Optimal. Leaf size=220 \[ -\frac {\left (231 a^3+840 a^2 b+1152 a b^2+1024 b^3\right ) \tanh ^{-1}(\cosh (c+d x))}{1024 d}+\frac {3 a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{1024 d}-\frac {a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}^3(c+d x)}{512 d}+\frac {7 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^5(c+d x)}{640 d}-\frac {3 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^7(c+d x)}{320 d}+\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d} \]
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Rubi [A]
time = 0.28, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3294, 1171,
1828, 393, 212} \begin {gather*} -\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}+\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}^3(c+d x)}{512 d}+\frac {3 a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{1024 d}-\frac {3 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^7(c+d x)}{320 d}+\frac {7 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^5(c+d x)}{640 d}-\frac {\left (231 a^3+840 a^2 b+1152 a b^2+1024 b^3\right ) \tanh ^{-1}(\cosh (c+d x))}{1024 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 393
Rule 1171
Rule 1828
Rule 3294
Rubi steps
\begin {align*} \int \text {csch}^{13}(c+d x) \left (a+b \sinh ^4(c+d x)\right )^3 \, dx &=-\frac {\text {Subst}\left (\int \frac {\left (a+b-2 b x^2+b x^4\right )^3}{\left (1-x^2\right )^7} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}+\frac {\text {Subst}\left (\int \frac {-11 a^3-36 a^2 b-36 a b^2-12 b^3+12 b \left (3 a^2+9 a b+5 b^2\right ) x^2-12 b^2 (9 a+10 b) x^4+12 b^2 (3 a+10 b) x^6-60 b^3 x^8+12 b^3 x^{10}}{\left (1-x^2\right )^6} \, dx,x,\cosh (c+d x)\right )}{12 d}\\ &=\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}-\frac {\text {Subst}\left (\int \frac {3 \left (33 a^3+120 a^2 b+120 a b^2+40 b^3\right )-240 b^2 (3 a+2 b) x^2+360 b^2 (a+2 b) x^4-480 b^3 x^6+120 b^3 x^8}{\left (1-x^2\right )^5} \, dx,x,\cosh (c+d x)\right )}{120 d}\\ &=-\frac {3 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^7(c+d x)}{320 d}+\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}+\frac {\text {Subst}\left (\int \frac {-3 \left (231 a^3+840 a^2 b+960 a b^2+320 b^3\right )+2880 b^2 (a+b) x^2-2880 b^3 x^4+960 b^3 x^6}{\left (1-x^2\right )^4} \, dx,x,\cosh (c+d x)\right )}{960 d}\\ &=\frac {7 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^5(c+d x)}{640 d}-\frac {3 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^7(c+d x)}{320 d}+\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}-\frac {\text {Subst}\left (\int \frac {45 \left (77 a^3+280 a^2 b+384 a b^2+128 b^3\right )-11520 b^3 x^2+5760 b^3 x^4}{\left (1-x^2\right )^3} \, dx,x,\cosh (c+d x)\right )}{5760 d}\\ &=-\frac {a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}^3(c+d x)}{512 d}+\frac {7 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^5(c+d x)}{640 d}-\frac {3 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^7(c+d x)}{320 d}+\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}+\frac {\text {Subst}\left (\int \frac {-45 \left (231 a^3+840 a^2 b+1152 a b^2+512 b^3\right )+23040 b^3 x^2}{\left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{23040 d}\\ &=\frac {3 a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{1024 d}-\frac {a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}^3(c+d x)}{512 d}+\frac {7 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^5(c+d x)}{640 d}-\frac {3 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^7(c+d x)}{320 d}+\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}-\frac {\left (231 a^3+840 a^2 b+1152 a b^2+1024 