Optimal. Leaf size=104 \[ -\frac {\cos (e+f x) (c+d \sin (e+f x))^n \left (\frac {c+d \sin (e+f x)}{c+d}\right )^{-n} F_1\left (\frac {1}{2};-n,3;\frac {3}{2};\frac {d (1-\sin (e+f x))}{c+d},\frac {1}{2} (1-\sin (e+f x))\right )}{4 a^2 f \sqrt {a \sin (e+f x)+a}} \]
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Rubi [A] time = 0.18, antiderivative size = 137, normalized size of antiderivative = 1.32, number of steps used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2788, 137, 136} \[ -\frac {d^2 \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{n+1} F_1\left (n+1;\frac {1}{2},3;n+2;\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right )}{f (n+1) (c-d)^3 \sqrt {a \sin (e+f x)+a} \left (a^2-a^2 \sin (e+f x)\right )} \]
Warning: Unable to verify antiderivative.
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Rule 136
Rule 137
Rule 2788
Rubi steps
\begin {align*} \int \frac {(c+d \sin (e+f x))^n}{(a+a \sin (e+f x))^{5/2}} \, dx &=\frac {\left (a^2 \cos (e+f x)\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^n}{\sqrt {a-a x} (a+a x)^3} \, dx,x,\sin (e+f x)\right )}{f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=\frac {\left (a^2 \cos (e+f x) \sqrt {\frac {d (a-a \sin (e+f x))}{a c+a d}}\right ) \operatorname {Subst}\left (\int \frac {(c+d x)^n}{(a+a x)^3 \sqrt {\frac {a d}{a c+a d}-\frac {a d x}{a c+a d}}} \, dx,x,\sin (e+f x)\right )}{f (a-a \sin (e+f x)) \sqrt {a+a \sin (e+f x)}}\\ &=-\frac {d^2 F_1\left (1+n;\frac {1}{2},3;2+n;\frac {c+d \sin (e+f x)}{c+d},\frac {c+d \sin (e+f x)}{c-d}\right ) \cos (e+f x) \sqrt {\frac {d (1-\sin (e+f x))}{c+d}} (c+d \sin (e+f x))^{1+n}}{(c-d)^3 f (1+n) \sqrt {a+a \sin (e+f x)} \left (a^2-a^2 \sin (e+f x)\right )}\\ \end {align*}
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Mathematica [B] time = 9.54, size = 414, normalized size = 3.98 \[ \frac {\sec (e+f x) (c+d \sin (e+f x))^n \left (a^3 \sqrt {2-2 \sin (e+f x)} (\sin (e+f x)+1)^3 \left (\frac {c+d \sin (e+f x)}{c-d}\right )^{-n} F_1\left (1;\frac {1}{2},-n;2;\frac {1}{2} (\sin (e+f x)+1),\frac {d (\sin (e+f x)+1)}{d-c}\right )-\frac {4 a^2 (\sin (e+f x)+1) \sqrt {1-\frac {2}{\sin (e+f x)+1}} \left (\frac {c-d}{d \sin (e+f x)+d}+1\right )^{-n} \left (a \left (4 n^2-8 n+3\right ) (\sin (e+f x)+1)^2 F_1\left (-n-\frac {1}{2};-\frac {1}{2},-n;\frac {1}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {d-c}{\sin (e+f x) d+d}\right )+2 (2 n+1) \left (2 a (2 n-1) F_1\left (\frac {3}{2}-n;-\frac {1}{2},-n;\frac {5}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {d-c}{\sin (e+f x) d+d}\right )+a (2 n-3) (\sin (e+f x)+1) F_1\left (\frac {1}{2}-n;-\frac {1}{2},-n;\frac {3}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {d-c}{\sin (e+f x) d+d}\right )\right )\right )}{(2 n-3) (2 n-1) (2 n+1)}\right )}{16 a^4 f (a (\sin (e+f x)+1))^{3/2}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.48, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {a \sin \left (f x + e\right ) + a} {\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]
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 \[ \int \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.25, size = 0, normalized size = 0.00 \[ \int \frac {\left (c +d \sin \left (f x +e \right )\right )^{n}}{\left (a +a \sin \left (f x +e \right )\right )^{\frac {5}{2}}}\, dx \]
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 \[ \int \frac {{\left (d \sin \left (f x + e\right ) + c\right )}^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {{\left (c+d\,\sin \left (e+f\,x\right )\right )}^n}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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