\(\int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx\) [276]

Optimal result

Integrand size = 21, antiderivative size = 424 \[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx=-\frac {3 \operatorname {AppellF1}\left (-\frac {1}{2},\frac {5}{2},-n,\frac {1}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \cot (c+d x) \sqrt {1+\sec (c+d x)} (a+b \sec (c+d x))^n \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{-n}}{2 \sqrt {2} d}-\frac {\operatorname {AppellF1}\left (-\frac {3}{2},\frac {5}{2},-n,-\frac {1}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) \cot ^3(c+d x) (1+\sec (c+d x))^{3/2} (a+b \sec (c+d x))^n \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{-n}}{6 \sqrt {2} d}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},\frac {3}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^n \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{-n} \tan (c+d x)}{\sqrt {2} d \sqrt {1+\sec (c+d x)}}+\frac {\operatorname {AppellF1}\left (\frac {1}{2},\frac {5}{2},-n,\frac {3}{2},\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right ) (a+b \sec (c+d x))^n \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{-n} \tan (c+d x)}{2 \sqrt {2} d \sqrt {1+\sec (c+d x)}} \]

[Out]

-1/12*AppellF1(-3/2,-n,5/2,-1/2,b*(1-sec(d*x+c))/(a+b),1/2-1/2*sec(d*x+c))*cot(d*x+c)^3*(1+sec(d*x+c))^(3/2)*(
a+b*sec(d*x+c))^n/d/(((a+b*sec(d*x+c))/(a+b))^n)*2^(1/2)-3/4*AppellF1(-1/2,-n,5/2,1/2,b*(1-sec(d*x+c))/(a+b),1
/2-1/2*sec(d*x+c))*cot(d*x+c)*(a+b*sec(d*x+c))^n*(1+sec(d*x+c))^(1/2)/d/(((a+b*sec(d*x+c))/(a+b))^n)*2^(1/2)+1
/2*AppellF1(1/2,-n,3/2,3/2,b*(1-sec(d*x+c))/(a+b),1/2-1/2*sec(d*x+c))*(a+b*sec(d*x+c))^n*tan(d*x+c)/d/(((a+b*s
ec(d*x+c))/(a+b))^n)*2^(1/2)/(1+sec(d*x+c))^(1/2)+1/4*AppellF1(1/2,-n,5/2,3/2,b*(1-sec(d*x+c))/(a+b),1/2-1/2*s
ec(d*x+c))*(a+b*sec(d*x+c))^n*tan(d*x+c)/d/(((a+b*sec(d*x+c))/(a+b))^n)*2^(1/2)/(1+sec(d*x+c))^(1/2)

Rubi [F]

\[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx \]

[In]

Int[Csc[c + d*x]^4*(a + b*Sec[c + d*x])^n,x]

[Out]

Defer[Int][Csc[c + d*x]^4*(a + b*Sec[c + d*x])^n, x]

Rubi steps \begin{align*} \text {integral}& = \int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx \\ \end{align*}

Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(5928\) vs. \(2(424)=848\).

Time = 23.44 (sec) , antiderivative size = 5928, normalized size of antiderivative = 13.98 \[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\text {Result too large to show} \]

[In]

Integrate[Csc[c + d*x]^4*(a + b*Sec[c + d*x])^n,x]

[Out]

Result too large to show

Maple [F]

\[\int \csc \left (d x +c \right )^{4} \left (a +b \sec \left (d x +c \right )\right )^{n}d x\]

[In]

int(csc(d*x+c)^4*(a+b*sec(d*x+c))^n,x)

[Out]

int(csc(d*x+c)^4*(a+b*sec(d*x+c))^n,x)

Fricas [F]

\[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(csc(d*x+c)^4*(a+b*sec(d*x+c))^n,x, algorithm="fricas")

[Out]

integral((b*sec(d*x + c) + a)^n*csc(d*x + c)^4, x)

Sympy [F(-1)]

Timed out. \[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\text {Timed out} \]

[In]

integrate(csc(d*x+c)**4*(a+b*sec(d*x+c))**n,x)

[Out]

Timed out

Maxima [F]

\[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(csc(d*x+c)^4*(a+b*sec(d*x+c))^n,x, algorithm="maxima")

[Out]

integrate((b*sec(d*x + c) + a)^n*csc(d*x + c)^4, x)

Giac [F]

\[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\int { {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4} \,d x } \]

[In]

integrate(csc(d*x+c)^4*(a+b*sec(d*x+c))^n,x, algorithm="giac")

[Out]

integrate((b*sec(d*x + c) + a)^n*csc(d*x + c)^4, x)

Mupad [F(-1)]

Timed out. \[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx=\int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n}{{\sin \left (c+d\,x\right )}^4} \,d x \]

[In]

int((a + b/cos(c + d*x))^n/sin(c + d*x)^4,x)

[Out]

int((a + b/cos(c + d*x))^n/sin(c + d*x)^4, x)