^m (g \tan (e+f x))^p \, dx \end {gather*}

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 = 0.22, 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 \begin {gather*} \text {Failed to integrate} \end {gather*}

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

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]

Integral((g*tan(e + f*x))**p*(a + b*sin(e + f*x))**m, 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((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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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \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 \end {gather*}

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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