b^3\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{1024 d}\\ &=-\frac {\left (231 a^3+840 a^2 b+1152 a b^2+1024 b^3\right ) \tanh ^{-1}(\cosh (c+d x))}{1024 d}+\frac {3 a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}(c+d x)}{1024 d}-\frac {a \left (77 a^2+280 a b+384 b^2\right ) \coth (c+d x) \text {csch}^3(c+d x)}{512 d}+\frac {7 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^5(c+d x)}{640 d}-\frac {3 a^2 (11 a+40 b) \coth (c+d x) \text {csch}^7(c+d x)}{320 d}+\frac {11 a^3 \coth (c+d x) \text {csch}^9(c+d x)}{120 d}-\frac {a^3 \coth (c+d x) \text {csch}^{11}(c+d x)}{12 d}\\ \end {align*}
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Mathematica [A]
time = 1.49, size = 246, normalized size = 1.12 \begin {gather*} \frac {-30 a \left (76555 a^2+75816 a b+45696 b^2\right ) \coth (c+d x) \text {csch}^{11}(c+d x)+2 a \left (750629 a^2+2074200 a b+1422720 b^2\right ) \cosh (3 (c+d x)) \text {csch}^{12}(c+d x)-9 a \left (77099 a^2+280360 a b+246400 b^2\right ) \cosh (5 (c+d x)) \text {csch}^{12}(c+d x)+63 a \left (3421 a^2+12440 a b+14720 b^2\right ) \cosh (7 (c+d x)) \text {csch}^{12}(c+d x)-525 a \left (77 a^2+280 a b+384 b^2\right ) \cosh (9 (c+d x)) \text {csch}^{12}(c+d x)+45 a \left (77 a^2+280 a b+384 b^2\right ) \cosh (11 (c+d x)) \text {csch}^{12}(c+d x)+15360 \left (231 a^3+840 a^2 b+1152 a b^2+1024 b^3\right ) \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{15728640 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. \(631\) vs.
\(2(206)=412\).
time = 1.50, size = 632, normalized size = 2.87
method | result | size |
risch | \(\frac {a \,{\mathrm e}^{d x +c} \left (12600 a b +3465 a^{2}-147000 a b \,{\mathrm e}^{2 d x +2 c}+17280 b^{2}-2274480 a b \,{\mathrm e}^{10 d x +10 c}-2523240 a b \,{\mathrm e}^{6 d x +6 c}+4148400 a b \,{\mathrm e}^{8 d x +8 c}+215523 a^{2} {\mathrm e}^{4 d x +4 c}+783720 a b \,{\mathrm e}^{18 d x +18 c}+12600 a b \,{\mathrm e}^{22 d x +22 c}-2523240 a b \,{\mathrm e}^{16 d x +16 c}+4148400 a b \,{\mathrm e}^{14 d x +14 c}-2274480 a b \,{\mathrm e}^{12 d x +12 c}-147000 a b \,{\mathrm e}^{20 d x +20 c}+783720 a b \,{\mathrm e}^{4 d x +4 c}-40425 a^{2} {\mathrm e}^{20 d x +20 c}-201600 b^{2} {\mathrm e}^{20 d x +20 c}+215523 a^{2} {\mathrm e}^{18 d x +18 c}+927360 b^{2} {\mathrm e}^{18 d x +18 c}-693891 a^{2} {\mathrm e}^{16 d x +16 c}-2217600 b^{2} {\mathrm e}^{16 d x +16 c}+1501258 a^{2} {\mathrm e}^{14 d x +14 c}+2845440 b^{2} {\mathrm e}^{14 d x +14 c}-2296650 a^{2} {\mathrm e}^{12 d x +12 c}+3465 a^{2} {\mathrm e}^{22 d x +22 c}+17280 b^{2} {\mathrm e}^{22 d x +22 c}-693891 a^{2} {\mathrm e}^{6 d x +6 c}+2845440 b^{2} {\mathrm e}^{8 d x +8 c}-201600 b^{2} {\mathrm e}^{2 d x +2 c}-2217600 b^{2} {\mathrm e}^{6 d x +6 c}+927360 b^{2} {\mathrm e}^{4 d x +4 c}-40425 a^{2} {\mathrm e}^{2 d x +2 c}-1370880 b^{2} {\mathrm e}^{12 d x +12 c}-2296650 a^{2} {\mathrm e}^{10 d x +10 c}-1370880 b^{2} {\mathrm e}^{10 d x +10 c}+1501258 a^{2} {\mathrm e}^{8 d x +8 c}\right )}{7680 d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{12}}-\frac {231 a^{3} \ln \left ({\mathrm e}^{d x +c}+1\right )}{1024 d}-\frac {105 a^{2} b \ln \left ({\mathrm e}^{d x +c}+1\right )}{128 d}-\frac {9 a \ln \left ({\mathrm e}^{d x +c}+1\right ) b^{2}}{8 d}-\frac {\ln \left ({\mathrm e}^{d x +c}+1\right ) b^{3}}{d}+\frac {231 a^{3} \ln \left ({\mathrm e}^{d x +c}-1\right )}{1024 d}+\frac {105 a^{2} b \ln \left ({\mathrm e}^{d x +c}-1\right )}{128 d}+\frac {9 a \ln \left ({\mathrm e}^{d x +c}-1\right ) b^{2}}{8 d}+\frac {\ln \left ({\mathrm e}^{d x +c}-1\right ) b^{3}}{d}\) | \(632\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 720 vs.
