Optimal. Leaf size=226 \[ -\frac {(-4 A n+A+B (4 n+3)) \cos (e+f x) \sin ^{-n}(e+f x) F_1\left (\frac {1}{2};-n,1;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) (d \sin (e+f x))^n}{4 a f \sqrt {a \sin (e+f x)+a}}-\frac {(2 n+1) (A-B) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right )}{2 a f \sqrt {a \sin (e+f x)+a}}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{2 d f (a \sin (e+f x)+a)^{3/2}} \]
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Rubi [A] time = 0.67, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2978, 2987, 2787, 2786, 2785, 130, 429, 2776, 67, 65} \[ -\frac {(-4 A n+A+B (4 n+3)) \cos (e+f x) \sin ^{-n}(e+f x) F_1\left (\frac {1}{2};-n,1;\frac {3}{2};1-\sin (e+f x),\frac {1}{2} (1-\sin (e+f x))\right ) (d \sin (e+f x))^n}{4 a f \sqrt {a \sin (e+f x)+a}}-\frac {(2 n+1) (A-B) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right )}{2 a f \sqrt {a \sin (e+f x)+a}}+\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{n+1}}{2 d f (a \sin (e+f x)+a)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 65
Rule 67
Rule 130
Rule 429
Rule 2776
Rule 2785
Rule 2786
Rule 2787
Rule 2978
Rule 2987
Rubi steps
\begin {align*} \int \frac {(d \sin (e+f x))^n (A+B \sin (e+f x))}{(a+a \sin (e+f x))^{3/2}} \, dx &=\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {(d \sin (e+f x))^n \left (a d (A+B-A n+B n)+\frac {1}{2} a (A-B) d (1+2 n) \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^2 d}\\ &=\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}+\frac {((A-B) (1+2 n)) \int (d \sin (e+f x))^n \sqrt {a+a \sin (e+f x)} \, dx}{4 a^2}+\frac {\left (-\frac {1}{2} a^2 (A-B) d (1+2 n)+a^2 d (A+B-A n+B n)\right ) \int \frac {(d \sin (e+f x))^n}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^3 d}\\ &=\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}+\frac {\left (\left (-\frac {1}{2} a^2 (A-B) d (1+2 n)+a^2 d (A+B-A n+B n)\right ) \sqrt {1+\sin (e+f x)}\right ) \int \frac {(d \sin (e+f x))^n}{\sqrt {1+\sin (e+f x)}} \, dx}{2 a^3 d \sqrt {a+a \sin (e+f x)}}+\frac {((A-B) (1+2 n) \cos (e+f x)) \operatorname {Subst}\left (\int \frac {(d x)^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{4 f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}+\frac {\left (\left (-\frac {1}{2} a^2 (A-B) d (1+2 n)+a^2 d (A+B-A n+B n)\right ) \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)}{\sqrt {1+\sin (e+f x)}} \, dx}{2 a^3 d \sqrt {a+a \sin (e+f x)}}+\frac {\left ((A-B) (1+2 n) \cos (e+f x) \sin ^{-n}(e+f x) (d \sin (e+f x))^n\right ) \operatorname {Subst}\left (\int \frac {x^n}{\sqrt {a-a x}} \, dx,x,\sin (e+f x)\right )}{4 f \sqrt {a-a \sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}-\frac {(A-B) (1+2 n) \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{2 a f \sqrt {a+a \sin (e+f x)}}-\frac {\left (\left (-\frac {1}{2} a^2 (A-B) d (1+2 n)+a^2 d (A+B-A n+B n)\right ) \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) \sqrt {x}} \, dx,x,1-\sin (e+f x)\right )}{2 a^3 d f \sqrt {1-\sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}-\frac {(A-B) (1+2 n) \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right ) \sin ^{-n}(e+f x) (d \sin (e+f x))^n}{2 a f \sqrt {a+a \sin (e+f x)}}-\frac {\left (\left (-\frac {1}{2} a^2 (A-B) d (1+2 n)+a^2 d (A+B-A n+B n)\right ) \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}{2-x^2} \, dx,x,\sqrt {1-\sin (e+f x)}\right )}{a^3 d f \sqrt {1-\sin (e+f x)} \sqrt {a+a \sin (e+f x)}}\\ &=\frac {(A-B) \cos (e+f x) (d \sin (e+f x))^{1+n}}{2 d f (a+a \sin (e+f x))^{3/2}}-\frac {(A+3 B-4 A n+4 B n) F_1\left (\frac {1}{2};-n,1;\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}{4 a f \sqrt {a+a \sin (e+f x)}}-\frac {(A-B) (1+2 n) \cos (e+f x) \, _2F_1\left (\frac {1}{2},-n;\frac {3}{2};1-\sin (e+f x)\right ) \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 = 21.90, size = 523, normalized size = 2.31 \[ \frac {\sec (e+f x) (d \sin (e+f x))^n \left (A \left (a^2 \sqrt {2-2 \sin (e+f x)} (\sin (e+f x)+1)^2 (-\sin (e+f x))^{-n} F_1\left (1;\frac {1}{2},-n;2;\frac {1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )-\frac {4 a \sqrt {\frac {\sin (e+f x)-1}{\sin (e+f x)+1}} (\sin (e+f x)+1) \left (1-\frac {1}{\sin (e+f x)+1}\right )^{-n} \left (2 a (2 n+1) F_1\left (\frac {1}{2}-n;-\frac {1}{2},-n;\frac {3}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {1}{\sin (e+f x)+1}\right )+a (2 n-1) (\sin (e+f x)+1) F_1\left (-n-\frac {1}{2};-\frac {1}{2},-n;\frac {1}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {1}{\sin (e+f x)+1}\right )\right )}{4 n^2-1}\right )+a B (\sin (e+f x)+1) \left (a \sqrt {2-2 \sin (e+f x)} (\sin (e+f x)+1) (-\sin (e+f x))^{-n} F_1\left (1;\frac {1}{2},-n;2;\frac {1}{2} (\sin (e+f x)+1),\sin (e+f x)+1\right )-\frac {4 \sqrt {\frac {\sin (e+f x)-1}{\sin (e+f x)+1}} \left (1-\frac {1}{\sin (e+f x)+1}\right )^{-n} \left (a (2 n-1) (\sin (e+f x)+1) F_1\left (-n-\frac {1}{2};-\frac {1}{2},-n;\frac {1}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {1}{\sin (e+f x)+1}\right )-2 a (2 n+1) F_1\left (\frac {1}{2}-n;-\frac {1}{2},-n;\frac {3}{2}-n;\frac {2}{\sin (e+f x)+1},\frac {1}{\sin (e+f x)+1}\right )\right )}{4 n^2-1}\right )\right )}{8 a^3 f \sqrt {a (\sin (e+f x)+1)}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.46, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (B \sin \left (f x + e\right ) + A\right )} \sqrt {a \sin \left (f x + e\right ) + a} \left (d \sin \left (f x + e\right )\right )^{n}}{a^{2} \cos \left (f x + e\right )^{2} - 2 \, a^{2} \sin \left (f x + e\right ) - 2 \, a^{2}}, x\right ) \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [F] time = 0.89, size = 0, normalized size = 0.00 \[ \int \frac {\left (d \sin \left (f x +e \right )\right )^{n} \left (A +B \sin \left (f x +e \right )\right )}{\left (a +a \sin \left (f x +e \right )\right )^{\frac {3}{2}}}\, dx \]
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 \[ \int \frac {{\left (B \sin \left (f x + e\right ) + A\right )} \left (d \sin \left (f x + e\right )\right )^{n}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\left (d\,\sin \left (e+f\,x\right )\right )}^n\,\left (A+B\,\sin \left (e+f\,x\right )\right )}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{3/2}} \,d x \]
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 \[ \int \frac {\left (d \sin {\left (e + f x \right )}\right )^{n} \left (A + B \sin {\left (e + f x \right )}\right )}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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