Optimal. Leaf size=138 \[ \frac {\sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{3 d}-\frac {3 F\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{\sqrt {7} d}-\frac {5 \Pi \left (2;\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{3 \sqrt {7} d}-\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {\sqrt {3-4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 d} \]
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Rubi [A]
time = 0.24, antiderivative size = 138, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {2875, 3134,
3138, 2733, 3081, 2741, 2885} \begin {gather*} -\frac {3 F\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{\sqrt {7} d}+\frac {\sqrt {7} E\left (\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{3 d}-\frac {5 \Pi \left (2;\frac {1}{2} (c+d x+\pi )|\frac {8}{7}\right )}{3 \sqrt {7} d}-\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x) \sec (c+d x)}{2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 2733
Rule 2741
Rule 2875
Rule 2885
Rule 3081
Rule 3134
Rule 3138
Rubi steps
\begin {align*} \int \sqrt {3-4 \cos (c+d x)} \sec ^3(c+d x) \, dx &=\frac {\sqrt {3-4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{2} \int \frac {\left (-2+3 \cos (c+d x)-2 \cos ^2(c+d x)\right ) \sec ^2(c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx\\ &=-\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {\sqrt {3-4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{6} \int \frac {\left (5-6 \cos (c+d x)-4 \cos ^2(c+d x)\right ) \sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx\\ &=-\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {\sqrt {3-4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 d}+\frac {1}{24} \int \frac {(20-36 \cos (c+d x)) \sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx+\frac {1}{6} \int \sqrt {3-4 \cos (c+d x)} \, dx\\ &=\frac {\sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{3 d}-\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {\sqrt {3-4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 d}+\frac {5}{6} \int \frac {\sec (c+d x)}{\sqrt {3-4 \cos (c+d x)}} \, dx-\frac {3}{2} \int \frac {1}{\sqrt {3-4 \cos (c+d x)}} \, dx\\ &=\frac {\sqrt {7} E\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{3 d}-\frac {3 F\left (\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{\sqrt {7} d}-\frac {5 \Pi \left (2;\frac {1}{2} (c+\pi +d x)|\frac {8}{7}\right )}{3 \sqrt {7} d}-\frac {\sqrt {3-4 \cos (c+d x)} \tan (c+d x)}{3 d}+\frac {\sqrt {3-4 \cos (c+d x)} \sec (c+d x) \tan (c+d x)}{2 d}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 12.02, size = 237, normalized size = 1.72 \begin {gather*} \frac {-\frac {12 \sqrt {-3+4 \cos (c+d x)} F\left (\left .\frac {1}{2} (c+d x)\right |8\right )}{\sqrt {3-4 \cos (c+d x)}}+\frac {6 \sqrt {-3+4 \cos (c+d x)} \Pi \left (2;\left .\frac {1}{2} (c+d x)\right |8\right )}{\sqrt {3-4 \cos (c+d x)}}+\frac {2 i \left (21 E\left (i \sinh ^{-1}\left (\sqrt {3-4 \cos (c+d x)}\right )|-\frac {1}{7}\right )-12 F\left (i \sinh ^{-1}\left (\sqrt {3-4 \cos (c+d x)}\right )|-\frac {1}{7}\right )-8 \Pi \left (-\frac {1}{3};i \sinh ^{-1}\left (\sqrt {3-4 \cos (c+d x)}\right )|-\frac {1}{7}\right )\right ) \sin (c+d x)}{3 \sqrt {7} \sqrt {\sin ^2(c+d x)}}-\sqrt {3-4 \cos (c+d x)} (-3+2 \cos (c+d x)) \sec (c+d x) \tan (c+d x)}{6 d} \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. \(407\) vs.
\(2(195)=390\).
time = 0.27, size = 408, normalized size = 2.96
method | result | size |
default | \(-\frac {\sqrt {-\left (8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7\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 {8 \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 )}}{\left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )^{2}}+\frac {2 \cos \left (\frac {d x}{2}+\frac {c}{2}\right ) \sqrt {8 \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 )}}{3 \left (2 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-1\right )}-\frac {3 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticF \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )}{7 \sqrt {8 \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 )}}+\frac {\sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticE \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), \frac {2 \sqrt {14}}{7}\right )}{3 \sqrt {8 \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 )}}-\frac {5 \sqrt {\frac {1}{2}-\frac {\cos \left (d x +c \right )}{2}}\, \sqrt {56 \left (\sin ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-7}\, \EllipticPi \left (\cos \left (\frac {d x}{2}+\frac {c}{2}\right ), 2, \frac {2 \sqrt {14}}{7}\right )}{21 \sqrt {8 \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 {-8 \left (\cos ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+7}\, d}\) | \(408\) |
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 [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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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {3 - 4 \cos {\left (c + d x \right )}} \sec ^{3}{\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 {\sqrt {3-4\,\cos \left (c+d\,x\right )}}{{\cos \left (c+d\,x\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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