\(2 (206) = 412\).
time = 0.30, size = 720, normalized size = 3.27 \begin {gather*} -\frac {1}{15360} \, a^{3} {\left (\frac {3465 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3465 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3465 \, e^{\left (-d x - c\right )} - 40425 \, e^{\left (-3 \, d x - 3 \, c\right )} + 215523 \, e^{\left (-5 \, d x - 5 \, c\right )} - 693891 \, e^{\left (-7 \, d x - 7 \, c\right )} + 1501258 \, e^{\left (-9 \, d x - 9 \, c\right )} - 2296650 \, e^{\left (-11 \, d x - 11 \, c\right )} - 2296650 \, e^{\left (-13 \, d x - 13 \, c\right )} + 1501258 \, e^{\left (-15 \, d x - 15 \, c\right )} - 693891 \, e^{\left (-17 \, d x - 17 \, c\right )} + 215523 \, e^{\left (-19 \, d x - 19 \, c\right )} - 40425 \, e^{\left (-21 \, d x - 21 \, c\right )} + 3465 \, e^{\left (-23 \, d x - 23 \, c\right )}\right )}}{d {\left (12 \, e^{\left (-2 \, d x - 2 \, c\right )} - 66 \, e^{\left (-4 \, d x - 4 \, c\right )} + 220 \, e^{\left (-6 \, d x - 6 \, c\right )} - 495 \, e^{\left (-8 \, d x - 8 \, c\right )} + 792 \, e^{\left (-10 \, d x - 10 \, c\right )} - 924 \, e^{\left (-12 \, d x - 12 \, c\right )} + 792 \, e^{\left (-14 \, d x - 14 \, c\right )} - 495 \, e^{\left (-16 \, d x - 16 \, c\right )} + 220 \, e^{\left (-18 \, d x - 18 \, c\right )} - 66 \, e^{\left (-20 \, d x - 20 \, c\right )} + 12 \, e^{\left (-22 \, d x - 22 \, c\right )} - e^{\left (-24 \, d x - 24 \, c\right )} - 1\right )}}\right )} - \frac {1}{128} \, a^{2} b {\left (\frac {105 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {105 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (105 \, e^{\left (-d x - c\right )} - 805 \, e^{\left (-3 \, d x - 3 \, c\right )} + 2681 \, e^{\left (-5 \, d x - 5 \, c\right )} - 5053 \, e^{\left (-7 \, d x - 7 \, c\right )} - 5053 \, e^{\left (-9 \, d x - 9 \, c\right )} + 2681 \, e^{\left (-11 \, d x - 11 \, c\right )} - 805 \, e^{\left (-13 \, d x - 13 \, c\right )} + 105 \, e^{\left (-15 \, d x - 15 \, c\right )}\right )}}{d {\left (8 \, e^{\left (-2 \, d x - 2 \, c\right )} - 28 \, e^{\left (-4 \, d x - 4 \, c\right )} + 56 \, e^{\left (-6 \, d x - 6 \, c\right )} - 70 \, e^{\left (-8 \, d x - 8 \, c\right )} + 56 \, e^{\left (-10 \, d x - 10 \, c\right )} - 28 \, e^{\left (-12 \, d x - 12 \, c\right )} + 8 \, e^{\left (-14 \, d x - 14 \, c\right )} - e^{\left (-16 \, d x - 16 \, c\right )} - 1\right )}}\right )} - \frac {3}{8} \, a b^{2} {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} - 11 \, e^{\left (-3 \, d x - 3 \, c\right )} - 11 \, e^{\left (-5 \, d x - 5 \, c\right )} + 3 \, e^{\left (-7 \, d x - 7 \, c\right )}\right )}}{d {\left (4 \, e^{\left (-2 \, d x - 2 \, c\right )} - 6 \, e^{\left (-4 \, d x - 4 \, c\right )} + 4 \, e^{\left (-6 \, d x - 6 \, c\right )} - e^{\left (-8 \, d x - 8 \, c\right )} - 1\right )}}\right )} - b^{3} {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d}\right )} \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. 17811 vs.
