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

Optimal. Leaf size=27 \[ \text{Unintegrable}\left ((d \sin (e+f x))^m \left (a+b \tan ^n(e+f x)\right )^p,x\right ) \]

[Out]

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

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Rubi [A]  time = 0.0537551, 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 = {} \[ \int (d \sin (e+f x))^m \left (a+b \tan ^n(e+f x)\right )^p \, dx \]

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*}

Mathematica [A]  time = 2.83118, size = 0, normalized size = 0. \[ \int (d \sin (e+f x))^m \left (a+b \tan ^n(e+f x)\right )^p \, dx \]

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 = 2.056, size = 0, normalized size = 0. \begin{align*} \int \left ( d\sin \left ( fx+e \right ) \right ) ^{m} \left ( a+b \left ( \tan \left ( fx+e \right ) \right ) ^{n} \right ) ^{p}\, dx \end{align*}

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

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

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

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)