Optimal. Leaf size=66 \[ \frac {2 a \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d \sqrt {e \sin (c+d x)}}+\frac {2 b \sqrt {e \sin (c+d x)}}{d e} \]
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Rubi [A] time = 0.05, antiderivative size = 66, 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, 2642, 2641} \[ \frac {2 a \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right )}{d \sqrt {e \sin (c+d x)}}+\frac {2 b \sqrt {e \sin (c+d x)}}{d e} \]
Antiderivative was successfully verified.
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Rule 2641
Rule 2642
Rule 2669
Rubi steps
\begin {align*} \int \frac {a+b \cos (c+d x)}{\sqrt {e \sin (c+d x)}} \, dx &=\frac {2 b \sqrt {e \sin (c+d x)}}{d e}+a \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx\\ &=\frac {2 b \sqrt {e \sin (c+d x)}}{d e}+\frac {\left (a \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{\sqrt {e \sin (c+d x)}}\\ &=\frac {2 a F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{d \sqrt {e \sin (c+d x)}}+\frac {2 b \sqrt {e \sin (c+d x)}}{d e}\\ \end {align*}
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Mathematica [A] time = 0.20, size = 54, normalized size = 0.82 \[ \frac {2 \left (b \sin (c+d x)-a \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )\right )}{d \sqrt {e \sin (c+d x)}} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.12, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b \cos \left (d x + c\right ) + a\right )} \sqrt {e \sin \left (d x + c\right )}}{e \sin \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {b \cos \left (d x + c\right ) + a}{\sqrt {e \sin \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.16, size = 92, normalized size = 1.39 \[ -\frac {a \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \EllipticF \left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (d x +c \right ) \cos \left (d x +c \right ) b}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d} \]
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 \[ \int \frac {b \cos \left (d x + c\right ) + a}{\sqrt {e \sin \left (d x + c\right )}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.72, size = 50, normalized size = 0.76 \[ -\frac {2\,\sqrt {\sin \left (c+d\,x\right )}\,\left (a\,\mathrm {F}\left (\frac {\pi }{4}-\frac {c}{2}-\frac {d\,x}{2}\middle |2\right )-b\,\sqrt {\sin \left (c+d\,x\right )}\right )}{d\,\sqrt {e\,\sin \left (c+d\,x\right )}} \]
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 \[ \int \frac {a + b \cos {\left (c + d x \right )}}{\sqrt {e \sin {\left (c + d x \right )}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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