Optimal. Leaf size=63 \[ \frac {2 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {2 b (e \cos (c+d x))^{3/2}}{3 d e} \]
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Rubi [A] time = 0.05, antiderivative size = 63, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {2669, 2640, 2639} \[ \frac {2 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{d \sqrt {\cos (c+d x)}}-\frac {2 b (e \cos (c+d x))^{3/2}}{3 d e} \]
Antiderivative was successfully verified.
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Rule 2639
Rule 2640
Rule 2669
Rubi steps
\begin {align*} \int \sqrt {e \cos (c+d x)} (a+b \sin (c+d x)) \, dx &=-\frac {2 b (e \cos (c+d x))^{3/2}}{3 d e}+a \int \sqrt {e \cos (c+d x)} \, dx\\ &=-\frac {2 b (e \cos (c+d x))^{3/2}}{3 d e}+\frac {\left (a \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{\sqrt {\cos (c+d x)}}\\ &=-\frac {2 b (e \cos (c+d x))^{3/2}}{3 d e}+\frac {2 a \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 56, normalized size = 0.89 \[ -\frac {2 \sqrt {e \cos (c+d x)} \left (b \cos ^{\frac {3}{2}}(c+d x)-3 a E\left (\left .\frac {1}{2} (c+d x)\right |2\right )\right )}{3 d \sqrt {\cos (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}, x\right ) \]
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 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 1.12, size = 123, normalized size = 1.95 \[ \frac {2 e \left (-4 b \left (\sin ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right ) a +4 b \left (\sin ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \sin \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) e +e}\, d} \]
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 \[ \int \sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.02 \[ \int \sqrt {e\,\cos \left (c+d\,x\right )}\,\left (a+b\,\sin \left (c+d\,x\right )\right ) \,d x \]
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 \[ \int \sqrt {e \cos {\left (c + d x \right )}} \left (a + b \sin {\left (c + d x \right )}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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