Optimal. Leaf size=42 \[ \frac {\, _2F_1(2,1+n;2+n;1+\sec (c+d x)) (a+a \sec (c+d x))^{1+n}}{a d (1+n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 42, 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, 67}
\begin {gather*} \frac {(a \sec (c+d x)+a)^{n+1} \, _2F_1(2,n+1;n+2;\sec (c+d x)+1)}{a d (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 3958
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^n \sin (c+d x) \, dx &=-\frac {\text {Subst}\left (\int \frac {(a-a x)^n}{x^2} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=\frac {\, _2F_1(2,1+n;2+n;1+\sec (c+d x)) (a+a \sec (c+d x))^{1+n}}{a d (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 42, normalized size = 1.00 \begin {gather*} \frac {\, _2F_1(2,1+n;2+n;1+\sec (c+d x)) (a (1+\sec (c+d x)))^{1+n}}{a d (1+n)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.04, size = 0, normalized size = 0.00 \[\int \left (a +a \sec \left (d x +c \right )\right )^{n} \sin \left (d x +c \right )\, dx\]
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} \sin {\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 [B]
time = 1.18, size = 64, normalized size = 1.52 \begin {gather*} \frac {\cos \left (c+d\,x\right )\,{\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right )}^n\,{{}}_2{\mathrm {F}}_1\left (1-n,-n;\ 2-n;\ -\cos \left (c+d\,x\right )\right )}{d\,{\left (\cos \left (c+d\,x\right )+1\right )}^n\,\left (n-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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