Optimal. Leaf size=72 \[ -\frac {\sqrt {2} 7^n \cos (c+d x) F_1\left (\frac {1}{2};\frac {1}{2},-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x)),\frac {4}{7} (1-\sin (c+d x))\right )}{d \sqrt {\sin (c+d x)+1}} \]
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Rubi [A] time = 0.04, antiderivative size = 72, 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} \[ -\frac {\sqrt {2} 7^n \cos (c+d x) F_1\left (\frac {1}{2};\frac {1}{2},-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x)),\frac {4}{7} (1-\sin (c+d x))\right )}{d \sqrt {\sin (c+d x)+1}} \]
Antiderivative was successfully verified.
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Rule 138
Rule 2665
Rubi steps
\begin {align*} \int (3+4 \sin (c+d x))^n \, dx &=\frac {\cos (c+d x) \operatorname {Subst}\left (\int \frac {(3+4 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};\frac {1}{2},-n;\frac {3}{2};\frac {1}{2} (1-\sin (c+d x)),\frac {4}{7} (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 = 0.16, size = 83, normalized size = 1.15 \[ \frac {\sqrt {\cos ^2(c+d x)} \sec (c+d x) (4 \sin (c+d x)+3)^{n+1} F_1\left (n+1;\frac {1}{2},\frac {1}{2};n+2;-4 \sin (c+d x)-3,\frac {1}{7} (4 \sin (c+d x)+3)\right )}{\sqrt {7} d (n+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.83, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (4 \, \sin \left (d x + c\right ) + 3\right )}^{n}, x\right ) \]
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 \[ \int {\left (4 \, \sin \left (d x + c\right ) + 3\right )}^{n}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.66, size = 0, normalized size = 0.00 \[ \int \left (3+4 \sin \left (d x +c \right )\right )^{n}\, dx \]
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 \[ \int {\left (4 \, \sin \left (d x + c\right ) + 3\right )}^{n}\,{d x} \]
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 \[ \int {\left (4\,\sin \left (c+d\,x\right )+3\right )}^n \,d x \]
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 \[ \int \left (4 \sin {\left (c + d x \right )} + 3\right )^{n}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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