Optimal. Leaf size=95 \[ \frac {\sqrt {2} 7^n \cos (c+d x) (4-3 \sin (c+d x))^{-n} (3 \sin (c+d x)-4)^n F_1\left (\frac {1}{2};-n,\frac {1}{2};\frac {3}{2};\frac {3}{7} (\sin (c+d x)+1),\frac {1}{2} (\sin (c+d x)+1)\right )}{d \sqrt {1-\sin (c+d x)}} \]
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Rubi [A] time = 0.05, antiderivative size = 95, 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} \[ \frac {\sqrt {2} 7^n \cos (c+d x) (4-3 \sin (c+d x))^{-n} (3 \sin (c+d x)-4)^n F_1\left (\frac {1}{2};-n,\frac {1}{2};\frac {3}{2};\frac {3}{7} (\sin (c+d x)+1),\frac {1}{2} (\sin (c+d x)+1)\right )}{d \sqrt {1-\sin (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 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 (\cos (c+d x) (4-3 \sin (c+d x))^{-n} (-4+3 \sin (c+d x))^n\right ) \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) (4-3 \sin (c+d x))^{-n} (-4+3 \sin (c+d x))^n}{d \sqrt {1-\sin (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.12, size = 95, normalized size = 1.00 \[ \frac {\sqrt {\sin (c+d x)-1} \sqrt {\sin (c+d x)+1} \sec (c+d x) (3 \sin (c+d x)-4)^{n+1} F_1\left (n+1;\frac {1}{2},\frac {1}{2};n+2;\frac {1}{7} (4-3 \sin (c+d x)),4-3 \sin (c+d x)\right )}{\sqrt {7} d (n+1)} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.87, 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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