Optimal. Leaf size=126 \[ -\frac {14 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^2 d \sqrt {\cos (c+d x)}}+\frac {2 b^3 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 b d \sqrt {b \cos (c+d x)}} \]
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Rubi [A]
time = 0.07, antiderivative size = 126, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {16, 2716, 2721,
2719} \begin {gather*} \frac {2 b^3 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}-\frac {14 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{15 b^2 d \sqrt {\cos (c+d x)}}+\frac {14 b \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 b d \sqrt {b \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2716
Rule 2719
Rule 2721
Rubi steps
\begin {align*} \int \frac {\sec ^4(c+d x)}{(b \cos (c+d x))^{3/2}} \, dx &=b^4 \int \frac {1}{(b \cos (c+d x))^{11/2}} \, dx\\ &=\frac {2 b^3 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {1}{9} \left (7 b^2\right ) \int \frac {1}{(b \cos (c+d x))^{7/2}} \, dx\\ &=\frac {2 b^3 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {7}{15} \int \frac {1}{(b \cos (c+d x))^{3/2}} \, dx\\ &=\frac {2 b^3 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 b d \sqrt {b \cos (c+d x)}}-\frac {7 \int \sqrt {b \cos (c+d x)} \, dx}{15 b^2}\\ &=\frac {2 b^3 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 b d \sqrt {b \cos (c+d x)}}-\frac {\left (7 \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 b^2 \sqrt {\cos (c+d x)}}\\ &=-\frac {14 \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^2 d \sqrt {\cos (c+d x)}}+\frac {2 b^3 \sin (c+d x)}{9 d (b \cos (c+d x))^{9/2}}+\frac {14 b \sin (c+d x)}{45 d (b \cos (c+d x))^{5/2}}+\frac {14 \sin (c+d x)}{15 b d \sqrt {b \cos (c+d x)}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 80, normalized size = 0.63 \begin {gather*} \frac {-42 \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+42 \sin (c+d x)+2 \sec (c+d x) \left (7+5 \sec ^2(c+d x)\right ) \tan (c+d x)}{45 b d \sqrt {b \cos (c+d x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(413\) vs.
\(2(134)=268\).
time = 0.13, size = 414, normalized size = 3.29
method | result | size |
default | \(-\frac {2 \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-\frac {\cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}{144 b \left (-\frac {1}{2}+\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{5}}-\frac {7 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}{180 b \left (-\frac {1}{2}+\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {14 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )}{15 \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}}+\frac {7 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )}{15 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}-\frac {7 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {-2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+1}\, \left (\EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-\EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{15 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}}\right )}{b \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}\) | \(414\) |
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 = 0.12, size = 131, normalized size = 1.04 \begin {gather*} \frac {-21 i \, \sqrt {2} \sqrt {b} \cos \left (d x + c\right )^{5} {\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} \cos \left (d x + c\right )^{5} {\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 (21 \, \cos \left (d x + c\right )^{4} + 7 \, \cos \left (d x + c\right )^{2} + 5\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{45 \, b^{2} d \cos \left (d x + c\right )^{5}} \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 ^{4}{\left (c + d x \right )}}{\left (b \cos {\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 )}^4\,{\left (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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