Optimal. Leaf size=17 \[ \frac {2 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d} \]
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Rubi [A] time = 0.01, antiderivative size = 17, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {21, 2639} \[ \frac {2 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d} \]
Antiderivative was successfully verified.
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Rule 21
Rule 2639
Rubi steps
\begin {align*} \int \frac {\sqrt {\cos (c+d x)} (a B+b B \cos (c+d x))}{a+b \cos (c+d x)} \, dx &=B \int \sqrt {\cos (c+d x)} \, dx\\ &=\frac {2 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 17, normalized size = 1.00 \[ \frac {2 B E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{d} \]
Antiderivative was successfully verified.
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fricas [F] time = 1.65, size = 0, normalized size = 0.00 \[ {\rm integral}\left (B \sqrt {\cos \left (d x + c\right )}, x\right ) \]
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 \[ \int \frac {{\left (B b \cos \left (d x + c\right ) + B a\right )} \sqrt {\cos \left (d x + c\right )}}{b \cos \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.96, size = 134, normalized size = 7.88 \[ \frac {2 \sqrt {\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{\sqrt {-2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, d} \]
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 \[ \int \frac {{\left (B b \cos \left (d x + c\right ) + B a\right )} \sqrt {\cos \left (d x + c\right )}}{b \cos \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.06 \[ \int \frac {\sqrt {\cos \left (c+d\,x\right )}\,\left (B\,a+B\,b\,\cos \left (c+d\,x\right )\right )}{a+b\,\cos \left (c+d\,x\right )} \,d x \]
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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