Optimal. Leaf size=61 \[ \frac {A b x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}}+\frac {b B \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {17, 2717}
\begin {gather*} \frac {A b x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}}+\frac {b B \sin (c+d x) \sqrt {b \cos (c+d x)}}{d \sqrt {\cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 17
Rule 2717
Rubi steps
\begin {align*} \int \frac {(b \cos (c+d x))^{3/2} (A+B \cos (c+d x))}{\cos ^{\frac {3}{2}}(c+d x)} \, dx &=\frac {\left (b \sqrt {b \cos (c+d x)}\right ) \int (A+B \cos (c+d x)) \, dx}{\sqrt {\cos (c+d x)}}\\ &=\frac {A b x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}}+\frac {\left (b B \sqrt {b \cos (c+d x)}\right ) \int \cos (c+d x) \, dx}{\sqrt {\cos (c+d x)}}\\ &=\frac {A b x \sqrt {b \cos (c+d x)}}{\sqrt {\cos (c+d x)}}+\frac {b B \sqrt {b \cos (c+d x)} \sin (c+d x)}{d \sqrt {\cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 42, normalized size = 0.69 \begin {gather*} \frac {(b \cos (c+d x))^{3/2} (A (c+d x)+B \sin (c+d x))}{d \cos ^{\frac {3}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.19, size = 39, normalized size = 0.64
method | result | size |
default | \(\frac {\left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \left (A \left (d x +c \right )+B \sin \left (d x +c \right )\right )}{d \cos \left (d x +c \right )^{\frac {3}{2}}}\) | \(39\) |
risch | \(\frac {A b x \sqrt {b \cos \left (d x +c \right )}}{\sqrt {\cos \left (d x +c \right )}}+\frac {b B \sin \left (d x +c \right ) \sqrt {b \cos \left (d x +c \right )}}{d \sqrt {\cos \left (d x +c \right )}}\) | \(54\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.55, size = 40, normalized size = 0.66 \begin {gather*} \frac {2 \, A b^{\frac {3}{2}} \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right ) + B b^{\frac {3}{2}} \sin \left (d x + c\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 184, normalized size = 3.02 \begin {gather*} \left [\frac {A \sqrt {-b} b \cos \left (d x + c\right ) \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 )} B b \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{2 \, d \cos \left (d x + c\right )}, \frac {A b^{\frac {3}{2}} \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 ) + \sqrt {b \cos \left (d x + c\right )} B b \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{d \cos \left (d x + c\right )}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 43.34, size = 80, normalized size = 1.31 \begin {gather*} \begin {cases} \frac {A x \left (b \cos {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\cos ^{\frac {3}{2}}{\left (c + d x \right )}} + \frac {B \left (b \cos {\left (c + d x \right )}\right )^{\frac {3}{2}} \sin {\left (c + d x \right )}}{d \cos ^{\frac {3}{2}}{\left (c + d x \right )}} & \text {for}\: d \neq 0 \\\frac {x \left (b \cos {\left (c \right )}\right )^{\frac {3}{2}} \left (A + B \cos {\left (c \right )}\right )}{\cos ^{\frac {3}{2}}{\left (c \right )}} & \text {otherwise} \end {cases} \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 = 0.85, size = 36, normalized size = 0.59 \begin {gather*} \frac {b\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (B\,\sin \left (c+d\,x\right )+A\,d\,x\right )}{d\,\sqrt {\cos \left (c+d\,x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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