Optimal. Leaf size=424 \[ -\frac {\cot ^3(c+d x) (\sec (c+d x)+1)^{3/2} (a+b \sec (c+d x))^n \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{-n} F_1\left (-\frac {3}{2};\frac {5}{2},-n;-\frac {1}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{6 \sqrt {2} d}-\frac {3 \cot (c+d x) \sqrt {\sec (c+d x)+1} (a+b \sec (c+d x))^n \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{-n} F_1\left (-\frac {1}{2};\frac {5}{2},-n;\frac {1}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{2 \sqrt {2} d}+\frac {\tan (c+d x) (a+b \sec (c+d x))^n \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{-n} F_1\left (\frac {1}{2};\frac {3}{2},-n;\frac {3}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{\sqrt {2} d \sqrt {\sec (c+d x)+1}}+\frac {\tan (c+d x) (a+b \sec (c+d x))^n \left (\frac {a+b \sec (c+d x)}{a+b}\right )^{-n} F_1\left (\frac {1}{2};\frac {5}{2},-n;\frac {3}{2};\frac {1}{2} (1-\sec (c+d x)),\frac {b (1-\sec (c+d x))}{a+b}\right )}{2 \sqrt {2} d \sqrt {\sec (c+d x)+1}} \]
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Rubi [F] time = 0.04, 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 = {} \[ \int \csc ^4(c+d x) (a+b \sec (c+d x))^n \, dx \]
Verification is Not applicable to the result.
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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] time = 23.72, size = 6403, normalized size = 15.10 \[ \text {Result too large to show} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.27, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.26, 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.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n}{{\sin \left (c+d\,x\right )}^4} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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