Optimal. Leaf size=95 \[ \frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 95, 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, 3854, 3856,
2719} \begin {gather*} \frac {2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 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 3854
Rule 3856
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x)}{\sqrt {b \sec (c+d x)}} \, dx &=b^4 \int \frac {1}{(b \sec (c+d x))^{9/2}} \, dx\\ &=\frac {2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {1}{9} \left (7 b^2\right ) \int \frac {1}{(b \sec (c+d x))^{5/2}} \, dx\\ &=\frac {2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {7}{15} \int \frac {1}{\sqrt {b \sec (c+d x)}} \, dx\\ &=\frac {2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}+\frac {7 \int \sqrt {\cos (c+d x)} \, dx}{15 \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}\\ &=\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 d \sqrt {\cos (c+d x)} \sqrt {b \sec (c+d x)}}+\frac {2 b^3 \sin (c+d x)}{9 d (b \sec (c+d x))^{7/2}}+\frac {14 b \sin (c+d x)}{45 d (b \sec (c+d x))^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 70, normalized size = 0.74 \begin {gather*} \frac {\frac {336 E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{\sqrt {\cos (c+d x)}}+4 \cos (c+d x) (33 \sin (c+d x)+5 \sin (3 (c+d x)))}{360 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 = 30.84, size = 328, normalized size = 3.45
method | result | size |
default | \(\frac {2 \left (21 i \sin \left (d x +c \right ) \cos \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \EllipticF \left (\frac {i \left (\cos \left (d x +c \right )-1\right )}{\sin \left (d x +c \right )}, i\right )-21 i \cos \left (d x +c \right ) \EllipticE \left (\frac {i \left (\cos \left (d x +c \right )-1\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sin \left (d x +c \right )-5 \left (\cos ^{6}\left (d x +c \right )\right )+21 i \EllipticF \left (\frac {i \left (\cos \left (d x +c \right )-1\right )}{\sin \left (d x +c \right )}, i\right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}\, \sin \left (d x +c \right )-21 i \EllipticE \left (\frac {i \left (\cos \left (d x +c \right )-1\right )}{\sin \left (d x +c \right )}, i\right ) \sin \left (d x +c \right ) \sqrt {\frac {1}{\cos \left (d x +c \right )+1}}\, \sqrt {\frac {\cos \left (d x +c \right )}{\cos \left (d x +c \right )+1}}-2 \left (\cos ^{4}\left (d x +c \right )\right )-14 \left (\cos ^{2}\left (d x +c \right )\right )+21 \cos \left (d x +c \right )\right ) \sqrt {\frac {b}{\cos \left (d x +c \right )}}}{45 d \sin \left (d x +c \right ) b}\) | \(328\) |
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 = 0.96, size = 108, normalized size = 1.14 \begin {gather*} \frac {2 \, {\left (5 \, \cos \left (d x + c\right )^{4} + 7 \, \cos \left (d x + c\right )^{2}\right )} \sqrt {\frac {b}{\cos \left (d x + c\right )}} \sin \left (d x + c\right ) + 21 i \, \sqrt {2} \sqrt {b} {\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 ) - 21 i \, \sqrt {2} \sqrt {b} {\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 )}{45 \, b d} \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 {\cos ^{4}{\left (c + d x \right )}}{\sqrt {b \sec {\left (c + d x \right )}}}\, 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 {{\cos \left (c+d\,x\right )}^4}{\sqrt {\frac {b}{\cos \left (c+d\,x\right )}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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