Optimal. Leaf size=80 \[ -\frac {F_1\left (\frac {1}{2};-n,2;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{2 f \sqrt {1+\sin (e+f x)}} \]
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Rubi [A]
time = 0.09, antiderivative size = 80, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {2865, 2864,
129, 440} \begin {gather*} -\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 f \sqrt {\sin (e+f x)+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 129
Rule 440
Rule 2864
Rule 2865
Rubi steps
\begin {align*} \int \frac {(d \sin (e+f x))^n}{(1+\sin (e+f x))^{3/2}} \, dx &=\left (\sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \int \frac {\sin ^n(e+f x)}{(1+\sin (e+f x))^{3/2}} \, dx\\ &=-\frac {\left (\cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \text {Subst}\left (\int \frac {(1-x)^n}{(2-x)^2 \sqrt {x}} \, dx,x,1-\sin (e+f x)\right )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\sin (e+f x)}}\\ &=-\frac {\left (2 \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \text {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 )}{f \sqrt {1-\sin (e+f x)} \sqrt {1+\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 f \sqrt {1+\sin (e+f x)}}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(265\) vs. \(2(80)=160\).
time = 0.62, size = 265, normalized size = 3.31 \begin {gather*} \frac {\sec (e+f x) (d \sin (e+f x))^n \left (F_1\left (1;\frac {1}{2},-n;2;\frac {1}{2} (1+\sin (e+f x)),1+\sin (e+f x)\right ) \sqrt {2-2 \sin (e+f x)} (-\sin (e+f x))^{-n} (1+\sin (e+f x))^2-\frac {4 (1+\sin (e+f x)) \sqrt {1-\frac {2}{1+\sin (e+f x)}} \left (1-\frac {1}{1+\sin (e+f x)}\right )^{-n} \left (2 (1+2 n) F_1\left (\frac {1}{2}-n;-\frac {1}{2},-n;\frac {3}{2}-n;\frac {2}{1+\sin (e+f x)},\frac {1}{1+\sin (e+f x)}\right )+(-1+2 n) F_1\left (-\frac {1}{2}-n;-\frac {1}{2},-n;\frac {1}{2}-n;\frac {2}{1+\sin (e+f x)},\frac {1}{1+\sin (e+f x)}\right ) (1+\sin (e+f x))\right )}{-1+4 n^2}\right )}{8 f \sqrt {1+\sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.07, size = 0, normalized size = 0.00 \[\int \frac {\left (d \sin \left (f x +e \right )\right )^{n}}{\left (1+\sin \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx\]
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {\left (d \sin {\left (e + f x \right )}\right )^{n}}{\left (\sin {\left (e + f x \right )} + 1\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
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 \begin {gather*} \int \frac {{\left (d\,\sin \left (e+f\,x\right )\right )}^n}{{\left (\sin \left (e+f\,x\right )+1\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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