Optimal. Leaf size=192 \[ -\frac {a^2 \log (\cos (c+d x))}{d}+\frac {2 a^2 \sec (c+d x)}{d}-\frac {3 a^2 \sec ^2(c+d x)}{2 d}-\frac {8 a^2 \sec ^3(c+d x)}{3 d}+\frac {a^2 \sec ^4(c+d x)}{2 d}+\frac {12 a^2 \sec ^5(c+d x)}{5 d}+\frac {a^2 \sec ^6(c+d x)}{3 d}-\frac {8 a^2 \sec ^7(c+d x)}{7 d}-\frac {3 a^2 \sec ^8(c+d x)}{8 d}+\frac {2 a^2 \sec ^9(c+d x)}{9 d}+\frac {a^2 \sec ^{10}(c+d x)}{10 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3964, 90}
\begin {gather*} \frac {a^2 \sec ^{10}(c+d x)}{10 d}+\frac {2 a^2 \sec ^9(c+d x)}{9 d}-\frac {3 a^2 \sec ^8(c+d x)}{8 d}-\frac {8 a^2 \sec ^7(c+d x)}{7 d}+\frac {a^2 \sec ^6(c+d x)}{3 d}+\frac {12 a^2 \sec ^5(c+d x)}{5 d}+\frac {a^2 \sec ^4(c+d x)}{2 d}-\frac {8 a^2 \sec ^3(c+d x)}{3 d}-\frac {3 a^2 \sec ^2(c+d x)}{2 d}+\frac {2 a^2 \sec (c+d x)}{d}-\frac {a^2 \log (\cos (c+d x))}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 3964
Rubi steps
\begin {align*} \int (a+a \sec (c+d x))^2 \tan ^9(c+d x) \, dx &=-\frac {\text {Subst}\left (\int \frac {(a-a x)^4 (a+a x)^6}{x^{11}} \, dx,x,\cos (c+d x)\right )}{a^8 d}\\ &=-\frac {\text {Subst}\left (\int \left (\frac {a^{10}}{x^{11}}+\frac {2 a^{10}}{x^{10}}-\frac {3 a^{10}}{x^9}-\frac {8 a^{10}}{x^8}+\frac {2 a^{10}}{x^7}+\frac {12 a^{10}}{x^6}+\frac {2 a^{10}}{x^5}-\frac {8 a^{10}}{x^4}-\frac {3 a^{10}}{x^3}+\frac {2 a^{10}}{x^2}+\frac {a^{10}}{x}\right ) \, dx,x,\cos (c+d x)\right )}{a^8 d}\\ &=-\frac {a^2 \log (\cos (c+d x))}{d}+\frac {2 a^2 \sec (c+d x)}{d}-\frac {3 a^2 \sec ^2(c+d x)}{2 d}-\frac {8 a^2 \sec ^3(c+d x)}{3 d}+\frac {a^2 \sec ^4(c+d x)}{2 d}+\frac {12 a^2 \sec ^5(c+d x)}{5 d}+\frac {a^2 \sec ^6(c+d x)}{3 d}-\frac {8 a^2 \sec ^7(c+d x)}{7 d}-\frac {3 a^2 \sec ^8(c+d x)}{8 d}+\frac {2 a^2 \sec ^9(c+d x)}{9 d}+\frac {a^2 \sec ^{10}(c+d x)}{10 d}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 140, normalized size = 0.73 \begin {gather*} -\frac {a^2 (1+\cos (c+d x))^2 \sec ^4\left (\frac {1}{2} (c+d x)\right ) \left (2520 \log (\cos (c+d x))-5040 \sec (c+d x)+3780 \sec ^2(c+d x)+6720 \sec ^3(c+d x)-1260 \sec ^4(c+d x)-6048 \sec ^5(c+d x)-840 \sec ^6(c+d x)+2880 \sec ^7(c+d x)+945 \sec ^8(c+d x)-560 \sec ^9(c+d x)-252 \sec ^{10}(c+d x)\right )}{10080 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.18, size = 225, normalized size = 1.17
method | result | size |
derivativedivides | \(\frac {\frac {a^{2} \left (\sin ^{10}\left (d x +c \right )\right )}{10 \cos \left (d x +c \right )^{10}}+2 a^{2} \left (\frac {\sin ^{10}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}-\frac {\sin ^{10}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{10}\left (d x +c \right )}{105 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{10}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{10}\left (d x +c \right )}{9 \cos \left (d x +c \right )}+\frac {\left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )}{9}\right )+a^{2} \left (\frac {\left (\tan ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\tan ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(225\) |
