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

Optimal. Leaf size=424 \[ -\frac{\cot ^3(c+d x) (\sec (c+d x)+1)^{3/2} (a+b \sec (c+d x))^n \left (\frac{a+b \sec (c+d x)}{a+b}\right )^{-n} F_1\left (-\frac{3}{2};\frac{5}{2},-n;-\frac{1}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right )}{6 \sqrt{2} d}-\frac{3 \cot (c+d x) \sqrt{\sec (c+d x)+1} (a+b \sec (c+d x))^n \left (\frac{a+b \sec (c+d x)}{a+b}\right )^{-n} F_1\left (-\frac{1}{2};\frac{5}{2},-n;\frac{1}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right )}{2 \sqrt{2} d}+\frac{\tan (c+d x) (a+b \sec (c+d x))^n \left (\frac{a+b \sec (c+d x)}{a+b}\right )^{-n} F_1\left (\frac{1}{2};\frac{3}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right )}{\sqrt{2} d \sqrt{\sec (c+d x)+1}}+\frac{\tan (c+d x) (a+b \sec (c+d x))^n \left (\frac{a+b \sec (c+d x)}{a+b}\right )^{-n} F_1\left (\frac{1}{2};\frac{5}{2},-n;\frac{3}{2};\frac{1}{2} (1-\sec (c+d x)),\frac{b (1-\sec (c+d x))}{a+b}\right )}{2 \sqrt{2} d \sqrt{\sec (c+d x)+1}} \]

[Out]

(-3*AppellF1[-1/2, 5/2, -n, 1/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*Cot[c + d*x]*Sqrt[1 + S
ec[c + d*x]]*(a + b*Sec[c + d*x])^n)/(2*Sqrt[2]*d*((a + b*Sec[c + d*x])/(a + b))^n) - (AppellF1[-3/2, 5/2, -n,
-1/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a + b)]*Cot[c + d*x]^3*(1 + Sec[c + d*x])^(3/2)*(a + b*Se
c[c + d*x])^n)/(6*Sqrt[2]*d*((a + b*Sec[c + d*x])/(a + b))^n) + (AppellF1[1/2, 3/2, -n, 3/2, (1 - Sec[c + d*x]
)/2, (b*(1 - Sec[c + d*x]))/(a + b)]*(a + b*Sec[c + d*x])^n*Tan[c + d*x])/(Sqrt[2]*d*Sqrt[1 + Sec[c + d*x]]*((
a + b*Sec[c + d*x])/(a + b))^n) + (AppellF1[1/2, 5/2, -n, 3/2, (1 - Sec[c + d*x])/2, (b*(1 - Sec[c + d*x]))/(a
+ b)]*(a + b*Sec[c + d*x])^n*Tan[c + d*x])/(2*Sqrt[2]*d*Sqrt[1 + Sec[c + d*x]]*((a + b*Sec[c + d*x])/(a + b))
^n)

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Rubi [F]  time = 0.0400988, antiderivative size = 0, normalized size of antiderivative = 0., number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0., Rules used = {} \[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx \]

Verification is Not applicable to the result.

[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*} \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\\ \end{align*}

Mathematica [B]  time = 23.726, size = 6403, normalized size = 15.1 \[ \text{Result too large to show} \]

Warning: Unable to verify antiderivative.

[In]

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

[Out]

Result too large to show

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Maple [F]  time = 0.296, size = 0, normalized size = 0. \begin{align*} \int \left ( \csc \left ( dx+c \right ) \right ) ^{4} \left ( a+b\sec \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (b \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (b \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[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)