Optimal. Leaf size=101 \[ \frac {3 b x \sqrt {b \sec (c+d x)}}{8 \sqrt {\sec (c+d x)}}+\frac {b \sqrt {b \sec (c+d x)} \sin (c+d x)}{4 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {3 b \sqrt {b \sec (c+d x)} \sin (c+d x)}{8 d \sec ^{\frac {3}{2}}(c+d x)} \]
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Rubi [A]
time = 0.02, antiderivative size = 101, 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 = {17, 2715, 8}
\begin {gather*} \frac {3 b x \sqrt {b \sec (c+d x)}}{8 \sqrt {\sec (c+d x)}}+\frac {3 b \sin (c+d x) \sqrt {b \sec (c+d x)}}{8 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {b \sin (c+d x) \sqrt {b \sec (c+d x)}}{4 d \sec ^{\frac {7}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 17
Rule 2715
Rubi steps
\begin {align*} \int \frac {(b \sec (c+d x))^{3/2}}{\sec ^{\frac {11}{2}}(c+d x)} \, dx &=\frac {\left (b \sqrt {b \sec (c+d x)}\right ) \int \cos ^4(c+d x) \, dx}{\sqrt {\sec (c+d x)}}\\ &=\frac {b \sqrt {b \sec (c+d x)} \sin (c+d x)}{4 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {\left (3 b \sqrt {b \sec (c+d x)}\right ) \int \cos ^2(c+d x) \, dx}{4 \sqrt {\sec (c+d x)}}\\ &=\frac {b \sqrt {b \sec (c+d x)} \sin (c+d x)}{4 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {3 b \sqrt {b \sec (c+d x)} \sin (c+d x)}{8 d \sec ^{\frac {3}{2}}(c+d x)}+\frac {\left (3 b \sqrt {b \sec (c+d x)}\right ) \int 1 \, dx}{8 \sqrt {\sec (c+d x)}}\\ &=\frac {3 b x \sqrt {b \sec (c+d x)}}{8 \sqrt {\sec (c+d x)}}+\frac {b \sqrt {b \sec (c+d x)} \sin (c+d x)}{4 d \sec ^{\frac {7}{2}}(c+d x)}+\frac {3 b \sqrt {b \sec (c+d x)} \sin (c+d x)}{8 d \sec ^{\frac {3}{2}}(c+d x)}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 55, normalized size = 0.54 \begin {gather*} \frac {(b \sec (c+d x))^{3/2} (12 (c+d x)+8 \sin (2 (c+d x))+\sin (4 (c+d x)))}{32 d \sec ^{\frac {3}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 37.97, size = 74, normalized size = 0.73
method | result | size |
default | \(\frac {\left (2 \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right )+3 \sin \left (d x +c \right ) \cos \left (d x +c \right )+3 d x +3 c \right ) \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}}}{8 d \cos \left (d x +c \right )^{4} \left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {11}{2}}}\) | \(74\) |
risch | \(\frac {3 b \sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, x}{8 \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}-\frac {i b \sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, {\mathrm e}^{2 i \left (d x +c \right )}}{8 \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}+\frac {i b \sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, {\mathrm e}^{-2 i \left (d x +c \right )}}{8 \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}+\frac {b \sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, \sin \left (4 d x +4 c \right )}{32 \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d}\) | \(257\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.62, size = 53, normalized size = 0.52 \begin {gather*} \frac {{\left (12 \, {\left (d x + c\right )} b + b \sin \left (4 \, d x + 4 \, c\right ) + 8 \, b \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 )\right )} \sqrt {b}}{32 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.69, size = 207, normalized size = 2.05 \begin {gather*} \left [\frac {3 \, \sqrt {-b} 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 ) + \frac {2 \, {\left (2 \, b \cos \left (d x + c\right )^{4} + 3 \, b \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 )}}}{16 \, d}, \frac {3 \, b^{\frac {3}{2}} \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 ) + \frac {{\left (2 \, b \cos \left (d x + c\right )^{4} + 3 \, b \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 )}}}{8 \, d}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = 0.59, size = 53, normalized size = 0.52 \begin {gather*} \frac {b\,\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\,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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