Optimal. Leaf size=169 \[ \frac {6 A b \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {10 b^2 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {10 b B \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d} \]
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Rubi [A]
time = 0.10, antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 6, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.207, Rules used = {16, 2827, 2715,
2721, 2719, 2720} \begin {gather*} \frac {2 A \sin (c+d x) (b \cos (c+d x))^{3/2}}{5 d}+\frac {6 A b E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {b \cos (c+d x)}}{5 d \sqrt {\cos (c+d x)}}+\frac {10 b^2 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {2 B \sin (c+d x) (b \cos (c+d x))^{5/2}}{7 b d}+\frac {10 b B \sin (c+d x) \sqrt {b \cos (c+d x)}}{21 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 16
Rule 2715
Rule 2719
Rule 2720
Rule 2721
Rule 2827
Rubi steps
\begin {align*} \int \cos (c+d x) (b \cos (c+d x))^{3/2} (A+B \cos (c+d x)) \, dx &=\frac {\int (b \cos (c+d x))^{5/2} (A+B \cos (c+d x)) \, dx}{b}\\ &=\frac {A \int (b \cos (c+d x))^{5/2} \, dx}{b}+\frac {B \int (b \cos (c+d x))^{7/2} \, dx}{b^2}\\ &=\frac {2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {1}{5} (3 A b) \int \sqrt {b \cos (c+d x)} \, dx+\frac {1}{7} (5 B) \int (b \cos (c+d x))^{3/2} \, dx\\ &=\frac {10 b B \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {1}{21} \left (5 b^2 B\right ) \int \frac {1}{\sqrt {b \cos (c+d x)}} \, dx+\frac {\left (3 A b \sqrt {b \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \sqrt {\cos (c+d x)}}\\ &=\frac {6 A b \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {10 b B \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}+\frac {\left (5 b^2 B \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{21 \sqrt {b \cos (c+d x)}}\\ &=\frac {6 A b \sqrt {b \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 d \sqrt {\cos (c+d x)}}+\frac {10 b^2 B \sqrt {\cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{21 d \sqrt {b \cos (c+d x)}}+\frac {10 b B \sqrt {b \cos (c+d x)} \sin (c+d x)}{21 d}+\frac {2 A (b \cos (c+d x))^{3/2} \sin (c+d x)}{5 d}+\frac {2 B (b \cos (c+d x))^{5/2} \sin (c+d x)}{7 b d}\\ \end {align*}
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Mathematica [A]
time = 0.32, size = 103, normalized size = 0.61 \begin {gather*} \frac {(b \cos (c+d x))^{5/2} \left (252 A E\left (\left .\frac {1}{2} (c+d x)\right |2\right )+100 B F\left (\left .\frac {1}{2} (c+d x)\right |2\right )+2 \sqrt {\cos (c+d x)} (65 B+42 A \cos (c+d x)+15 B \cos (2 (c+d x))) \sin (c+d x)\right )}{210 b d \cos ^{\frac {5}{2}}(c+d x)} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.25, size = 301, normalized size = 1.78
method | result | size |
default | \(-\frac {2 \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}\, b^{2} \left (240 B \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \left (\sin ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+\left (-168 A -360 B \right ) \left (\sin ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (168 A +280 B \right ) \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )+\left (-42 A -80 B \right ) \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) \cos \left (\frac {d x}{2}+\frac {c}{2}\right )-63 A \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )+25 B \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {2 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \sqrt {2}\right )\right )}{105 \sqrt {-b \left (2 \left (\sin ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-\left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )\right )}\, \sin \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {b \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}\, d}\) | \(301\) |
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 = 0.11, size = 165, normalized size = 0.98 \begin {gather*} \frac {-25 i \, \sqrt {2} B b^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 25 i \, \sqrt {2} B b^{\frac {3}{2}} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 63 i \, \sqrt {2} A b^{\frac {3}{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 ) - 63 i \, \sqrt {2} A b^{\frac {3}{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 (15 \, B b \cos \left (d x + c\right )^{2} + 21 \, A b \cos \left (d x + c\right ) + 25 \, B b\right )} \sqrt {b \cos \left (d x + c\right )} \sin \left (d x + c\right )}{105 \, d} \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 [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \cos \left (c+d\,x\right )\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{3/2}\,\left (A+B\,\cos \left (c+d\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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