Optimal. Leaf size=59 \[ \frac {C x \sqrt {\cos (c+d x)}}{\sqrt {b \cos (c+d x)}}+\frac {A \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.086, Rules used = {18, 3091, 8}
\begin {gather*} \frac {A \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}+\frac {C x \sqrt {\cos (c+d x)}}{\sqrt {b \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 18
Rule 3091
Rubi steps
\begin {align*} \int \frac {A+C \cos ^2(c+d x)}{\cos ^{\frac {3}{2}}(c+d x) \sqrt {b \cos (c+d x)}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \left (A+C \cos ^2(c+d x)\right ) \sec ^2(c+d x) \, dx}{\sqrt {b \cos (c+d x)}}\\ &=\frac {A \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}+\frac {\left (C \sqrt {\cos (c+d x)}\right ) \int 1 \, dx}{\sqrt {b \cos (c+d x)}}\\ &=\frac {C x \sqrt {\cos (c+d x)}}{\sqrt {b \cos (c+d x)}}+\frac {A \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.05, size = 45, normalized size = 0.76 \begin {gather*} \frac {C d x \cos (c+d x)+A \sin (c+d x)}{d \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.29, size = 45, normalized size = 0.76
method | result | size |
default | \(\frac {C \cos \left (d x +c \right ) \left (d x +c \right )+A \sin \left (d x +c \right )}{d \sqrt {b \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}}\) | \(45\) |
risch | \(\frac {C x \left (\sqrt {\cos }\left (d x +c \right )\right )}{\sqrt {b \cos \left (d x +c \right )}}+\frac {i {\mathrm e}^{-i \left (d x +c \right )} A}{\sqrt {b \cos \left (d x +c \right )}\, \sqrt {\cos \left (d x +c \right )}\, d}\) | \(57\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.57, size = 85, normalized size = 1.44 \begin {gather*} \frac {2 \, {\left (\frac {C \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{\sqrt {b}} + \frac {A \sqrt {b} \sin \left (2 \, d x + 2 \, c\right )}{b \cos \left (2 \, d x + 2 \, c\right )^{2} + b \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, b \cos \left (2 \, d x + 2 \, c\right ) + b}\right )}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.43, size = 191, normalized size = 3.24 \begin {gather*} \left [-\frac {C \sqrt {-b} \cos \left (d x + c\right )^{2} \log \left (2 \, b \cos \left (d x + c\right )^{2} + 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - b\right ) - 2 \, \sqrt {b \cos \left (d x + c\right )} A \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, b d \cos \left (d x + c\right )^{2}}, \frac {C \sqrt {b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{\sqrt {b} \cos \left (d x + c\right )^{\frac {3}{2}}}\right ) \cos \left (d x + c\right )^{2} + \sqrt {b \cos \left (d x + c\right )} A \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{b d \cos \left (d x + c\right )^{2}}\right ] \end {gather*}
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 \begin {gather*} \int \frac {A + C \cos ^{2}{\left (c + d x \right )}}{\sqrt {b \cos {\left (c + d x \right )}} \cos ^{\frac {3}{2}}{\left (c + d x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.25, size = 84, normalized size = 1.42 \begin {gather*} \frac {\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (A\,\sin \left (2\,c+2\,d\,x\right )+C\,d\,x+C\,d\,x\,\cos \left (2\,c+2\,d\,x\right )+A\,1{}\mathrm {i}+A\,\cos \left (2\,c+2\,d\,x\right )\,1{}\mathrm {i}\right )}{b\,d\,\sqrt {\cos \left (c+d\,x\right )}\,\left (\cos \left (2\,c+2\,d\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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