Optimal. Leaf size=74 \[ \frac {b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {b \sec (c+d x)}}{2 d}+\frac {b \sqrt {b \sec (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt {\sec (c+d x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {17, 3768, 3770} \[ \frac {b \sin (c+d x) \sec ^{\frac {3}{2}}(c+d x) \sqrt {b \sec (c+d x)}}{2 d}+\frac {b \sqrt {b \sec (c+d x)} \tanh ^{-1}(\sin (c+d x))}{2 d \sqrt {\sec (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3768
Rule 3770
Rubi steps
\begin {align*} \int \sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^{3/2} \, dx &=\frac {\left (b \sqrt {b \sec (c+d x)}\right ) \int \sec ^3(c+d x) \, dx}{\sqrt {\sec (c+d x)}}\\ &=\frac {b \sec ^{\frac {3}{2}}(c+d x) \sqrt {b \sec (c+d x)} \sin (c+d x)}{2 d}+\frac {\left (b \sqrt {b \sec (c+d x)}\right ) \int \sec (c+d x) \, dx}{2 \sqrt {\sec (c+d x)}}\\ &=\frac {b \tanh ^{-1}(\sin (c+d x)) \sqrt {b \sec (c+d x)}}{2 d \sqrt {\sec (c+d x)}}+\frac {b \sec ^{\frac {3}{2}}(c+d x) \sqrt {b \sec (c+d x)} \sin (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 50, normalized size = 0.68 \[ \frac {(b \sec (c+d x))^{3/2} \left (\tanh ^{-1}(\sin (c+d x))+\tan (c+d x) \sec (c+d x)\right )}{2 d \sec ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.85, size = 202, normalized size = 2.73 \[ \left [\frac {b^{\frac {3}{2}} \cos \left (d x + c\right ) \log \left (-\frac {b \cos \left (d x + c\right )^{2} - 2 \, \sqrt {b} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, b}{\cos \left (d x + c\right )^{2}}\right ) + \frac {2 \, b \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{4 \, d \cos \left (d x + c\right )}, -\frac {\sqrt {-b} b \arctan \left (\frac {\sqrt {-b} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{b}\right ) \cos \left (d x + c\right ) - \frac {b \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}}}{2 \, d \cos \left (d x + c\right )}\right ] \]
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 \[ \int \left (b \sec \left (d x + c\right )\right )^{\frac {3}{2}} \sec \left (d x + c\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.90, size = 114, normalized size = 1.54 \[ -\frac {\left (\left (\cos ^{2}\left (d x +c \right )\right ) \ln \left (-\frac {-1+\cos \left (d x +c \right )+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-\left (\cos ^{2}\left (d x +c \right )\right ) \ln \left (-\frac {-\sin \left (d x +c \right )-1+\cos \left (d x +c \right )}{\sin \left (d x +c \right )}\right )-\sin \left (d x +c \right )\right ) \cos \left (d x +c \right ) \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}}}{2 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.92, size = 691, normalized size = 9.34 \[ \text {result too large to display} \]
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 \[ \int {\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}\,{\left (\frac {1}{\cos \left (c+d\,x\right )}\right )}^{3/2} \,d x \]
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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