Optimal. Leaf size=87 \[ \frac {2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {10 a^2 \tan (c+d x)}{3 d}-\frac {2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))}-\frac {a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2} \]
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Rubi [A]
time = 0.22, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 8, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.381, Rules used = {3957, 2948,
2845, 3057, 2827, 3852, 8, 3855} \begin {gather*} -\frac {a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}+\frac {10 a^2 \tan (c+d x)}{3 d}+\frac {2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 2827
Rule 2845
Rule 2948
Rule 3057
Rule 3852
Rule 3855
Rule 3957
Rubi steps
\begin {align*} \int \csc ^4(c+d x) (a+a \sec (c+d x))^2 \, dx &=\int (-a-a \cos (c+d x))^2 \csc ^4(c+d x) \sec ^2(c+d x) \, dx\\ &=a^4 \int \frac {\sec ^2(c+d x)}{(-a+a \cos (c+d x))^2} \, dx\\ &=-\frac {a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}+\frac {1}{3} a^2 \int \frac {(-4 a-2 a \cos (c+d x)) \sec ^2(c+d x)}{-a+a \cos (c+d x)} \, dx\\ &=-\frac {2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))}-\frac {a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}+\frac {1}{3} \int \left (10 a^2+6 a^2 \cos (c+d x)\right ) \sec ^2(c+d x) \, dx\\ &=-\frac {2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))}-\frac {a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}+\left (2 a^2\right ) \int \sec (c+d x) \, dx+\frac {1}{3} \left (10 a^2\right ) \int \sec ^2(c+d x) \, dx\\ &=\frac {2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}-\frac {2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))}-\frac {a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}-\frac {\left (10 a^2\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{3 d}\\ &=\frac {2 a^2 \tanh ^{-1}(\sin (c+d x))}{d}+\frac {10 a^2 \tan (c+d x)}{3 d}-\frac {2 a^2 \tan (c+d x)}{d (1-\cos (c+d x))}-\frac {a^4 \tan (c+d x)}{3 d (a-a \cos (c+d x))^2}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(228\) vs. \(2(87)=174\).
time = 1.27, size = 228, normalized size = 2.62 \begin {gather*} \frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (-\cot \left (\frac {c}{2}\right ) \csc ^2\left (\frac {1}{2} (c+d x)\right )-(-8+7 \cos (c+d x)) \csc \left (\frac {c}{2}\right ) \csc ^3\left (\frac {1}{2} (c+d x)\right ) \sin \left (\frac {d x}{2}\right )+6 \left (-2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+2 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\frac {\sin (d x)}{\left (\cos \left (\frac {c}{2}\right )-\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {c}{2}\right )+\sin \left (\frac {c}{2}\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}\right )\right )}{24 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.09, size = 117, normalized size = 1.34
method | result | size |
norman | \(\frac {\frac {a^{2}}{6 d}+\frac {7 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 d}-\frac {9 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}+\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(116\) |
derivativedivides | \(\frac {a^{2} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+2 a^{2} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )}{d}\) | \(117\) |
default | \(\frac {a^{2} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3} \cos \left (d x +c \right )}+\frac {4}{3 \sin \left (d x +c \right ) \cos \left (d x +c \right )}-\frac {8 \cot \left (d x +c \right )}{3}\right )+2 a^{2} \left (-\frac {1}{3 \sin \left (d x +c \right )^{3}}-\frac {1}{\sin \left (d x +c \right )}+\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )\right )+a^{2} \left (-\frac {2}{3}-\frac {\left (\csc ^{2}\left (d x +c \right )\right )}{3}\right ) \cot \left (d x +c \right )}{d}\) | \(117\) |
risch | \(-\frac {4 i a^{2} \left (3 \,{\mathrm e}^{4 i \left (d x +c \right )}-9 \,{\mathrm e}^{3 i \left (d x +c \right )}+11 \,{\mathrm e}^{2 i \left (d x +c \right )}-12 \,{\mathrm e}^{i \left (d x +c \right )}+5\right )}{3 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{3} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{d}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 113, normalized size = 1.30 \begin {gather*} -\frac {a^{2} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{2} + 1\right )}}{\sin \left (d x + c\right )^{3}} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + a^{2} {\left (\frac {6 \, \tan \left (d x + c\right )^{2} + 1}{\tan \left (d x + c\right )^{3}} - 3 \, \tan \left (d x + c\right )\right )} + \frac {{\left (3 \, \tan \left (d x + c\right )^{2} + 1\right )} a^{2}}{\tan \left (d x + c\right )^{3}}}{3 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.74, size = 159, normalized size = 1.83 \begin {gather*} -\frac {10 \, a^{2} \cos \left (d x + c\right )^{3} - 4 \, a^{2} \cos \left (d x + c\right )^{2} - 11 \, a^{2} \cos \left (d x + c\right ) - 3 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 3 \, {\left (a^{2} \cos \left (d x + c\right )^{2} - a^{2} \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) \sin \left (d x + c\right ) + 3 \, a^{2}}{3 \, {\left (d \cos \left (d x + c\right )^{2} - d \cos \left (d x + c\right )\right )} \sin \left (d x + c\right )} \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*} a^{2} \left (\int 2 \csc ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}\, dx + \int \csc ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}\, dx + \int \csc ^{4}{\left (c + d x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.47, size = 104, normalized size = 1.20 \begin {gather*} \frac {12 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) + 1 \right |}\right ) - 12 \, a^{2} \log \left ({\left | \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right ) - 1 \right |}\right ) - \frac {12 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} - 1} - \frac {15 \, a^{2} \tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{2} + a^{2}}{\tan \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )^{3}}}{6 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.48, size = 91, normalized size = 1.05 \begin {gather*} \frac {4\,a^2\,\mathrm {atanh}\left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}{d}-\frac {-9\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+\frac {14\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{3}+\frac {a^2}{3}}{d\,\left (2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3-2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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