Optimal. Leaf size=64 \[ \frac{\sqrt{2} \cos (c+d x) F_1\left (\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1),-3 (\sin (c+d x)+1)\right )}{d \sqrt{1-\sin (c+d x)}} \]
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Rubi [A] time = 0.0346196, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {2665, 138} \[ \frac{\sqrt{2} \cos (c+d x) F_1\left (\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (\sin (c+d x)+1),-3 (\sin (c+d x)+1)\right )}{d \sqrt{1-\sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 2665
Rule 138
Rubi steps
\begin{align*} \int (4+3 \sin (c+d x))^n \, dx &=\frac{\cos (c+d x) \operatorname{Subst}\left (\int \frac{(4+3 x)^n}{\sqrt{1-x} \sqrt{1+x}} \, dx,x,\sin (c+d x)\right )}{d \sqrt{1-\sin (c+d x)} \sqrt{1+\sin (c+d x)}}\\ &=\frac{\sqrt{2} F_1\left (\frac{1}{2};\frac{1}{2},-n;\frac{3}{2};\frac{1}{2} (1+\sin (c+d x)),-3 (1+\sin (c+d x))\right ) \cos (c+d x)}{d \sqrt{1-\sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.125495, size = 99, normalized size = 1.55 \[ \frac{\sqrt{-\sin (c+d x)-1} \sqrt{1-\sin (c+d x)} \sec (c+d x) (3 \sin (c+d x)+4)^{n+1} F_1\left (n+1;\frac{1}{2},\frac{1}{2};n+2;3 \sin (c+d x)+4,\frac{1}{7} (3 \sin (c+d x)+4)\right )}{\sqrt{7} d (n+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.346, size = 0, normalized size = 0. \begin{align*} \int \left ( 4+3\,\sin \left ( dx+c \right ) \right ) ^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (3 \, \sin \left (d x + c\right ) + 4\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left ({\left (3 \, \sin \left (d x + c\right ) + 4\right )}^{n}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \left (3 \sin{\left (c + d x \right )} + 4\right )^{n}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (3 \, \sin \left (d x + c\right ) + 4\right )}^{n}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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