Optimal. Leaf size=85 \[ -\frac {\cos (e+f x) \sin ^{-n}(e+f x) F_1\left (\frac {1}{2};-n,2;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) (d \sin (e+f x))^n}{2 a f \sqrt {a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.20, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2787, 2786, 2785, 130, 429} \[ -\frac {\cos (e+f x) \sin ^{-n}(e+f x) F_1\left (\frac {1}{2};-n,2;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) (d \sin (e+f x))^n}{2 a f \sqrt {a \sin (e+f x)+a}} \]
Antiderivative was successfully verified.
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Rule 130
Rule 429
Rule 2785
Rule 2786
Rule 2787
Rubi steps
\begin {align*} \int \frac {(d \sin (e+f x))^n}{(a+a \sin (e+f x))^{3/2}} \, dx &=\frac {\sqrt {1+\sin (e+f x)} \int \frac {(d \sin (e+f x))^n}{(1+\sin (e+f x))^{3/2}} \, dx}{a \sqrt {a+a \sin (e+f x)}}\\ &=\frac {\left (\sin ^{-n}(e+f x) (d \sin (e+f x))^n \sqrt {1+\sin (e+f x)}\right ) \int \frac {\sin ^n(e+f x)}{(1+\sin (e+f x))^{3/2}} \, dx}{a \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {\left (\cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \operatorname {Subst}\left (\int \frac {(1-x)^n}{(2-x)^2 \sqrt {x}} \, dx,x,1-\sin (e+f x)\right )}{a f \sqrt {1-\sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {\left (2 \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \operatorname {Subst}\left (\int \frac {\left (1-x^2\right )^n}{\left (2-x^2\right )^2} \, dx,x,\sqrt {1-\sin (e+f x)}\right )}{a f \sqrt {1-\sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {F_1\left (\frac {1}{2};-n,2;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{2 a f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
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Mathematica [B] time = 1.26, size = 276, normalized size = 3.25 \[ \frac {\sec (e+f x) (d \sin (e+f x))^n \left (a^2 \sqrt {2-2 \sin (e+f x)} (\sin (e+f x)+1)^2 (-\sin (e+f x))^{-n} F_1\left (1;\frac {1}{2},-n;2;\frac {1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )-\frac {4 a (\sin (e+f x)-1) \left (1-\frac {1}{\sin (e+f x)+1}\right )^{-n} \left (2 a (2 n+1) F_1\left (\frac {1}{2}-n;-\frac {1}{2},-n;\frac {3}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {1}{\sin (e+f x)+1}\right )+a (2 n-1) (\sin (e+f x)+1) F_1\left (-n-\frac {1}{2};-\frac {1}{2},-n;\frac {1}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {1}{\sin (e+f x)+1}\right )\right )}{\left (4 n^2-1\right ) \sqrt {1-\frac {2}{\sin (e+f x)+1}}}\right )}{8 a^3 f \sqrt {a (\sin (e+f x)+1)}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.53, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {a \sin \left (f x + e\right ) + a} \left (d \sin \left (f x + e\right )\right )^{n}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \]
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 \[ \int \frac {\left (d \sin \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.22, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sin \left (f x +e \right )\right )^{n}}{\left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {\left (d \sin \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (d\,\sin \left (e+f\,x\right )\right )}^n}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (d \sin {\left (e + f x \right )}\right )^{n}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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