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

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 = 0.05, 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 = {} \[ \int (d \sin (e+f x))^m \left (a+b \tan ^n(e+f x)\right )^p \, dx \]

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*}

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Mathematica [A]  time = 2.80, size = 0, normalized size = 0.00 \[ \int (d \sin (e+f x))^m \left (a+b \tan ^n(e+f x)\right )^p \, dx \]

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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fricas [A]  time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \tan \left (f x + e\right )^{n} + a\right )}^{p} \left (d \sin \left (f x + e\right )\right )^{m}, x\right ) \]

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

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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maple [A]  time = 7.66, 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 \[ \int {\left (b \tan \left (f x + e\right )^{n} + a\right )}^{p} \left (d \sin \left (f x + e\right )\right )^{m}\,{d x} \]

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

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

[Out]

Timed out

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