Optimal. Leaf size=48 \[ \frac {b \, _2F_1\left (2,1+n;2+n;1+\frac {b \sec (c+d x)}{a}\right ) (a+b \sec (c+d x))^{1+n}}{a^2 d (1+n)} \]
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Rubi [A]
time = 0.03, antiderivative size = 48, 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 = {3959, 67}
\begin {gather*} \frac {b (a+b \sec (c+d x))^{n+1} \, _2F_1\left (2,n+1;n+2;\frac {b \sec (c+d x)}{a}+1\right )}{a^2 d (n+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 67
Rule 3959
Rubi steps
\begin {align*} \int (a+b \sec (c+d x))^n \sin (c+d x) \, dx &=-\frac {\text {Subst}\left (\int \frac {(a-b x)^n}{x^2} \, dx,x,-\sec (c+d x)\right )}{d}\\ &=\frac {b \, _2F_1\left (2,1+n;2+n;1+\frac {b \sec (c+d x)}{a}\right ) (a+b \sec (c+d x))^{1+n}}{a^2 d (1+n)}\\ \end {align*}
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Mathematica [A]
time = 0.36, size = 72, normalized size = 1.50 \begin {gather*} \frac {b \cos (c+d x) \, _2F_1\left (2,1-n;2-n;\frac {a \cos (c+d x)}{b+a \cos (c+d x)}\right ) (a+b \sec (c+d x))^n}{d (-1+n) (b+a \cos (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (a +b \sec \left (d x +c \right )\right )^{n} \sin \left (d x +c \right )\, dx\]
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 + b \sec {\left (c + d x \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 = 1.39, size = 73, normalized size = 1.52 \begin {gather*} \frac {\cos \left (c+d\,x\right )\,{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^n\,{{}}_2{\mathrm {F}}_1\left (1-n,-n;\ 2-n;\ -\frac {a\,\cos \left (c+d\,x\right )}{b}\right )}{d\,{\left (\frac {a\,\cos \left (c+d\,x\right )}{b}+1\right )}^n\,\left (n-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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