3.5.87 \(\int \sec ^3(e+f x) (a+b (c \tan (e+f x))^n)^p \, dx\) [487]

Optimal. Leaf size=28 \[ \text {Int}\left (\sec ^3(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p,x\right ) \]

[Out]

Unintegrable(sec(f*x+e)^3*(a+b*(c*tan(f*x+e))^n)^p,x)

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Rubi [A]
time = 0.04, 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 ^3(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[Sec[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p,x]

[Out]

Defer[Int][Sec[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p, x]

Rubi steps

\begin {align*} \int \sec ^3(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx &=\int \sec ^3(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx\\ \end {align*}

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Mathematica [A]
time = 2.97, size = 0, normalized size = 0.00 \begin {gather*} \int \sec ^3(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sec[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p,x]

[Out]

Integrate[Sec[e + f*x]^3*(a + b*(c*Tan[e + f*x])^n)^p, x]

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Maple [A]
time = 0.34, size = 0, normalized size = 0.00 \[\int \left (\sec ^{3}\left (f x +e \right )\right ) \left (a +b \left (c \tan \left (f x +e \right )\right )^{n}\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^3*(a+b*(c*tan(f*x+e))^n)^p,x)

[Out]

int(sec(f*x+e)^3*(a+b*(c*tan(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)^3*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm="maxima")

[Out]

integrate(((c*tan(f*x + e))^n*b + a)^p*sec(f*x + e)^3, x)

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Fricas [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)^3*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm="fricas")

[Out]

integral(((c*tan(f*x + e))^n*b + a)^p*sec(f*x + e)^3, x)

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)**3*(a+b*(c*tan(f*x+e))**n)**p,x)

[Out]

Timed out

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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)^3*(a+b*(c*tan(f*x+e))^n)^p,x, algorithm="giac")

[Out]

integrate(((c*tan(f*x + e))^n*b + a)^p*sec(f*x + e)^3, x)

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Mupad [A]
time = 0.00, size = -1, normalized size = -0.04 \begin {gather*} \int \frac {{\left (a+b\,{\left (c\,\mathrm {tan}\left (e+f\,x\right )\right )}^n\right )}^p}{{\cos \left (e+f\,x\right )}^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*(c*tan(e + f*x))^n)^p/cos(e + f*x)^3,x)

[Out]

int((a + b*(c*tan(e + f*x))^n)^p/cos(e + f*x)^3, x)

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