Optimal. Leaf size=76 \[ \frac {\sin ^3(c+d x)}{3 b^2 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {\sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}} \]
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Rubi [A] time = 0.02, antiderivative size = 76, 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 = {18, 3767} \[ \frac {\sin ^3(c+d x)}{3 b^2 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {\sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}} \]
Antiderivative was successfully verified.
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Rule 18
Rule 3767
Rubi steps
\begin {align*} \int \frac {1}{\cos ^{\frac {3}{2}}(c+d x) (b \cos (c+d x))^{5/2}} \, dx &=\frac {\sqrt {\cos (c+d x)} \int \sec ^4(c+d x) \, dx}{b^2 \sqrt {b \cos (c+d x)}}\\ &=-\frac {\sqrt {\cos (c+d x)} \operatorname {Subst}\left (\int \left (1+x^2\right ) \, dx,x,-\tan (c+d x)\right )}{b^2 d \sqrt {b \cos (c+d x)}}\\ &=\frac {\sin (c+d x)}{b^2 d \sqrt {\cos (c+d x)} \sqrt {b \cos (c+d x)}}+\frac {\sin ^3(c+d x)}{3 b^2 d \cos ^{\frac {5}{2}}(c+d x) \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 45, normalized size = 0.59 \[ \frac {\cos ^{\frac {5}{2}}(c+d x) \left (\frac {1}{3} \tan ^3(c+d x)+\tan (c+d x)\right )}{d (b \cos (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.95, size = 44, normalized size = 0.58 \[ \frac {\sqrt {b \cos \left (d x + c\right )} {\left (2 \, \cos \left (d x + c\right )^{2} + 1\right )} \sin \left (d x + c\right )}{3 \, b^{3} d \cos \left (d x + c\right )^{\frac {7}{2}}} \]
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 \[ \int \frac {1}{\left (b \cos \left (d x + c\right )\right )^{\frac {5}{2}} \cos \left (d x + c\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.11, size = 42, normalized size = 0.55 \[ \frac {\sin \left (d x +c \right ) \left (2 \left (\cos ^{2}\left (d x +c \right )\right )+1\right )}{3 d \left (b \cos \left (d x +c \right )\right )^{\frac {5}{2}} \sqrt {\cos \left (d x +c \right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.51, size = 343, normalized size = 4.51 \[ \frac {4 \, {\left ({\left (3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (6 \, d x + 6 \, c\right ) + 3 \, {\left (3 \, \cos \left (2 \, d x + 2 \, c\right ) + 1\right )} \sin \left (4 \, d x + 4 \, c\right ) - 3 \, \cos \left (6 \, d x + 6 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) - 9 \, \cos \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right )\right )}}{3 \, {\left (b^{2} \cos \left (6 \, d x + 6 \, c\right )^{2} + 9 \, b^{2} \cos \left (4 \, d x + 4 \, c\right )^{2} + 9 \, b^{2} \cos \left (2 \, d x + 2 \, c\right )^{2} + b^{2} \sin \left (6 \, d x + 6 \, c\right )^{2} + 9 \, b^{2} \sin \left (4 \, d x + 4 \, c\right )^{2} + 18 \, b^{2} \sin \left (4 \, d x + 4 \, c\right ) \sin \left (2 \, d x + 2 \, c\right ) + 9 \, b^{2} \sin \left (2 \, d x + 2 \, c\right )^{2} + 6 \, b^{2} \cos \left (2 \, d x + 2 \, c\right ) + b^{2} + 2 \, {\left (3 \, b^{2} \cos \left (4 \, d x + 4 \, c\right ) + 3 \, b^{2} \cos \left (2 \, d x + 2 \, c\right ) + b^{2}\right )} \cos \left (6 \, d x + 6 \, c\right ) + 6 \, {\left (3 \, b^{2} \cos \left (2 \, d x + 2 \, c\right ) + b^{2}\right )} \cos \left (4 \, d x + 4 \, c\right ) + 6 \, {\left (b^{2} \sin \left (4 \, d x + 4 \, c\right ) + b^{2} \sin \left (2 \, d x + 2 \, c\right )\right )} \sin \left (6 \, d x + 6 \, c\right )\right )} \sqrt {b} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 131, normalized size = 1.72 \[ \frac {2\,\sqrt {b\,\cos \left (c+d\,x\right )}\,\left (\cos \left (2\,c+2\,d\,x\right )\,15{}\mathrm {i}+\cos \left (4\,c+4\,d\,x\right )\,6{}\mathrm {i}+\cos \left (6\,c+6\,d\,x\right )\,1{}\mathrm {i}+9\,\sin \left (2\,c+2\,d\,x\right )+6\,\sin \left (4\,c+4\,d\,x\right )+\sin \left (6\,c+6\,d\,x\right )+10{}\mathrm {i}\right )}{3\,b^3\,d\,\sqrt {\cos \left (c+d\,x\right )}\,\left (15\,\cos \left (2\,c+2\,d\,x\right )+6\,\cos \left (4\,c+4\,d\,x\right )+\cos \left (6\,c+6\,d\,x\right )+10\right )} \]
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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