Optimal. Leaf size=117 \[ \frac {2 F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) (3 a B+3 A b+b C)}{3 f}+\frac {2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) (b B-a (A-C))}{f}-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}-\frac {2 b C \sqrt {\sin (e+f x)} \cos (e+f x)}{3 f} \]
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Rubi [A] time = 0.22, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.122, Rules used = {3031, 3023, 2748, 2641, 2639} \[ \frac {2 F\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) (3 a B+3 A b+b C)}{3 f}+\frac {2 E\left (\left .\frac {1}{2} \left (e+f x-\frac {\pi }{2}\right )\right |2\right ) (b B-a (A-C))}{f}-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}-\frac {2 b C \sqrt {\sin (e+f x)} \cos (e+f x)}{3 f} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2641
Rule 2748
Rule 3023
Rule 3031
Rubi steps
\begin {align*} \int \frac {(a+b \sin (e+f x)) \left (A+B \sin (e+f x)+C \sin ^2(e+f x)\right )}{\sin ^{\frac {3}{2}}(e+f x)} \, dx &=-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}-2 \int \frac {\frac {1}{2} (-A b-a B)-\frac {1}{2} (b B-a (A-C)) \sin (e+f x)-\frac {1}{2} b C \sin ^2(e+f x)}{\sqrt {\sin (e+f x)}} \, dx\\ &=-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}-\frac {2 b C \cos (e+f x) \sqrt {\sin (e+f x)}}{3 f}-\frac {4}{3} \int \frac {\frac {1}{4} (-3 A b-3 a B-b C)-\frac {3}{4} (b B-a (A-C)) \sin (e+f x)}{\sqrt {\sin (e+f x)}} \, dx\\ &=-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}-\frac {2 b C \cos (e+f x) \sqrt {\sin (e+f x)}}{3 f}-(-b B+a (A-C)) \int \sqrt {\sin (e+f x)} \, dx-\frac {1}{3} (-3 A b-3 a B-b C) \int \frac {1}{\sqrt {\sin (e+f x)}} \, dx\\ &=\frac {2 (b B-a (A-C)) E\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{f}+\frac {2 (3 A b+3 a B+b C) F\left (\left .\frac {1}{2} \left (e-\frac {\pi }{2}+f x\right )\right |2\right )}{3 f}-\frac {2 a A \cos (e+f x)}{f \sqrt {\sin (e+f x)}}-\frac {2 b C \cos (e+f x) \sqrt {\sin (e+f x)}}{3 f}\\ \end {align*}
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Mathematica [A] time = 0.88, size = 97, normalized size = 0.83 \[ -\frac {2 F\left (\left .\frac {1}{4} (-2 e-2 f x+\pi )\right |2\right ) (3 a B+3 A b+b C)+6 E\left (\left .\frac {1}{4} (-2 e-2 f x+\pi )\right |2\right ) (a (C-A)+b B)+\frac {2 \cos (e+f x) (3 a A+b C \sin (e+f x))}{\sqrt {\sin (e+f x)}}}{3 f} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.47, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left ({\left (C a + B b\right )} \cos \left (f x + e\right )^{2} - {\left (A + C\right )} a - B b + {\left (C b \cos \left (f x + e\right )^{2} - B a - {\left (A + C\right )} b\right )} \sin \left (f x + e\right )\right )} \sqrt {\sin \left (f x + e\right )}}{\cos \left (f x + e\right )^{2} - 1}, x\right ) \]
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 \[ \int \frac {{\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (b \sin \left (f x + e\right ) + a\right )}}{\sin \left (f x + e\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 1.46, size = 516, normalized size = 4.41 \[ \frac {-A \sqrt {-\sin \left (f x +e \right )}\, \EllipticF \left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, a +A b \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \EllipticF \left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+2 A \sqrt {-\sin \left (f x +e \right )}\, \EllipticE \left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, a +a B \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \EllipticF \left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+B b \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \EllipticF \left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )-2 B b \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \EllipticE \left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+a C \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \EllipticF \left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )+\frac {C \sqrt {-\sin \left (f x +e \right )}\, \EllipticF \left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right ) \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, b}{3}-2 a C \sqrt {1+\sin \left (f x +e \right )}\, \sqrt {-2 \sin \left (f x +e \right )+2}\, \sqrt {-\sin \left (f x +e \right )}\, \EllipticE \left (\sqrt {1+\sin \left (f x +e \right )}, \frac {\sqrt {2}}{2}\right )-\frac {2 C \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) b}{3}-2 A a \left (\cos ^{2}\left (f x +e \right )\right )}{\cos \left (f x +e \right ) \sqrt {\sin \left (f x +e \right )}\, f} \]
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 \[ \int \frac {{\left (C \sin \left (f x + e\right )^{2} + B \sin \left (f x + e\right ) + A\right )} {\left (b \sin \left (f x + e\right ) + a\right )}}{\sin \left (f x + e\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 14.85, size = 169, normalized size = 1.44 \[ \frac {2\,B\,b\,\mathrm {E}\left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\middle |2\right )}{f}-\frac {2\,B\,a\,\mathrm {F}\left (\frac {\pi }{4}-\frac {e}{2}-\frac {f\,x}{2}\middle |2\right )}{f}-\frac {2\,A\,b\,\mathrm {F}\left (\frac {\pi }{4}-\frac {e}{2}-\frac {f\,x}{2}\middle |2\right )}{f}+\frac {2\,C\,a\,\mathrm {E}\left (\frac {e}{2}-\frac {\pi }{4}+\frac {f\,x}{2}\middle |2\right )}{f}-\frac {A\,a\,\cos \left (e+f\,x\right )\,{\left ({\sin \left (e+f\,x\right )}^2\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {5}{4};\ \frac {3}{2};\ {\cos \left (e+f\,x\right )}^2\right )}{f\,\sqrt {\sin \left (e+f\,x\right )}}-\frac {C\,b\,\cos \left (e+f\,x\right )\,{\sin \left (e+f\,x\right )}^{5/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {1}{2};\ \frac {3}{2};\ {\cos \left (e+f\,x\right )}^2\right )}{f\,{\left ({\sin \left (e+f\,x\right )}^2\right )}^{5/4}} \]
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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