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

Optimal. Leaf size=55 \[ (d \sin (e+f x))^{-n p} \left (c (d \sin (e+f x))^p\right )^n \text{Unintegrable}\left ((a+b \sin (e+f x))^m (d \sin (e+f x))^{n p},x\right ) \]

[Out]

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

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Rubi [A]  time = 0.112432, 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 = {} \[ \int \left (c (d \sin (e+f x))^p\right )^n (a+b \sin (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

Rubi steps

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

Mathematica [A]  time = 2.50163, size = 0, normalized size = 0. \[ \int \left (c (d \sin (e+f x))^p\right )^n (a+b \sin (e+f x))^m \, dx \]

Verification is Not applicable to the result.

[In]

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

[Out]

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

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Maple [A]  time = 0.214, size = 0, normalized size = 0. \begin{align*} \int \left ( c \left ( d\sin \left ( fx+e \right ) \right ) ^{p} \right ) ^{n} \left ( a+b\sin \left ( fx+e \right ) \right ) ^{m}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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