\(\int \csc (c+d x) (a+a \sec (c+d x))^n \, dx\) [149]

Optimal result

Integrand size = 19, antiderivative size = 40 \[ \int \csc (c+d x) (a+a \sec (c+d x))^n \, dx=-\frac {\operatorname {Hypergeometric2F1}\left (1,n,1+n,\frac {1}{2} (1+\sec (c+d x))\right ) (a+a \sec (c+d x))^n}{2 d n} \]

[Out]

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

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 40, 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, 70} \[ \int \csc (c+d x) (a+a \sec (c+d x))^n \, dx=-\frac {(a \sec (c+d x)+a)^n \operatorname {Hypergeometric2F1}\left (1,n,n+1,\frac {1}{2} (\sec (c+d x)+1)\right )}{2 d n} \]

[In]

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

[Out]

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

Rule 70

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

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

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(92\) vs. \(2(40)=80\).

Time = 0.51 (sec) , antiderivative size = 92, normalized size of antiderivative = 2.30 \[ \int \csc (c+d x) (a+a \sec (c+d x))^n \, dx=\frac {2^{-1+n} \operatorname {Hypergeometric2F1}\left (1,1-n,2-n,\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^{-1+n} (1+\sec (c+d x))^{-n} (a (1+\sec (c+d x)))^n}{d (-1+n)} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

Timed out. \[ \int \csc (c+d x) (a+a \sec (c+d x))^n \, dx=\int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n}{\sin \left (c+d\,x\right )} \,d x \]

[In]

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

[Out]

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