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 = 0.07, antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, 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 138
Rule 139
Rule 2665
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*}
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Mathematica [A] time = 0.14, 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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fricas [F] time = 0.64, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (-3 \, \sin \left (d x + c\right ) - 4\right )}^{n}, x\right ) \]
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 \[ \int {\left (-3 \, \sin \left (d x + c\right ) - 4\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.44, size = 0, normalized size = 0.00 \[ \int \left (-4-3 \sin \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int {\left (-3 \, \sin \left (d x + c\right ) - 4\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\left (-3\,\sin \left (c+d\,x\right )-4\right )}^n \,d x \]
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 \[ \int \left (- 3 \sin {\left (c + d x \right )} - 4\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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