\(\int (a+a \sec (c+d x))^n \sin (c+d x) \, dx\) [148]

Optimal result

Integrand size = 19, antiderivative size = 42 \[ \int (a+a \sec (c+d x))^n \sin (c+d x) \, dx=\frac {\operatorname {Hypergeometric2F1}(2,1+n,2+n,1+\sec (c+d x)) (a+a \sec (c+d x))^{1+n}}{a d (1+n)} \]

[Out]

hypergeom([2, 1+n],[2+n],1+sec(d*x+c))*(a+a*sec(d*x+c))^(1+n)/a/d/(1+n)

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3958, 67} \[ \int (a+a \sec (c+d x))^n \sin (c+d x) \, dx=\frac {(a \sec (c+d x)+a)^{n+1} \operatorname {Hypergeometric2F1}(2,n+1,n+2,\sec (c+d x)+1)}{a d (n+1)} \]

[In]

Int[(a + a*Sec[c + d*x])^n*Sin[c + d*x],x]

[Out]

(Hypergeometric2F1[2, 1 + n, 2 + n, 1 + Sec[c + d*x]]*(a + a*Sec[c + d*x])^(1 + n))/(a*d*(1 + n))

Rule 67

Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x)^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))
*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] &&  !IntegerQ[n] && (Intege
rQ[m] || GtQ[-d/(b*c), 0])

Rule 3958

Int[cos[(e_.) + (f_.)*(x_)]^(p_.)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Dist[-(f*b^(p - 1)
)^(-1), Subst[Int[(-a + b*x)^((p - 1)/2)*((a + b*x)^(m + (p - 1)/2)/x^(p + 1)), x], x, Csc[e + f*x]], x] /; Fr
eeQ[{a, b, e, f, m}, x] && IntegerQ[(p - 1)/2] && EqQ[a^2 - b^2, 0]

Rubi steps \begin{align*} \text {integral}& = -\frac {\text {Subst}\left (\int \frac {(a-a x)^n}{x^2} \, dx,x,-\sec (c+d x)\right )}{d} \\ & = \frac {\operatorname {Hypergeometric2F1}(2,1+n,2+n,1+\sec (c+d x)) (a+a \sec (c+d x))^{1+n}}{a d (1+n)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.04 (sec) , antiderivative size = 42, normalized size of antiderivative = 1.00 \[ \int (a+a \sec (c+d x))^n \sin (c+d x) \, dx=\frac {\operatorname {Hypergeometric2F1}(2,1+n,2+n,1+\sec (c+d x)) (a (1+\sec (c+d x)))^{1+n}}{a d (1+n)} \]

[In]

Integrate[(a + a*Sec[c + d*x])^n*Sin[c + d*x],x]

[Out]

(Hypergeometric2F1[2, 1 + n, 2 + n, 1 + Sec[c + d*x]]*(a*(1 + Sec[c + d*x]))^(1 + n))/(a*d*(1 + n))

Maple [F]

\[\int \left (a +a \sec \left (d x +c \right )\right )^{n} \sin \left (d x +c \right )d x\]

[In]

int((a+a*sec(d*x+c))^n*sin(d*x+c),x)

[Out]

int((a+a*sec(d*x+c))^n*sin(d*x+c),x)

Fricas [F]

\[ \int (a+a \sec (c+d x))^n \sin (c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right ) \,d x } \]

[In]

integrate((a+a*sec(d*x+c))^n*sin(d*x+c),x, algorithm="fricas")

[Out]

integral((a*sec(d*x + c) + a)^n*sin(d*x + c), x)

Sympy [F]

\[ \int (a+a \sec (c+d x))^n \sin (c+d x) \, dx=\int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \sin {\left (c + d x \right )}\, dx \]

[In]

integrate((a+a*sec(d*x+c))**n*sin(d*x+c),x)

[Out]

Integral((a*(sec(c + d*x) + 1))**n*sin(c + d*x), x)

Maxima [F]

\[ \int (a+a \sec (c+d x))^n \sin (c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right ) \,d x } \]

[In]

integrate((a+a*sec(d*x+c))^n*sin(d*x+c),x, algorithm="maxima")

[Out]

integrate((a*sec(d*x + c) + a)^n*sin(d*x + c), x)

Giac [F]

\[ \int (a+a \sec (c+d x))^n \sin (c+d x) \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \sin \left (d x + c\right ) \,d x } \]

[In]

integrate((a+a*sec(d*x+c))^n*sin(d*x+c),x, algorithm="giac")

[Out]

integrate((a*sec(d*x + c) + a)^n*sin(d*x + c), x)

Mupad [B] (verification not implemented)

Time = 14.43 (sec) , antiderivative size = 64, normalized size of antiderivative = 1.52 \[ \int (a+a \sec (c+d x))^n \sin (c+d x) \, dx=\frac {\cos \left (c+d\,x\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n\,{{}}_2{\mathrm {F}}_1\left (1-n,-n;\ 2-n;\ -\cos \left (c+d\,x\right )\right )}{d\,{\left (\cos \left (c+d\,x\right )+1\right )}^n\,\left (n-1\right )} \]

[In]

int(sin(c + d*x)*(a + a/cos(c + d*x))^n,x)

[Out]

(cos(c + d*x)*(a + a/cos(c + d*x))^n*hypergeom([1 - n, -n], 2 - n, -cos(c + d*x)))/(d*(cos(c + d*x) + 1)^n*(n
- 1))