3.5.40 \(\int \sec ^4(e+f x) (a+b \sin ^n(e+f x))^p \, dx\) [440]

Optimal. Leaf size=26 \[ \text {Int}\left (\sec ^4(e+f x) \left (a+b \sin ^n(e+f x)\right )^p,x\right ) \]

[Out]

Unintegrable(sec(f*x+e)^4*(a+b*sin(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 \sec ^4(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^n)^p,x]

[Out]

Defer[Int][Sec[e + f*x]^4*(a + b*Sin[e + f*x]^n)^p, x]

Rubi steps

\begin {align*} \int \sec ^4(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx &=\int \sec ^4(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx\\ \end {align*}

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Mathematica [A]
time = 17.58, size = 0, normalized size = 0.00 \begin {gather*} \int \sec ^4(e+f x) \left (a+b \sin ^n(e+f x)\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^n)^p,x]

[Out]

Integrate[Sec[e + f*x]^4*(a + b*Sin[e + f*x]^n)^p, x]

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Maple [A]
time = 0.19, size = 0, normalized size = 0.00 \[\int \left (\sec ^{4}\left (f x +e \right )\right ) \left (a +b \left (\sin ^{n}\left (f x +e \right )\right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^4*(a+b*sin(f*x+e)^n)^p,x)

[Out]

int(sec(f*x+e)^4*(a+b*sin(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(sec(f*x+e)^4*(a+b*sin(f*x+e)^n)^p,x, algorithm="maxima")

[Out]

integrate((b*sin(f*x + e)^n + a)^p*sec(f*x + e)^4, x)

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Fricas [A]
time = 0.39, size = 25, normalized size = 0.96 \begin {gather*} {\rm integral}\left ({\left (b \sin \left (f x + e\right )^{n} + a\right )}^{p} \sec \left (f x + e\right )^{4}, x\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^4*(a+b*sin(f*x+e)^n)^p,x, algorithm="fricas")

[Out]

integral((b*sin(f*x + e)^n + a)^p*sec(f*x + e)^4, x)

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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)**4*(a+b*sin(f*x+e)**n)**p,x)

[Out]

Exception raised: SystemError >> excessive stack use: stack is 6438 deep

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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(sec(f*x+e)^4*(a+b*sin(f*x+e)^n)^p,x, algorithm="giac")

[Out]

integrate((b*sin(f*x + e)^n + a)^p*sec(f*x + e)^4, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (a+b\,{\sin \left (e+f\,x\right )}^n\right )}^p}{{\cos \left (e+f\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*sin(e + f*x)^n)^p/cos(e + f*x)^4,x)

[Out]

int((a + b*sin(e + f*x)^n)^p/cos(e + f*x)^4, x)

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