3.479 \(\int \sec ^2(e+f x) (b (c \tan (e+f x))^n)^p \, dx\)

Optimal. Leaf size=31 \[ \frac{\tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1)} \]

[Out]

(Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))

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Rubi [A]  time = 0.0925313, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3659, 2607, 32} \[ \frac{\tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1)} \]

Antiderivative was successfully verified.

[In]

Int[Sec[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p,x]

[Out]

(Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))

Rule 3659

Int[(u_.)*((b_.)*((c_.)*tan[(e_.) + (f_.)*(x_)])^(n_))^(p_), x_Symbol] :> Dist[(b^IntPart[p]*(b*(c*Tan[e + f*x
])^n)^FracPart[p])/(c*Tan[e + f*x])^(n*FracPart[p]), Int[ActivateTrig[u]*(c*Tan[e + f*x])^(n*p), x], x] /; Fre
eQ[{b, c, e, f, n, p}, x] &&  !IntegerQ[p] &&  !IntegerQ[n] && (EqQ[u, 1] || MatchQ[u, ((d_.)*(trig_)[e + f*x]
)^(m_.) /; FreeQ[{d, m}, x] && MemberQ[{sin, cos, tan, cot, sec, csc}, trig]])

Rule 2607

Int[sec[(e_.) + (f_.)*(x_)]^(m_)*((b_.)*tan[(e_.) + (f_.)*(x_)])^(n_.), x_Symbol] :> Dist[1/f, Subst[Int[(b*x)
^n*(1 + x^2)^(m/2 - 1), x], x, Tan[e + f*x]], x] /; FreeQ[{b, e, f, n}, x] && IntegerQ[m/2] &&  !(IntegerQ[(n
- 1)/2] && LtQ[0, n, m - 1])

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps

\begin{align*} \int \sec ^2(e+f x) \left (b (c \tan (e+f x))^n\right )^p \, dx &=\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \int \sec ^2(e+f x) (c \tan (e+f x))^{n p} \, dx\\ &=\frac{\left ((c \tan (e+f x))^{-n p} \left (b (c \tan (e+f x))^n\right )^p\right ) \operatorname{Subst}\left (\int (c x)^{n p} \, dx,x,\tan (e+f x)\right )}{f}\\ &=\frac{\tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (1+n p)}\\ \end{align*}

Mathematica [A]  time = 0.0228926, size = 31, normalized size = 1. \[ \frac{\tan (e+f x) \left (b (c \tan (e+f x))^n\right )^p}{f (n p+1)} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[e + f*x]^2*(b*(c*Tan[e + f*x])^n)^p,x]

[Out]

(Tan[e + f*x]*(b*(c*Tan[e + f*x])^n)^p)/(f*(1 + n*p))

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Maple [C]  time = 2.749, size = 25070, normalized size = 808.7 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(f*x+e)^2*(b*(c*tan(f*x+e))^n)^p,x)

[Out]

result too large to display

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Maxima [A]  time = 1.03106, size = 47, normalized size = 1.52 \begin{align*} \frac{b^{p}{\left (c^{n}\right )}^{p}{\left (\tan \left (f x + e\right )^{n}\right )}^{p} \tan \left (f x + e\right )}{{\left (n p + 1\right )} f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2*(b*(c*tan(f*x+e))^n)^p,x, algorithm="maxima")

[Out]

b^p*(c^n)^p*(tan(f*x + e)^n)^p*tan(f*x + e)/((n*p + 1)*f)

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Fricas [A]  time = 1.42176, size = 126, normalized size = 4.06 \begin{align*} \frac{e^{\left (n p \log \left (\frac{c \sin \left (f x + e\right )}{\cos \left (f x + e\right )}\right ) + p \log \left (b\right )\right )} \sin \left (f x + e\right )}{{\left (f n p + f\right )} \cos \left (f x + e\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2*(b*(c*tan(f*x+e))^n)^p,x, algorithm="fricas")

[Out]

e^(n*p*log(c*sin(f*x + e)/cos(f*x + e)) + p*log(b))*sin(f*x + e)/((f*n*p + f)*cos(f*x + e))

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (b \left (c \tan{\left (e + f x \right )}\right )^{n}\right )^{p} \sec ^{2}{\left (e + f x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)**2*(b*(c*tan(f*x+e))**n)**p,x)

[Out]

Integral((b*(c*tan(e + f*x))**n)**p*sec(e + f*x)**2, x)

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(f*x+e)^2*(b*(c*tan(f*x+e))^n)^p,x, algorithm="giac")

[Out]

Exception raised: NotImplementedError