Optimal. Leaf size=115 \[ \frac {2 (9 A+7 C) \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{15 b^2 d \sqrt {\cos (c+d x)}}+\frac {2 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 b^3 d}+\frac {2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^5 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 115, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.152, Rules used = {16, 3093, 2715,
2721, 2719} \begin {gather*} \frac {2 (9 A+7 C) \sin (c+d x) (b \cos (c+d x))^{3/2}}{45 b^3 d}+\frac {2 (9 A+7 C) 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 {2 C \sin (c+d x) (b \cos (c+d x))^{7/2}}{9 b^5 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2715
Rule 2719
Rule 2721
Rule 3093
Rubi steps
\begin {align*} \int \frac {\cos ^4(c+d x) \left (A+C \cos ^2(c+d x)\right )}{(b \cos (c+d x))^{3/2}} \, dx &=\frac {\int (b \cos (c+d x))^{5/2} \left (A+C \cos ^2(c+d x)\right ) \, dx}{b^4}\\ &=\frac {2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^5 d}+\frac {(9 A+7 C) \int (b \cos (c+d x))^{5/2} \, dx}{9 b^4}\\ &=\frac {2 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 b^3 d}+\frac {2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^5 d}+\frac {(9 A+7 C) \int \sqrt {b \cos (c+d x)} \, dx}{15 b^2}\\ &=\frac {2 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 b^3 d}+\frac {2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^5 d}+\frac {\left ((9 A+7 C) \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{15 b^2 \sqrt {\cos (c+d x)}}\\ &=\frac {2 (9 A+7 C) \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 (9 A+7 C) (b \cos (c+d x))^{3/2} \sin (c+d x)}{45 b^3 d}+\frac {2 C (b \cos (c+d x))^{7/2} \sin (c+d x)}{9 b^5 d}\\ \end {align*}
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Mathematica [A]
time = 0.45, size = 86, normalized size = 0.75 \begin {gather*} \frac {6 (9 A+7 C) \sqrt {\cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+\cos ^2(c+d x) (18 A+19 C+5 C \cos (2 (c+d x))) \sin (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. \(323\) vs.
\(2(127)=254\).
time = 0.39, size = 324, normalized size = 2.82
method | result | size |
default | \(-\frac {2 \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, \left (-160 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{10}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+320 C \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-72 A -296 C \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (72 A +136 C \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-18 A -24 C \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-27 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )-21 C \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{45 b \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}\) | \(324\) |
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 = 0.14, size = 125, normalized size = 1.09 \begin {gather*} -\frac {3 \, \sqrt {2} {\left (-9 i \, A - 7 i \, C\right )} \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 ) + 3 \, \sqrt {2} {\left (9 i \, A + 7 i \, C\right )} \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 ) - 2 \, {\left (5 \, C \cos \left (d x + c\right )^{3} + {\left (9 \, A + 7 \, C\right )} \cos \left (d x + c\right )\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{45 \, b^{2} d} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\cos \left (c+d\,x\right )}^4\,\left (C\,{\cos \left (c+d\,x\right )}^2+A\right )}{{\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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