3.917 \(\int \sec ^{1+m}(x) \sin (x) \, dx\)

Optimal. Leaf size=8 \[ \frac{\sec ^m(x)}{m} \]

[Out]

Sec[x]^m/m

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Rubi [A]  time = 0.0232947, antiderivative size = 8, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {2622, 30} \[ \frac{\sec ^m(x)}{m} \]

Antiderivative was successfully verified.

[In]

Int[Sec[x]^(1 + m)*Sin[x],x]

[Out]

Sec[x]^m/m

Rule 2622

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

Rule 30

Int[(x_)^(m_.), x_Symbol] :> Simp[x^(m + 1)/(m + 1), x] /; FreeQ[m, x] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \sec ^{1+m}(x) \sin (x) \, dx &=\operatorname{Subst}\left (\int x^{-1+m} \, dx,x,\sec (x)\right )\\ &=\frac{\sec ^m(x)}{m}\\ \end{align*}

Mathematica [A]  time = 0.016455, size = 8, normalized size = 1. \[ \frac{\sec ^m(x)}{m} \]

Antiderivative was successfully verified.

[In]

Integrate[Sec[x]^(1 + m)*Sin[x],x]

[Out]

Sec[x]^m/m

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Maple [A]  time = 0.019, size = 11, normalized size = 1.4 \begin{align*}{\frac{ \left ( \left ( \cos \left ( x \right ) \right ) ^{-1} \right ) ^{m}}{m}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(sec(x)^(1+m)*sin(x),x)

[Out]

1/m*(1/cos(x))^m

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Maxima [A]  time = 0.946384, size = 14, normalized size = 1.75 \begin{align*} \frac{\cos \left (x\right )^{-m}}{m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^(1+m)*sin(x),x, algorithm="maxima")

[Out]

cos(x)^(-m)/m

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Fricas [A]  time = 2.11288, size = 39, normalized size = 4.88 \begin{align*} \frac{\frac{1}{\cos \left (x\right )}^{m + 1} \cos \left (x\right )}{m} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^(1+m)*sin(x),x, algorithm="fricas")

[Out]

(1/cos(x))^(m + 1)*cos(x)/m

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sin{\left (x \right )} \sec ^{m + 1}{\left (x \right )}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)**(1+m)*sin(x),x)

[Out]

Integral(sin(x)*sec(x)**(m + 1), x)

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \sec \left (x\right )^{m + 1} \sin \left (x\right )\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(sec(x)^(1+m)*sin(x),x, algorithm="giac")

[Out]

integrate(sec(x)^(m + 1)*sin(x), x)