default | \(\frac {\frac {a^{2} \left (\sin ^{10}\left (d x +c \right )\right )}{10 \cos \left (d x +c \right )^{10}}+2 a^{2} \left (\frac {\sin ^{10}\left (d x +c \right )}{9 \cos \left (d x +c \right )^{9}}-\frac {\sin ^{10}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{7}}+\frac {\sin ^{10}\left (d x +c \right )}{105 \cos \left (d x +c \right )^{5}}-\frac {\sin ^{10}\left (d x +c \right )}{63 \cos \left (d x +c \right )^{3}}+\frac {\sin ^{10}\left (d x +c \right )}{9 \cos \left (d x +c \right )}+\frac {\left (\frac {128}{35}+\sin ^{8}\left (d x +c \right )+\frac {8 \left (\sin ^{6}\left (d x +c \right )\right )}{7}+\frac {48 \left (\sin ^{4}\left (d x +c \right )\right )}{35}+\frac {64 \left (\sin ^{2}\left (d x +c \right )\right )}{35}\right ) \cos \left (d x +c \right )}{9}\right )+a^{2} \left (\frac {\left (\tan ^{8}\left (d x +c \right )\right )}{8}-\frac {\left (\tan ^{6}\left (d x +c \right )\right )}{6}+\frac {\left (\tan ^{4}\left (d x +c \right )\right )}{4}-\frac {\left (\tan ^{2}\left (d x +c \right )\right )}{2}-\ln \left (\cos \left (d x +c \right )\right )\right )}{d}\) | \(225\) |
risch | \(i a^{2} x +\frac {2 i a^{2} c}{d}+\frac {2 a^{2} \left (630 \,{\mathrm e}^{19 i \left (d x +c \right )}-945 \,{\mathrm e}^{18 i \left (d x +c \right )}+2310 \,{\mathrm e}^{17 i \left (d x +c \right )}-6300 \,{\mathrm e}^{16 i \left (d x +c \right )}+11256 \,{\mathrm e}^{15 i \left (d x +c \right )}-15540 \,{\mathrm e}^{14 i \left (d x +c \right )}+19800 \,{\mathrm e}^{13 i \left (d x +c \right )}-35700 \,{\mathrm e}^{12 i \left (d x +c \right )}+31540 \,{\mathrm e}^{11 i \left (d x +c \right )}-34902 \,{\mathrm e}^{10 i \left (d x +c \right )}+31540 \,{\mathrm e}^{9 i \left (d x +c \right )}-35700 \,{\mathrm e}^{8 i \left (d x +c \right )}+19800 \,{\mathrm e}^{7 i \left (d x +c \right )}-15540 \,{\mathrm e}^{6 i \left (d x +c \right )}+11256 \,{\mathrm e}^{5 i \left (d x +c \right )}-6300 \,{\mathrm e}^{4 i \left (d x +c \right )}+2310 \,{\mathrm e}^{3 i \left (d x +c \right )}-945 \,{\mathrm e}^{2 i \left (d x +c \right )}+630 \,{\mathrm e}^{i \left (d x +c \right )}\right )}{315 d \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{10}}-\frac {a^{2} \ln \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )}{d}\) | \(270\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 149, normalized size = 0.78 \begin {gather*} -\frac {2520 \, a^{2} \log \left (\cos \left (d x + c\right )\right ) - \frac {5040 \, a^{2} \cos \left (d x + c\right )^{9} - 3780 \, a^{2} \cos \left (d x + c\right )^{8} - 6720 \, a^{2} \cos \left (d x + c\right )^{7} + 1260 \, a^{2} \cos \left (d x + c\right )^{6} + 6048 \, a^{2} \cos \left (d x + c\right )^{5} + 840 \, a^{2} \cos \left (d x + c\right )^{4} - 2880 \, a^{2} \cos \left (d x + c\right )^{3} - 945 \, a^{2} \cos \left (d x + c\right )^{2} + 560 \, a^{2} \cos \left (d x + c\right ) + 252 \, a^{2}}{\cos \left (d x + c\right )^{10}}}{2520 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 4.48, size = 156, normalized size = 0.81 \begin {gather*} -\frac {2520 \, a^{2} \cos \left (d x + c\right )^{10} \log \left (-\cos \left (d x + c\right )\right ) - 5040 \, a^{2} \cos \left (d x + c\right )^{9} + 3780 \, a^{2} \cos \left (d x + c\right )^{8} + 6720 \, a^{2} \cos \left (d x + c\right )^{7} - 1260 \, a^{2} \cos \left (d x + c\right )^{6} - 6048 \, a^{2} \cos \left (d x + c\right )^{5} - 840 \, a^{2} \cos \left (d x + c\right )^{4} + 2880 \, a^{2} \cos \left (d x + c\right )^{3} + 945 \, a^{2} \cos \left (d x + c\right )^{2} - 560 \, a^{2} \cos \left (d x + c\right ) - 252 \, a^{2}}{2520 \, d \cos \left (d x + c\right )^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.24, size = 314, normalized size = 1.64 \begin {gather*} \begin {cases} \frac {a^{2} \log {\left (\tan ^{2}{\left (c + d x \right )} + 1 \right )}}{2 d} + \frac {a^{2} \tan ^{8}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac {2 a^{2} \tan ^{8}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{9 d} + \frac {a^{2} \tan ^{8}{\left (c + d x \right )}}{8 