3.494 \(\int (b \sec (e+f x))^n \sin (e+f x) \, dx\)

Optimal. Leaf size=25 \[ -\frac{b (b \sec (e+f x))^{n-1}}{f (1-n)} \]

[Out]

-((b*(b*Sec[e + f*x])^(-1 + n))/(f*(1 - n)))

________________________________________________________________________________________

Rubi [A]  time = 0.0333282, antiderivative size = 25, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118, Rules used = {2622, 30} \[ -\frac{b (b \sec (e+f x))^{n-1}}{f (1-n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((b*(b*Sec[e + f*x])^(-1 + n))/(f*(1 - n)))

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

Mathematica [A]  time = 0.0211982, size = 22, normalized size = 0.88 \[ \frac{b (b \sec (e+f x))^{n-1}}{f (n-1)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*(b*Sec[e + f*x])^(-1 + n))/(f*(-1 + n))

________________________________________________________________________________________

Maple [B]  time = 0.029, size = 120, normalized size = 4.8 \begin{align*}{ \left ({\frac{1}{f \left ( -1+n \right ) }{{\rm e}^{n\ln \left ({b \left ( 1+ \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) \left ( 1- \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) ^{-1}} \right ) }}}-{\frac{1}{f \left ( -1+n \right ) } \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2}{{\rm e}^{n\ln \left ({b \left ( 1+ \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) \left ( 1- \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) ^{-1}} \right ) }}} \right ) \left ( 1+ \left ( \tan \left ({\frac{fx}{2}}+{\frac{e}{2}} \right ) \right ) ^{2} \right ) ^{-1}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(1/f/(-1+n)*exp(n*ln(b*(1+tan(1/2*f*x+1/2*e)^2)/(1-tan(1/2*f*x+1/2*e)^2)))-1/f/(-1+n)*tan(1/2*f*x+1/2*e)^2*exp
(n*ln(b*(1+tan(1/2*f*x+1/2*e)^2)/(1-tan(1/2*f*x+1/2*e)^2))))/(1+tan(1/2*f*x+1/2*e)^2)

________________________________________________________________________________________

Maxima [A]  time = 1.00783, size = 38, normalized size = 1.52 \begin{align*} \frac{b^{n} \cos \left (f x + e\right )^{-n} \cos \left (f x + e\right )}{f{\left (n - 1\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*sec(f*x+e))^n*sin(f*x+e),x, algorithm="maxima")

[Out]

b^n*cos(f*x + e)^(-n)*cos(f*x + e)/(f*(n - 1))

________________________________________________________________________________________

Fricas [A]  time = 1.64906, size = 58, normalized size = 2.32 \begin{align*} \frac{\left (\frac{b}{\cos \left (f x + e\right )}\right )^{n} \cos \left (f x + e\right )}{f n - f} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b/cos(f*x + e))^n*cos(f*x + e)/(f*n - f)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral((b*sec(e + f*x))**n*sin(e + f*x), x)

________________________________________________________________________________________

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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