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 = 0.03, antiderivative size = 64, normalized size of antiderivative = 1.00, 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 138
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 {\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*}
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Mathematica [A] time = 0.13, 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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fricas [F] time = 0.76, 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 = 0.51, 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 = -0.02 \[ \int {\left (3\,\sin \left (c+d\,x\right )+4\right )}^n \,d x \]
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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