Optimal. Leaf size=110 \[ -\frac{\sqrt{-\sin (c+d x)-1} \cos (c+d x) (-3 \sin (c+d x)-4)^{n+1} F_1\left (n+1;\frac{1}{2},\frac{1}{2};n+2;3 \sin (c+d x)+4,\frac{1}{7} (3 \sin (c+d x)+4)\right )}{\sqrt{7} d (n+1) \sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)} \]
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Rubi [A] time = 0.0665753, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25, Rules used = {2665, 139, 138} \[ -\frac{\sqrt{-\sin (c+d x)-1} \cos (c+d x) (-3 \sin (c+d x)-4)^{n+1} F_1\left (n+1;\frac{1}{2},\frac{1}{2};n+2;3 \sin (c+d x)+4,\frac{1}{7} (3 \sin (c+d x)+4)\right )}{\sqrt{7} d (n+1) \sqrt{1-\sin (c+d x)} (\sin (c+d x)+1)} \]
Antiderivative was successfully verified.
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Rule 2665
Rule 139
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{\left (\sqrt{3} \cos (c+d x) \sqrt{-1-\sin (c+d x)}\right ) \operatorname{Subst}\left (\int \frac{(-4-3 x)^n}{\sqrt{-3-3 x} \sqrt{1-x}} \, dx,x,\sin (c+d x)\right )}{d \sqrt{1-\sin (c+d x)} (1+\sin (c+d x))}\\ &=-\frac{F_1\left (1+n;\frac{1}{2},\frac{1}{2};2+n;4+3 \sin (c+d x),\frac{1}{7} (4+3 \sin (c+d x))\right ) \cos (c+d x) (-4-3 \sin (c+d x))^{1+n} \sqrt{-1-\sin (c+d x)}}{\sqrt{7} d (1+n) \sqrt{1-\sin (c+d x)} (1+\sin (c+d x))}\\ \end{align*}
Mathematica [A] time = 0.133339, size = 100, normalized size = 0.91 \[ -\frac{\sqrt{-\sin (c+d x)-1} \sqrt{1-\sin (c+d x)} \sec (c+d x) (-3 \sin (c+d x)-4)^{n+1} F_1\left (n+1;\frac{1}{2},\frac{1}{2};n+2;3 \sin (c+d x)+4,\frac{1}{7} (3 \sin (c+d x)+4)\right )}{\sqrt{7} d (n+1)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.264, 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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