Optimal. Leaf size=205 \[ -\frac {a^6}{32 d (a-a \cos (c+d x))^4}-\frac {7 a^5}{48 d (a-a \cos (c+d x))^3}-\frac {15 a^4}{32 d (a-a \cos (c+d x))^2}-\frac {51 a^3}{32 d (a-a \cos (c+d x))}+\frac {a^4}{64 d (a+a \cos (c+d x))^2}+\frac {9 a^3}{64 d (a+a \cos (c+d x))}+\frac {303 a^2 \log (1-\cos (c+d x))}{128 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}-\frac {47 a^2 \log (1+\cos (c+d x))}{128 d}+\frac {a^2 \sec (c+d x)}{d} \]
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Rubi [A]
time = 0.18, antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2915, 12,
90} \begin {gather*} -\frac {a^6}{32 d (a-a \cos (c+d x))^4}-\frac {7 a^5}{48 d (a-a \cos (c+d x))^3}-\frac {15 a^4}{32 d (a-a \cos (c+d x))^2}+\frac {a^4}{64 d (a \cos (c+d x)+a)^2}-\frac {51 a^3}{32 d (a-a \cos (c+d x))}+\frac {9 a^3}{64 d (a \cos (c+d x)+a)}+\frac {a^2 \sec (c+d x)}{d}+\frac {303 a^2 \log (1-\cos (c+d x))}{128 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}-\frac {47 a^2 \log (\cos (c+d x)+1)}{128 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 90
Rule 2915
Rule 3957
Rubi steps
\begin {align*} \int \csc ^9(c+d x) (a+a \sec (c+d x))^2 \, dx &=\int (-a-a \cos (c+d x))^2 \csc ^9(c+d x) \sec ^2(c+d x) \, dx\\ &=\frac {a^9 \text {Subst}\left (\int \frac {a^2}{(-a-x)^5 x^2 (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^{11} \text {Subst}\left (\int \frac {1}{(-a-x)^5 x^2 (-a+x)^3} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {a^{11} \text {Subst}\left (\int \left (\frac {1}{32 a^7 (a-x)^3}+\frac {9}{64 a^8 (a-x)^2}+\frac {47}{128 a^9 (a-x)}+\frac {1}{a^8 x^2}-\frac {2}{a^9 x}+\frac {1}{8 a^5 (a+x)^5}+\frac {7}{16 a^6 (a+x)^4}+\frac {15}{16 a^7 (a+x)^3}+\frac {51}{32 a^8 (a+x)^2}+\frac {303}{128 a^9 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=-\frac {a^6}{32 d (a-a \cos (c+d x))^4}-\frac {7 a^5}{48 d (a-a \cos (c+d x))^3}-\frac {15 a^4}{32 d (a-a \cos (c+d x))^2}-\frac {51 a^3}{32 d (a-a \cos (c+d x))}+\frac {a^4}{64 d (a+a \cos (c+d x))^2}+\frac {9 a^3}{64 d (a+a \cos (c+d x))}+\frac {303 a^2 \log (1-\cos (c+d x))}{128 d}-\frac {2 a^2 \log (\cos (c+d x))}{d}-\frac {47 a^2 \log (1+\cos (c+d x))}{128 d}+\frac {a^2 \sec (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 2.28, size = 164, normalized size = 0.80 \begin {gather*} -\frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (1224 \csc ^2\left (\frac {1}{2} (c+d x)\right )+180 \csc ^4\left (\frac {1}{2} (c+d x)\right )+28 \csc ^6\left (\frac {1}{2} (c+d x)\right )+3 \csc ^8\left (\frac {1}{2} (c+d x)\right )-6 \left (18 \sec ^2\left (\frac {1}{2} (c+d x)\right )+\sec ^4\left (\frac {1}{2} (c+d x)\right )+4 \left (-47 \log \left (\cos \left (\frac {1}{2} (c+d x)\right )\right )-128 \log (\cos (c+d x))+303 \log \left (\sin \left (\frac {1}{2} (c+d x)\right )\right )+64 \sec (c+d x)\right )\right )\right )}{6144 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.16, size = 112, normalized size = 0.55
method | result | size |
derivativedivides | \(-\frac {a^{2} \left (-\sec \left (d x +c \right )+\frac {1}{32 \left (-1+\sec \left (d x +c \right )\right )^{4}}+\frac {13}{48 \left (-1+\sec \left (d x +c \right )\right )^{3}}+\frac {35}{32 \left (-1+\sec \left (d x +c \right )\right )^{2}}+\frac {99}{32 \left (-1+\sec \left (d x +c \right )\right )}-\frac {303 \ln \left (-1+\sec \left (d x +c \right )\right )}{128}-\frac {1}{64 \left (1+\sec \left (d x +c \right )\right )^{2}}+\frac {11}{64 \left (1+\sec \left (d x +c \right )\right )}+\frac {47 \ln \left (1+\sec \left (d x +c \right )\right )}{128}\right )}{d}\) | \(112\) |
