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

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

[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)

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Rubi [F]
time = 0.03, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx \end {gather*}

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(5928\) vs. \(2(424)=848\).
time = 21.38, size = 5928, normalized size = 13.98 \begin {gather*} \text {Result too large to show} \end {gather*}

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 = 0.10, size = 0, normalized size = 0.00 \[\int \left (\csc ^{4}\left (d x +c \right )\right ) \left (a +b \sec \left (d x +c \right )\right )^{n}\, dx\]

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

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)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n}{{\sin \left (c+d\,x\right )}^4} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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)

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