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

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 = 0.05, 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 = {} \[ \int \sec ^3(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

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 = 6.08, size = 0, normalized size = 0.00 \[ \int \sec ^3(e+f x) \left (a+b (c \tan (e+f x))^n\right )^p \, dx \]

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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fricas [A]  time = 0.59, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \sec \left (f x + e\right )^{3}, x\right ) \]

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

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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maple [A]  time = 1.73, 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 \[ \int {\left (\left (c \tan \left (f x + e\right )\right )^{n} b + a\right )}^{p} \sec \left (f x + e\right )^{3}\,{d x} \]

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

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

[Out]

Timed out

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