Optimal. Leaf size=164 \[ \frac {(2 A+3 C) \sin (c+d x)}{3 b d \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}+\frac {A \sin (c+d x)}{3 b d \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {B \sin (c+d x)}{2 b d \cos ^{\frac {3}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {B \sqrt {\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b d \sqrt {b \cos (c+d x)}} \]
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Rubi [A] time = 0.12, antiderivative size = 164, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {18, 3021, 2748, 3768, 3770, 3767, 8} \[ \frac {(2 A+3 C) \sin (c+d x)}{3 b d \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}+\frac {A \sin (c+d x)}{3 b d \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {B \sin (c+d x)}{2 b d \cos ^{\frac {3}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {B \sqrt {\cos (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 b d \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 8
Rule 18
Rule 2748
Rule 3021
Rule 3767
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \frac {A+B \cos (c+d x)+C \cos ^2(c+d x)}{\cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \left (A+B \cos (c+d x)+C \cos ^2(c+d x)\right ) \sec ^4(c+d x) \, dx}{b \sqrt {b \cos (c+d x)}}\\ &=\frac {A \sin (c+d x)}{3 b d \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {\sqrt {\cos (c+d x)} \int (3 B+(2 A+3 C) \cos (c+d x)) \sec ^3(c+d x) \, dx}{3 b \sqrt {b \cos (c+d x)}}\\ &=\frac {A \sin (c+d x)}{3 b d \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {\left (B \sqrt {\cos (c+d x)}\right ) \int \sec ^3(c+d x) \, dx}{b \sqrt {b \cos (c+d x)}}+\frac {\left ((2 A+3 C) \sqrt {\cos (c+d x)}\right ) \int \sec ^2(c+d x) \, dx}{3 b \sqrt {b \cos (c+d x)}}\\ &=\frac {A \sin (c+d x)}{3 b d \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {B \sin (c+d x)}{2 b d \cos ^{\frac {3}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {\left (B \sqrt {\cos (c+d x)}\right ) \int \sec (c+d x) \, dx}{2 b \sqrt {b \cos (c+d x)}}-\frac {\left ((2 A+3 C) \sqrt {\cos (c+d x)}\right ) \operatorname {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 b d \sqrt {b \cos (c+d x)}}\\ &=\frac {B \tanh ^{-1}(\sin (c+d x)) \sqrt {\cos (c+d x)}}{2 b d \sqrt {b \cos (c+d x)}}+\frac {A \sin (c+d x)}{3 b d \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {B \sin (c+d x)}{2 b d \cos ^{\frac {3}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {(2 A+3 C) \sin (c+d x)}{3 b d \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 87, normalized size = 0.53 \[ \frac {\tan (c+d x) ((2 A+3 C) \cos (2 (c+d x))+4 A+3 B \cos (c+d x)+3 C)+3 B \cos ^2(c+d x) \tanh ^{-1}(\sin (c+d x))}{6 d \sqrt {\cos (c+d x)} (b \cos (c+d x))^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 271, normalized size = 1.65 \[ \left [\frac {3 \, B \sqrt {b} \cos \left (d x + c\right )^{4} \log \left (-\frac {b \cos \left (d x + c\right )^{3} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, b \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{3}}\right ) + 2 \, {\left (2 \, {\left (2 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, B \cos \left (d x + c\right ) + 2 \, A\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{12 \, b^{2} d \cos \left (d x + c\right )^{4}}, -\frac {3 \, B \sqrt {-b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sin \left (d x + c\right )}{b \sqrt {\cos \left (d x + c\right )}}\right ) \cos \left (d x + c\right )^{4} - {\left (2 \, {\left (2 \, A + 3 \, C\right )} \cos \left (d x + c\right )^{2} + 3 \, B \cos \left (d x + c\right ) + 2 \, A\right )} \sqrt {b \cos \left (d x + c\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{6 \, b^{2} d \cos \left (d x + c\right )^{4}}\right ] \]
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 \[ \int \frac {C \cos \left (d x + c\right )^{2} + B \cos \left (d x + c\right ) + A}{\left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.33, size = 156, normalized size = 0.95 \[ \frac {3 B \left (\cos ^{3}\left (d x +c \right )\right ) \ln \left (\frac {1-\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-3 B \left (\cos ^{3}\left (d x +c \right )\right ) \ln \left (-\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}\right )+4 A \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )+6 C \sin \left (d x +c \right ) \left (\cos ^{2}\left (d x +c \right )\right )+3 B \cos \left (d x +c \right ) \sin \left (d x +c \right )+2 A \sin \left (d x +c \right )}{6 d \left (b \cos \left (d x +c \right )\right )^{\frac {3}{2}} \cos \left (d x +c \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.78, size = 1048, normalized size = 6.39 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {C\,{\cos \left (c+d\,x\right )}^2+B\,\cos \left (c+d\,x\right )+A}{{\cos \left (c+d\,x\right )}^{5/2}\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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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