default | \(-\frac {a^{2} \left (-\sec \left (d x +c \right )+\frac {1}{32 \left (-1+\sec \left (d x +c \right )\right )^{4}}+\frac {13}{48 \left (-1+\sec \left (d x +c \right )\right )^{3}}+\frac {35}{32 \left (-1+\sec \left (d x +c \right )\right )^{2}}+\frac {99}{32 \left (-1+\sec \left (d x +c \right )\right )}-\frac {303 \ln \left (-1+\sec \left (d x +c \right )\right )}{128}-\frac {1}{64 \left (1+\sec \left (d x +c \right )\right )^{2}}+\frac {11}{64 \left (1+\sec \left (d x +c \right )\right )}+\frac {47 \ln \left (1+\sec \left (d x +c \right )\right )}{128}\right )}{d}\) | \(112\) |
norman | \(\frac {\frac {a^{2}}{512 d}+\frac {37 a^{2} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{1536 d}+\frac {121 a^{2} \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{768 d}+\frac {233 a^{2} \left (\tan ^{6}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 d}+\frac {19 a^{2} \left (\tan ^{12}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 d}+\frac {a^{2} \left (\tan ^{14}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{256 d}-\frac {203 a^{2} \left (\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{8} \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {303 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{64 d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d}-\frac {2 a^{2} \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d}\) | \(210\) |
risch | \(\frac {a^{2} \left (525 \,{\mathrm e}^{13 i \left (d x +c \right )}-1716 \,{\mathrm e}^{12 i \left (d x +c \right )}+214 \,{\mathrm e}^{11 i \left (d x +c \right )}+4652 \,{\mathrm e}^{10 i \left (d x +c \right )}-4173 \,{\mathrm e}^{9 i \left (d x +c \right )}-2552 \,{\mathrm e}^{8 i \left (d x +c \right )}+4564 \,{\mathrm e}^{7 i \left (d x +c \right )}-2552 \,{\mathrm e}^{6 i \left (d x +c \right )}-4173 \,{\mathrm e}^{5 i \left (d x +c \right )}+4652 \,{\mathrm e}^{4 i \left (d x +c \right )}+214 \,{\mathrm e}^{3 i \left (d x +c \right )}-1716 \,{\mathrm e}^{2 i \left (d x +c \right )}+525 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{96 d \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )^{8} \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )^{4} \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}+\frac {303 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}-1\right )}{64 d}-\frac {47 a^{2} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+1\right )}{64 d}-\frac {2 a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(253\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 197, normalized size = 0.96 \begin {gather*} -\frac {141 \, a^{2} \log \left (\cos \left (d x + c\right ) + 1\right ) - 909 \, a^{2} \log \left (\cos \left (d x + c\right ) - 1\right ) + 768 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {2 \, {\left (525 \, a^{2} \cos \left (d x + c\right )^{6} - 858 \, a^{2} \cos \left (d x + c\right )^{5} - 734 \, a^{2} \cos \left (d x + c\right )^{4} + 1654 \, a^{2} \cos \left (d x + c\right )^{3} - 19 \, a^{2} \cos \left (d x + c\right )^{2} - 784 \, a^{2} \cos \left (d x + c\right ) + 192 \, a^{2}\right )}}{\cos \left (d x + c\right )^{7} - 2 \, \cos \left (d x + c\right )^{6} - \cos \left (d x + c\right )^{5} + 4 \, \cos \left (d x + c\right )^{4} - \cos \left (d x + c\right )^{3} - 2 \, \cos \left (d x + c\right )^{2} + \cos \left (d x + c\right )}}{384 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 461 vs.
