Optimal. Leaf size=107 \[ \frac {3 x \sqrt {\sec (c+d x)}}{8 b^2 \sqrt {b \sec (c+d x)}}+\frac {\sin (c+d x)}{4 b^2 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {b \sec (c+d x)}}+\frac {3 \sin (c+d x)}{8 b^2 d \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)}} \]
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Rubi [A]
time = 0.02, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {18, 2715, 8}
\begin {gather*} \frac {3 x \sqrt {\sec (c+d x)}}{8 b^2 \sqrt {b \sec (c+d x)}}+\frac {\sin (c+d x)}{4 b^2 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {b \sec (c+d x)}}+\frac {3 \sin (c+d x)}{8 b^2 d \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 18
Rule 2715
Rubi steps
\begin {align*} \int \frac {1}{\sec ^{\frac {3}{2}}(c+d x) (b \sec (c+d x))^{5/2}} \, dx &=\frac {\sqrt {\sec (c+d x)} \int \cos ^4(c+d x) \, dx}{b^2 \sqrt {b \sec (c+d x)}}\\ &=\frac {\sin (c+d x)}{4 b^2 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {b \sec (c+d x)}}+\frac {\left (3 \sqrt {\sec (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 b^2 \sqrt {b \sec (c+d x)}}\\ &=\frac {\sin (c+d x)}{4 b^2 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {b \sec (c+d x)}}+\frac {3 \sin (c+d x)}{8 b^2 d \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {\left (3 \sqrt {\sec (c+d x)}\right ) \int 1 \, dx}{8 b^2 \sqrt {b \sec (c+d x)}}\\ &=\frac {3 x \sqrt {\sec (c+d x)}}{8 b^2 \sqrt {b \sec (c+d x)}}+\frac {\sin (c+d x)}{4 b^2 d \sec ^{\frac {5}{2}}(c+d x) \sqrt {b \sec (c+d x)}}+\frac {3 \sin (c+d x)}{8 b^2 d \sqrt {\sec (c+d x)} \sqrt {b \sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 58, normalized size = 0.54 \begin {gather*} \frac {\sqrt {\sec (c+d x)} (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))}{32 b^2 d \sqrt {b \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 35.69, size = 74, normalized size = 0.69
method | result | size |
default | \(\frac {2 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 \sin \left (d x +c \right ) \cos \left (d x +c \right )+3 d x +3 c}{8 d \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \cos \left (d x +c \right )^{4}}\) | \(74\) |
risch | \(\frac {3 \,{\mathrm e}^{i \left (d x +c \right )} x}{8 b^{2} \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {i {\mathrm e}^{5 i \left (d x +c \right )}}{64 b^{2} \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}+\frac {i {\mathrm e}^{-i \left (d x +c \right )}}{8 b^{2} \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}-\frac {7 i \cos \left (3 d x +3 c \right )}{64 b^{2} \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}+\frac {9 \sin \left (3 d x +3 c \right )}{64 b^{2} \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right ) \sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}\) | \(408\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.66, size = 49, normalized size = 0.46 \begin {gather*} \frac {12 \, d x + 12 \, c + \sin \left (4 \, d x + 4 \, c\right ) + 8 \, \sin \left (\frac {1}{2} \, \arctan \left (\sin \left (4 \, d x + 4 \, c\right ), \cos \left (4 \, d x + 4 \, c\right )\right )\right )}{32 \, b^{\frac {5}{2}} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.62, size = 208, normalized size = 1.94 \begin {gather*} \left [\frac {\frac {2 \, {\left (2 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}} - 3 \, \sqrt {-b} \log \left (2 \, \sqrt {-b} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \cos \left (d x + c\right )^{\frac {3}{2}} \sin \left (d x + c\right ) + 2 \, b \cos \left (d x + c\right )^{2} - b\right )}{16 \, b^{3} d}, \frac {\frac {{\left (2 \, \cos \left (d x + c\right )^{4} + 3 \, \cos \left (d x + c\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {\cos \left (d x + c\right )}} + 3 \, \sqrt {b} \arctan \left (\frac {\sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{\sqrt {b} \sqrt {\cos \left (d x + c\right )}}\right )}{8 \, 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 [B]
time = 0.46, size = 55, normalized size = 0.51 \begin {gather*} \frac {\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}\,\left (8\,\sin \left (2\,c+2\,d\,x\right )+\sin \left (4\,c+4\,d\,x\right )+12\,d\,x\right )}{32\,b^3\,d\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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