Optimal. Leaf size=72 \[ \frac {b \sin ^3(c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {b \sec (c+d x)}}{3 d}+\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)}}{d} \]
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Rubi [A] time = 0.02, antiderivative size = 72, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {17, 3767} \[ \frac {b \sin ^3(c+d x) \sec ^{\frac {5}{2}}(c+d x) \sqrt {b \sec (c+d x)}}{3 d}+\frac {b \sin (c+d x) \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)}}{d} \]
Antiderivative was successfully verified.
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Rule 17
Rule 3767
Rubi steps
\begin {align*} \int \sec ^{\frac {5}{2}}(c+d x) (b \sec (c+d x))^{3/2} \, dx &=\frac {\left (b \sqrt {b \sec (c+d x)}\right ) \int \sec ^4(c+d x) \, dx}{\sqrt {\sec (c+d x)}}\\ &=-\frac {\left (b \sqrt {b \sec (c+d x)}\right ) \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{d \sqrt {\sec (c+d x)}}\\ &=\frac {b \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)} \sin (c+d x)}{d}+\frac {b \sec ^{\frac {5}{2}}(c+d x) \sqrt {b \sec (c+d x)} \sin ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 45, normalized size = 0.62 \[ \frac {\left (\frac {1}{3} \tan ^3(c+d x)+\tan (c+d x)\right ) (b \sec (c+d x))^{3/2}}{d \sec ^{\frac {3}{2}}(c+d x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.82, size = 44, normalized size = 0.61 \[ \frac {{\left (2 \, b \cos \left (d x + c\right )^{2} + b\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{3 \, d \cos \left (d x + c\right )^{\frac {5}{2}}} \]
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 \[ \int \left (b \sec \left (d x + c\right )\right )^{\frac {3}{2}} \sec \left (d x + c\right )^{\frac {5}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.88, size = 52, normalized size = 0.72 \[ \frac {\left (2 \left (\cos ^{2}\left (d x +c \right )\right )+1\right ) \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{3 d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.00, size = 299, normalized size = 4.15 \[ -\frac {4 \, {\left (3 \, b \cos \left (6 \, d x + 6 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 9 \, b \cos \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) - {\left (3 \, b \cos \left (2 \, d x + 2 \, c\right ) + b\right )} \sin \left (6 \, d x + 6 \, c\right ) - 3 \, {\left (3 \, b \cos \left (2 \, d x + 2 \, c\right ) + b\right )} \sin \left (4 \, d x + 4 \, c\right )\right )} \sqrt {b}}{3 \, {\left (2 \, {\left (3 \, \cos \left (4 \, d x + 4 \, c\right ) + 3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \cos \left (6 \, d x + 6 \, c\right ) + \cos \left (6 \, d x + 6 \, c\right )^{2} + 6 \, {\left (3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \cos \left (4 \, d x + 4 \, c\right ) + 9 \, \cos \left (4 \, d x + 4 \, c\right )^{2} + 9 \, \cos \left (2 \, d x + 2 \, c\right )^{2} + 6 \, {\left (\sin \left (4 \, d x + 4 \, c\right ) + \sin \left (2 \, d x + 2 \, c\right )\right )} \sin \left (6 \, d x + 6 \, c\right ) + \sin \left (6 \, d x + 6 \, c\right )^{2} + 9 \, \sin \left (4 \, d x + 4 \, c\right )^{2} + 18 \, \sin \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 9 \, \sin \left (2 \, d x + 2 \, c\right )^{2} + 6 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.36, size = 127, normalized size = 1.76 \[ \frac {2\,b\,\cos \left (c+d\,x\right )\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,\left (4\,\sin \left (c+d\,x\right )+5\,\sin \left (3\,c+3\,d\,x\right )+\sin \left (5\,c+5\,d\,x\right )+\cos \left (c+d\,x\right )\,10{}\mathrm {i}+\cos \left (3\,c+3\,d\,x\right )\,5{}\mathrm {i}+\cos \left (5\,c+5\,d\,x\right )\,1{}\mathrm {i}\right )}{3\,d\,\left (10\,\cos \left (c+d\,x\right )+5\,\cos \left (3\,c+3\,d\,x\right )+\cos \left (5\,c+5\,d\,x\right )\right )} \]
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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