\(2 (206) = 412\).
time = 0.53, size = 17811, normalized size = 80.96 \begin {gather*} \text {too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 537 vs.
\(2 (206) = 412\).
time = 0.62, size = 537, normalized size = 2.44 \begin {gather*} -\frac {15 \, {\left (231 \, a^{3} + 840 \, a^{2} b + 1152 \, a b^{2} + 1024 \, b^{3}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - 15 \, {\left (231 \, a^{3} + 840 \, a^{2} b + 1152 \, a b^{2} + 1024 \, b^{3}\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (3465 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{11} + 12600 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{11} + 17280 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{11} - 78540 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} - 285600 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} - 391680 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{9} + 731808 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + 2661120 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} + 3502080 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{7} - 3560832 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 12948480 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} - 15482880 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{5} + 9391360 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 32839680 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} + 33914880 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 12180480 \, a^{3} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 34283520 \, a^{2} b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )} - 29491200 \, a b^{2} {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 4\right )}^{6}}}{30720 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.10, size = 1314, normalized size = 5.97 \begin {gather*} \frac {3\,{\mathrm {e}}^{c+d\,x}\,\left (77\,a^3+280\,a^2\,b+384\,a\,b^2\right )}{512\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {5632\,a^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (11\,{\mathrm {e}}^{2\,c+2\,d\,x}-55\,{\mathrm {e}}^{4\,c+4\,d\,x}+165\,{\mathrm {e}}^{6\,c+6\,d\,x}-330\,{\mathrm {e}}^{8\,c+8\,d\,x}+462\,{\mathrm {e}}^{10\,c+10\,d\,x}-462\,{\mathrm {e}}^{12\,c+12\,d\,x}+330\,{\mathrm {e}}^{14\,c+14\,d\,x}-165\,{\mathrm {e}}^{16\,c+16\,d\,x}+55\,{\mathrm {e}}^{18\,c+18\,d\,x}-11\,{\mathrm {e}}^{20\,c+20\,d\,x}+{\mathrm {e}}^{22\,c+22\,d\,x}-1\right )}-\frac {1024\,a^3\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (66\,{\mathrm {e}}^{4\,c+4\,d\,x}-12\,{\mathrm {e}}^{2\,c+2\,d\,x}-220\,{\mathrm {e}}^{6\,c+6\,d\,x}+495\,{\mathrm {e}}^{8\,c+8\,d\,x}-792\,{\mathrm {e}}^{10\,c+10\,d\,x}+924\,{\mathrm {e}}^{12\,c+12\,d\,x}-792\,{\mathrm {e}}^{14\,c+14\,d\,x}+495\,{\mathrm {e}}^{16\,c+16\,d\,x}-220\,{\mathrm {e}}^{18\,c+18\,d\,x}+66\,{\mathrm {e}}^{20\,c+20\,d\,x}-12\,{\mathrm {e}}^{22\,c+22\,d\,x}+{\mathrm {e}}^{24\,c+24\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3+2424\,b\,a^2\right )}{6\,d\,\left (15\,{\mathrm {e}}^{4\,c+4\,d\,x}-6\,{\mathrm {e}}^{2\,c+2\,d\,x}-20\,{\mathrm {e}}^{6\,c+6\,d\,x}+15\,{\mathrm {e}}^{8\,c+8\,d\,x}-6\,{\mathrm {e}}^{10\,c+10\,d\,x}+{\mathrm {e}}^{12\,c+12\,d\,x}+1\right )}-\frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (231\,a^3\,\sqrt {-d^2}+1024\,b^3\,\sqrt {-d^2}+1152\,a\,b^2\,\sqrt {-d^2}+840\,a^2\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {53361\,a^6+388080\,a^5\,b+1237824\,a^4\,b^2+2408448\,a^3\,b^3+3047424\,a^2\,b^4+2359296\,a\,b^5+1048576\,b^6}}\right )\,\sqrt {53361\,a^6+388080\,a^5\,b+1237824\,a^4\,b^2+2408448\,a^3\,b^3+3047424\,a^2\,b^4+2359296\,a\,b^5+1048576\,b^6}}{512\,\sqrt {-d^2}}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (10200\,a^2\,b-11\,a^3\right )}{60\,d\,\left (5\,{\mathrm {e}}^{2\,c+2\,d\,x}-10\,{\mathrm {e}}^{4\,c+4\,d\,x}+10\,{\mathrm {e}}^{6\,c+6\,d\,x}-5\,{\mathrm {e}}^{8\,c+8\,d\,x}+{\mathrm {e}}^{10\,c+10\,d\,x}-1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (77\,a^3+280\,a^2\,b+384\,a\,b^2\right )}{256\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (77\,a^3+280\,a^2\,b-5760\,a\,b^2\right )}{320\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {42\,{\mathrm {e}}^{c+d\,x}\,\left (15\,a^3+8\,b\,a^2\right )}{d\,\left (7\,{\mathrm {e}}^{2\,c+2\,d\,x}-21\,{\mathrm {e}}^{4\,c+4\,d\,x}+35\,{\mathrm {e}}^{6\,c+6\,d\,x}-35\,{\mathrm {e}}^{8\,c+8\,d\,x}+21\,{\mathrm {e}}^{10\,c+10\,d\,x}-7\,{\mathrm {e}}^{12\,c+12\,d\,x}+{\mathrm {e}}^{14\,c+14\,d\,x}-1\right )}-\frac {23488\,a^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (9\,{\mathrm {e}}^{2\,c+2\,d\,x}-36\,{\mathrm {e}}^{4\,c+4\,d\,x}+84\,{\mathrm {e}}^{6\,c+6\,d\,x}-126\,{\mathrm {e}}^{8\,c+8\,d\,x}+126\,{\mathrm {e}}^{10\,c+10\,d\,x}-84\,{\mathrm {e}}^{12\,c+12\,d\,x}+36\,{\mathrm {e}}^{14\,c+14\,d\,x}-9\,{\mathrm {e}}^{16\,c+16\,d\,x}+{\mathrm {e}}^{18\,c+18\,d\,x}-1\right )}-\frac {3\,{\mathrm {e}}^{c+d\,x}\,\left (11\,a^3+40\,a^2\,b+640\,a\,b^2\right )}{160\,d\,\left (6\,{\mathrm {e}}^{4\,c+4\,d\,x}-4\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )}-\frac {4\,{\mathrm {e}}^{c+d\,x}\,\left (3361\,a^3+120\,b\,a^2\right )}{5\,d\,\left (28\,{\mathrm {e}}^{4\,c+4\,d\,x}-8\,{\mathrm {e}}^{2\,c+2\,d\,x}-56\,{\mathrm {e}}^{6\,c+6\,d\,x}+70\,{\mathrm {e}}^{8\,c+8\,d\,x}-56\,{\mathrm {e}}^{10\,c+10\,d\,x}+28\,{\mathrm {e}}^{12\,c+12\,d\,x}-8\,{\mathrm {e}}^{14\,c+14\,d\,x}+{\mathrm {e}}^{16\,c+16\,d\,x}+1\right )}-\frac {20864\,a^3\,{\mathrm {e}}^{c+d\,x}}{5\,d\,\left (45\,{\mathrm {e}}^{4\,c+4\,d\,x}-10\,{\mathrm {e}}^{2\,c+2\,d\,x}-120\,{\mathrm {e}}^{6\,c+6\,d\,x}+210\,{\mathrm {e}}^{8\,c+8\,d\,x}-252\,{\mathrm {e}}^{10\,c+10\,d\,x}+210\,{\mathrm {e}}^{12\,c+12\,d\,x}-120\,{\mathrm {e}}^{14\,c+14\,d\,x}+45\,{\mathrm {e}}^{16\,c+16\,d\,x}-10\,{\mathrm {e}}^{18\,c+18\,d\,x}+{\mathrm {e}}^{20\,c+20\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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