Optimal. Leaf size=69 \[ \frac{\sqrt{2} 7^n \cos (c+d x) F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};\frac{3}{7} (\sin (c+d x)+1),\frac{1}{2} (\sin (c+d x)+1)\right )}{d \sqrt{1-\sin (c+d x)}} \]
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Rubi [A] time = 0.0339597, antiderivative size = 69, 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} \[ \frac{\sqrt{2} 7^n \cos (c+d x) F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};\frac{3}{7} (\sin (c+d x)+1),\frac{1}{2} (\sin (c+d x)+1)\right )}{d \sqrt{1-\sin (c+d x)}} \]
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} 7^n F_1\left (\frac{1}{2};-n,\frac{1}{2};\frac{3}{2};\frac{3}{7} (1+\sin (c+d x)),\frac{1}{2} (1+\sin (c+d x))\right ) \cos (c+d x)}{d \sqrt{1-\sin (c+d x)}}\\ \end{align*}
Mathematica [A] time = 0.128832, size = 96, normalized size = 1.39 \[ -\frac{\sqrt{\sin (c+d x)-1} \sqrt{\sin (c+d x)+1} \sec (c+d x) (4-3 \sin (c+d x))^{n+1} F_1\left (n+1;\frac{1}{2},\frac{1}{2};n+2;\frac{1}{7} (4-3 \sin (c+d x)),4-3 \sin (c+d x)\right )}{\sqrt{7} d (n+1)} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 0.369, 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 (4 - 3 \sin{\left (c + d x \right )}\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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