Optimal. Leaf size=112 \[ -\frac {(n+2) (a \sec (c+d x)+a)^n \, _2F_1\left (1,n;n+1;\frac {1}{2} (\sec (c+d x)+1)\right )}{8 d n}-\frac {a (2-n) (a \sec (c+d x)+a)^{n-1}}{4 d (1-n)}+\frac {a (a \sec (c+d x)+a)^{n-1}}{2 d (1-\sec (c+d x))} \]
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Rubi [A] time = 0.10, antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3873, 89, 79, 68} \[ -\frac {(n+2) (a \sec (c+d x)+a)^n \, _2F_1\left (1,n;n+1;\frac {1}{2} (\sec (c+d x)+1)\right )}{8 d n}-\frac {a (2-n) (a \sec (c+d x)+a)^{n-1}}{4 d (1-n)}+\frac {a (a \sec (c+d x)+a)^{n-1}}{2 d (1-\sec (c+d x))} \]
Antiderivative was successfully verified.
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Rule 68
Rule 79
Rule 89
Rule 3873
Rubi steps
\begin {align*} \int \csc ^3(c+d x) (a+a \sec (c+d x))^n \, dx &=-\frac {a^4 \operatorname {Subst}\left (\int \frac {x^2 (a-a x)^{-2+n}}{(-a-a x)^2} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=\frac {a (a+a \sec (c+d x))^{-1+n}}{2 d (1-\sec (c+d x))}+\frac {\operatorname {Subst}\left (\int \frac {(a-a x)^{-2+n} \left (-a^3 n+2 a^3 x\right )}{-a-a x} \, dx,x,-\sec (c+d x)\right )}{2 d}\\ &=-\frac {a (2-n) (a+a \sec (c+d x))^{-1+n}}{4 d (1-n)}+\frac {a (a+a \sec (c+d x))^{-1+n}}{2 d (1-\sec (c+d x))}-\frac {\left (a^2 (2+n)\right ) \operatorname {Subst}\left (\int \frac {(a-a x)^{-1+n}}{-a-a x} \, dx,x,-\sec (c+d x)\right )}{4 d}\\ &=-\frac {a (2-n) (a+a \sec (c+d x))^{-1+n}}{4 d (1-n)}+\frac {a (a+a \sec (c+d x))^{-1+n}}{2 d (1-\sec (c+d x))}-\frac {(2+n) \, _2F_1\left (1,n;1+n;\frac {1}{2} (1+\sec (c+d x))\right ) (a+a \sec (c+d x))^n}{8 d n}\\ \end {align*}
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Mathematica [A] time = 2.17, size = 179, normalized size = 1.60 \[ \frac {\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right ) (\sec (c+d x)+1)^{-n} (a (\sec (c+d x)+1))^n \left (2^{n+1} \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^n \, _2F_1\left (1,1-n;2-n;\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )+2^n \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^n \, _2F_1\left (2,1-n;2-n;\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right )+(\sec (c+d x)+1)^n\right )}{8 d (n-1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 1.89, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{3}, x\right ) \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 1.40, size = 0, normalized size = 0.00 \[ \int \left (\csc ^{3}\left (d x +c \right )\right ) \left (a +a \sec \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int {\left (a \sec \left (d x + c\right ) + a\right )}^{n} \csc \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n}{{\sin \left (c+d\,x\right )}^3} \,d x \]
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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