Optimal. Leaf size=66 \[ \frac {B x \sqrt {\cos (c+d x)}}{b^2 \sqrt {b \cos (c+d x)}}+\frac {A \tanh ^{-1}(\sin (c+d x)) \sqrt {\cos (c+d x)}}{b^2 d \sqrt {b \cos (c+d x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {17, 2814, 3855}
\begin {gather*} \frac {A \sqrt {\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{b^2 d \sqrt {b \cos (c+d x)}}+\frac {B x \sqrt {\cos (c+d x)}}{b^2 \sqrt {b \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 17
Rule 2814
Rule 3855
Rubi steps
\begin {align*} \int \frac {\cos ^{\frac {3}{2}}(c+d x) (A+B \cos (c+d x))}{(b \cos (c+d x))^{5/2}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int (A+B \cos (c+d x)) \sec (c+d x) \, dx}{b^2 \sqrt {b \cos (c+d x)}}\\ &=\frac {B x \sqrt {\cos (c+d x)}}{b^2 \sqrt {b \cos (c+d x)}}+\frac {\left (A \sqrt {\cos (c+d x)}\right ) \int \sec (c+d x) \, dx}{b^2 \sqrt {b \cos (c+d x)}}\\ &=\frac {B x \sqrt {\cos (c+d x)}}{b^2 \sqrt {b \cos (c+d x)}}+\frac {A \tanh ^{-1}(\sin (c+d x)) \sqrt {\cos (c+d x)}}{b^2 d \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 43, normalized size = 0.65 \begin {gather*} \frac {\left (B d x+A \tanh ^{-1}(\sin (c+d x))\right ) \sqrt {\cos (c+d x)}}{b^2 d \sqrt {b \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.20, size = 54, normalized size = 0.82
method | result | size |
default | \(-\frac {\left (\cos ^{\frac {5}{2}}\left (d x +c \right )\right ) \left (2 A \arctanh \left (\frac {-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-B \left (d x +c \right )\right )}{d \left (b \cos \left (d x +c \right )\right )^{\frac {5}{2}}}\) | \(54\) |
risch | \(\frac {B x \left (\sqrt {\cos }\left (d x +c \right )\right )}{b^{2} \sqrt {b \cos \left (d x +c \right )}}+\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) A \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{b^{2} \sqrt {b \cos \left (d x +c \right )}\, d}-\frac {\left (\sqrt {\cos }\left (d x +c \right )\right ) A \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{b^{2} \sqrt {b \cos \left (d x +c \right )}\, d}\) | \(105\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.57, size = 92, normalized size = 1.39 \begin {gather*} \frac {\frac {A {\left (\log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) - \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right )\right )}}{b^{\frac {5}{2}}} + \frac {4 \, B \arctan \left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1}\right )}{b^{\frac {5}{2}}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.45, size = 215, normalized size = 3.26 \begin {gather*} \left [-\frac {2 \, A \sqrt {-b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sin \left (d x + c\right )}{b \sqrt {\cos \left (d x + c\right )}}\right ) + B \sqrt {-b} \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 \, b^{3} d}, \frac {2 \, B \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 ) + A \sqrt {b} \log \left (-\frac {b \cos \left (d x + c\right )^{3} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, b \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{3}}\right )}{2 \, b^{3} d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^{3/2}\,\left (A+B\,\cos \left (c+d\,x\right )\right )}{{\left (b\,\cos \left (c+d\,x\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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