Optimal. Leaf size=40 \[ -\frac {\, _2F_1\left (1,n;1+n;\frac {1}{2} (1+\sec (c+d x))\right ) (a+a \sec (c+d x))^n}{2 d n} \]
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Rubi [A]
time = 0.04, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {3958, 70}
\begin {gather*} -\frac {(a \sec (c+d x)+a)^n \, _2F_1\left (1,n;n+1;\frac {1}{2} (\sec (c+d x)+1)\right )}{2 d n} \end {gather*}
Antiderivative was successfully verified.
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Rule 70
Rule 3958
Rubi steps
\begin {align*} \int \csc (c+d x) (a+a \sec (c+d x))^n \, dx &=-\frac {a^2 \text {Subst}\left (\int \frac {(a-a x)^{-1+n}}{-a-a x} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=-\frac {\, _2F_1\left (1,n;1+n;\frac {1}{2} (1+\sec (c+d x))\right ) (a+a \sec (c+d x))^n}{2 d n}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(92\) vs. \(2(40)=80\).
time = 0.42, size = 92, normalized size = 2.30 \begin {gather*} \frac {2^{-1+n} \, _2F_1\left (1,1-n;2-n;\cos (c+d x) \sec ^2\left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos ^2\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x)\right )^{-1+n} (1+\sec (c+d x))^{-n} (a (1+\sec (c+d x)))^n}{d (-1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.09, size = 0, normalized size = 0.00 \[\int \csc \left (d x +c \right ) \left (a +a \sec \left (d x +c \right )\right )^{n}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (a \left (\sec {\left (c + d x \right )} + 1\right )\right )^{n} \csc {\left (c + d x \right )}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n}{\sin \left (c+d\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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