Optimal. Leaf size=100 \[ -\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {6 \sqrt {b \sec (c+d x)} \sin (c+d x)}{5 b^2 d}+\frac {2 (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 b^4 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 3853, 3856,
2719} \begin {gather*} \frac {2 \sin (c+d x) (b \sec (c+d x))^{5/2}}{5 b^4 d}+\frac {6 \sin (c+d x) \sqrt {b \sec (c+d x)}}{5 b^2 d}-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2719
Rule 3853
Rule 3856
Rubi steps
\begin {align*} \int \frac {\sec ^5(c+d x)}{(b \sec (c+d x))^{3/2}} \, dx &=\frac {\int (b \sec (c+d x))^{7/2} \, dx}{b^5}\\ &=\frac {2 (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 b^4 d}+\frac {3 \int (b \sec (c+d x))^{3/2} \, dx}{5 b^3}\\ &=\frac {6 \sqrt {b \sec (c+d x)} \sin (c+d x)}{5 b^2 d}+\frac {2 (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 b^4 d}-\frac {3 \int \frac {1}{\sqrt {b \sec (c+d x)}} \, dx}{5 b}\\ &=\frac {6 \sqrt {b \sec (c+d x)} \sin (c+d x)}{5 b^2 d}+\frac {2 (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 b^4 d}-\frac {3 \int \sqrt {\cos (c+d x)} \, dx}{5 b \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}\\ &=-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 b d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {6 \sqrt {b \sec (c+d x)} \sin (c+d x)}{5 b^2 d}+\frac {2 (b \sec (c+d x))^{5/2} \sin (c+d x)}{5 b^4 d}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 64, normalized size = 0.64 \begin {gather*} \frac {-\frac {6 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)}}+2 \left (3+\sec ^2(c+d x)\right ) \tan (c+d x)}{5 b d \sqrt {b \sec (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 48.59, size = 356, normalized size = 3.56
method | result | size |
default | \(\frac {2 \left (\cos \left (d x +c \right )+1\right )^{2} \left (\cos \left (d x +c \right )-1\right )^{2} \left (3 i \left (\cos ^{3}\left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \EllipticE \left (\frac {i \left (\cos \left (d x +c \right )-1\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-3 i \left (\cos ^{3}\left (d x +c \right )\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (d x +c \right )-1\right )}{\sin \left (d x +c \right )}, i\right )+3 i \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \EllipticE \left (\frac {i \left (\cos \left (d x +c \right )-1\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-3 i \left (\cos ^{2}\left (d x +c \right )\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (d x +c \right )-1\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right )-3 \left (\cos ^{3}\left (d x +c \right )\right )+2 \left (\cos ^{2}\left (d x +c \right )\right )+1\right )}{5 d \sin \left (d x +c \right )^{5} \left (\frac {b}{\cos \left (d x +c \right )}\right )^{\frac {3}{2}} \cos \left (d x +c \right )^{4}}\) | \(356\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order
4.
time = 0.92, size = 123, normalized size = 1.23 \begin {gather*} \frac {-3 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 3 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{2} {\rm weierstrassZeta}\left (-4, 0, {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) + 2 \, {\left (3 \, \cos \left (d x + c\right )^{2} + 1\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right )}{5 \, b^{2} d \cos \left (d x + c\right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sec ^{5}{\left (c + d x \right )}}{\left (b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {1}{{\cos \left (c+d\,x\right )}^5\,{\left (\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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