3.438 \(\int (a+b \sin ^n(e+f x))^p \, dx\)

Optimal. Leaf size=17 \[ \text {Int}\left (\left (a+b \sin ^n(e+f x)\right )^p,x\right ) \]

[Out]

Unintegrable((a+b*sin(f*x+e)^n)^p,x)

________________________________________________________________________________________

Rubi [A]  time = 0.01, 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 = {} \[ \int \left (a+b \sin ^n(e+f x)\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a + b*Sin[e + f*x]^n)^p,x]

[Out]

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

Rubi steps

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

________________________________________________________________________________________

Mathematica [A]  time = 1.57, size = 0, normalized size = 0.00 \[ \int \left (a+b \sin ^n(e+f x)\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a + b*Sin[e + f*x]^n)^p,x]

[Out]

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

________________________________________________________________________________________

fricas [A]  time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sin \left (f x + e\right )^{n} + a\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integral((b*sin(f*x + e)^n + a)^p, x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maple [A]  time = 0.62, size = 0, normalized size = 0.00 \[ \int \left (a +b \left (\sin ^{n}\left (f x +e \right )\right )\right )^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+b*sin(f*x+e)^n)^p,x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [A]  time = 0.00, size = -1, normalized size = -0.06 \[ \int {\left (a+b\,{\sin \left (e+f\,x\right )}^n\right )}^p \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sin(e + f*x)^n)^p,x)

[Out]

int((a + b*sin(e + f*x)^n)^p, x)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________