\(2 (193) = 386\).
time = 2.82, size = 461, normalized size = 2.25 \begin {gather*} \frac {1050 \, a^{2} \cos \left (d x + c\right )^{6} - 1716 \, a^{2} \cos \left (d x + c\right )^{5} - 1468 \, a^{2} \cos \left (d x + c\right )^{4} + 3308 \, a^{2} \cos \left (d x + c\right )^{3} - 38 \, a^{2} \cos \left (d x + c\right )^{2} - 1568 \, a^{2} \cos \left (d x + c\right ) + 384 \, a^{2} - 768 \, {\left (a^{2} \cos \left (d x + c\right )^{7} - 2 \, a^{2} \cos \left (d x + c\right )^{6} - a^{2} \cos \left (d x + c\right )^{5} + 4 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (-\cos \left (d x + c\right )\right ) - 141 \, {\left (a^{2} \cos \left (d x + c\right )^{7} - 2 \, a^{2} \cos \left (d x + c\right )^{6} - a^{2} \cos \left (d x + c\right )^{5} + 4 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + 909 \, {\left (a^{2} \cos \left (d x + c\right )^{7} - 2 \, a^{2} \cos \left (d x + c\right )^{6} - a^{2} \cos \left (d x + c\right )^{5} + 4 \, a^{2} \cos \left (d x + c\right )^{4} - a^{2} \cos \left (d x + c\right )^{3} - 2 \, a^{2} \cos \left (d x + c\right )^{2} + a^{2} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{384 \, {\left (d \cos \left (d x + c\right )^{7} - 2 \, d \cos \left (d x + c\right )^{6} - d \cos \left (d x + c\right )^{5} + 4 \, d \cos \left (d x + c\right )^{4} - d \cos \left (d x + c\right )^{3} - 2 \, d \cos \left (d x + c\right )^{2} + d \cos \left (d x + c\right )\right )}} \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 [A]
time = 0.76, size = 291, normalized size = 1.42 \begin {gather*} \frac {3636 \, a^{2} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right ) - 3072 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) - \frac {120 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {6 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {{\left (3 \, a^{2} - \frac {40 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {282 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1680 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {7575 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) - 1\right )}^{4}} + \frac {3072 \, {\left (2 \, a^{2} + \frac {a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}}{\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1}}{1536 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.17, size = 203, normalized size = 0.99 \begin {gather*} \frac {-\frac {175\,a^2\,{\cos \left (c+d\,x\right )}^6}{64}+\frac {143\,a^2\,{\cos \left (c+d\,x\right )}^5}{32}+\frac {367\,a^2\,{\cos \left (c+d\,x\right )}^4}{96}-\frac {827\,a^2\,{\cos \left (c+d\,x\right )}^3}{96}+\frac {19\,a^2\,{\cos \left (c+d\,x\right )}^2}{192}+\frac {49\,a^2\,\cos \left (c+d\,x\right )}{12}-a^2}{d\,\left (-{\cos \left (c+d\,x\right )}^7+2\,{\cos \left (c+d\,x\right )}^6+{\cos \left (c+d\,x\right )}^5-4\,{\cos \left (c+d\,x\right )}^4+{\cos \left (c+d\,x\right )}^3+2\,{\cos \left (c+d\,x\right )}^2-\cos \left (c+d\,x\right )\right )}+\frac {303\,a^2\,\ln \left (\cos \left (c+d\,x\right )-1\right )}{128\,d}-\frac {47\,a^2\,\ln \left (\cos \left (c+d\,x\right )+1\right )}{128\,d}-\frac {2\,a^2\,\ln \left (\cos \left (c+d\,x\right )\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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