3.2.74 \(\int (d \sin (e+f x))^m (a+b \tan ^n(e+f x))^p \, dx\) [174]

Optimal. Leaf size=28 \[ \text {Int}\left ((d \sin (e+f x))^m \left (a+b \tan ^n(e+f x)\right )^p,x\right ) \]

[Out]

Unintegrable((d*sin(f*x+e))^m*(a+b*tan(f*x+e)^n)^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 (d \sin (e+f x))^m \left (a+b \tan ^n(e+f x)\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]
time = 1.73, size = 0, normalized size = 0.00 \begin {gather*} \int (d \sin (e+f x))^m \left (a+b \tan ^n(e+f x)\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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Maple [A]
time = 0.27, size = 0, normalized size = 0.00 \[\int \left (d \sin \left (f x +e \right )\right )^{m} \left (a +b \left (\tan ^{n}\left (f x +e \right )\right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

int((d*sin(f*x+e))^m*(a+b*tan(f*x+e)^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((d*sin(f*x+e))^m*(a+b*tan(f*x+e)^n)^p,x, algorithm="maxima")

[Out]

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

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*sin(f*x+e))**m*(a+b*tan(f*x+e)**n)**p,x)

[Out]

Timed out

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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