d} - \frac {a^{2} \tan ^{6}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} - \frac {16 a^{2} \tan ^{6}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{63 d} - \frac {a^{2} \tan ^{6}{\left (c + d x \right )}}{6 d} + \frac {a^{2} \tan ^{4}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac {32 a^{2} \tan ^{4}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{105 d} + \frac {a^{2} \tan ^{4}{\left (c + d x \right )}}{4 d} - \frac {a^{2} \tan ^{2}{\left (c + d x \right )} \sec ^{2}{\left (c + d x \right )}}{10 d} - \frac {128 a^{2} \tan ^{2}{\left (c + d x \right )} \sec {\left (c + d x \right )}}{315 d} - \frac {a^{2} \tan ^{2}{\left (c + d x \right )}}{2 d} + \frac {a^{2} \sec ^{2}{\left (c + d x \right )}}{10 d} + \frac {256 a^{2} \sec {\left (c + d x \right )}}{315 d} & \text {for}\: d \neq 0 \\x \left (a \sec {\left (c \right )} + a\right )^{2} \tan ^{9}{\left (c \right )} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 6.40, size = 342, normalized size = 1.78 \begin {gather*} \frac {2520 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1 \right |}\right ) - 2520 \, a^{2} \log \left ({\left | -\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} - 1 \right |}\right ) + \frac {11477 \, a^{2} + \frac {119810 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {566865 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {1605720 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} + \frac {3031770 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {2995020 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{5}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{5}} + \frac {2171610 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1114200 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} + \frac {382545 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {78850 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} + \frac {7381 \, a^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}}{{\left (\frac {\cos \left (d x + c\right ) - 1}{\cos \left (d x + c\right ) + 1} + 1\right )}^{10}}}{2520 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.14, size = 308, normalized size = 1.60 \begin {gather*} \frac {2\,a^2\,\mathrm {atanh}\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\right )}{d}-\frac {2\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}-20\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}+\frac {272\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}}{3}-\frac {740\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}}{3}+\frac {2252\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}}{5}-588\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+\frac {2000\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6}{7}-\frac {652\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4}{7}+\frac {1150\,a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2}{63}-\frac {512\,a^2}{315}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{20}-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{18}+45\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{16}-120\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}+210\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}-252\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}+210\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8-120\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6+45\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4-10\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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