Optimal. Leaf size=32 \[ \frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d \sqrt {b \sec (c+d x)}} \]
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Rubi [A]
time = 0.01, antiderivative size = 32, 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, 3852, 8}
\begin {gather*} \frac {\sin (c+d x) \sec ^{\frac {3}{2}}(c+d x)}{d \sqrt {b \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 17
Rule 3852
Rubi steps
\begin {align*} \int \frac {\sec ^{\frac {5}{2}}(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx &=\frac {\sqrt {\sec (c+d x)} \int \sec ^2(c+d x) \, dx}{\sqrt {b \sec (c+d x)}}\\ &=-\frac {\sqrt {\sec (c+d x)} \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{d \sqrt {b \sec (c+d x)}}\\ &=\frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d \sqrt {b \sec (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.02, size = 32, normalized size = 1.00 \begin {gather*} \frac {\sec ^{\frac {3}{2}}(c+d x) \sin (c+d x)}{d \sqrt {b \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 35.92, size = 39, normalized size = 1.22
method | result | size |
default | \(\frac {\left (\frac {1}{\cos \left (d x +c \right )}\right )^{\frac {5}{2}} \cos \left (d x +c \right ) \sin \left (d x +c \right )}{d \sqrt {\frac {b}{\cos \left (d x +c \right )}}}\) | \(39\) |
risch | \(\frac {2 i \sqrt {\frac {{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}}{\sqrt {\frac {b \,{\mathrm e}^{i \left (d x +c \right )}}{{\mathrm e}^{2 i \left (d x +c \right )}+1}}\, d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 59 vs.
\(2 (28) = 56\).
time = 0.59, size = 59, normalized size = 1.84 \begin {gather*} \frac {2 \, \sqrt {b} \sin \left (2 \, d x + 2 \, c\right )}{{\left (b \cos \left (2 \, d x + 2 \, c\right )^{2} + b \sin \left (2 \, d x + 2 \, c\right )^{2} + 2 \, b \cos \left (2 \, d x + 2 \, c\right ) + b\right )} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.17, size = 33, normalized size = 1.03 \begin {gather*} \frac {\sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{b d \sqrt {\cos \left (d x + c\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 = 0.27, size = 51, normalized size = 1.59 \begin {gather*} \frac {\left (\cos \left (d\,x\right )-\sin \left (d\,x\right )\,1{}\mathrm {i}\right )\,\left (\cos \left (c\right )-\sin \left (c\right )\,1{}\mathrm {i}\right )\,\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}\,\sqrt {\frac {1}{\cos \left (c+d\,x\right )}}\,1{}\mathrm {i}}{b\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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