Integrand size = 14, antiderivative size = 160 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx=(a+b)^5 x-\frac {b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac {b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^3 \left (10 a^2+5 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^4 (5 a+b) \tanh ^7(c+d x)}{7 d}-\frac {b^5 \tanh ^9(c+d x)}{9 d} \]
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Time = 0.07 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {3742, 398, 212} \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx=-\frac {b^3 \left (10 a^2+5 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^3(c+d x)}{3 d}-\frac {b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac {b^4 (5 a+b) \tanh ^7(c+d x)}{7 d}+x (a+b)^5-\frac {b^5 \tanh ^9(c+d x)}{9 d} \]
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Rule 212
Rule 398
Rule 3742
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^5}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left (-b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right )-b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) x^2-b^3 \left (10 a^2+5 a b+b^2\right ) x^4-b^4 (5 a+b) x^6-b^5 x^8+\frac {(a+b)^5}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d} \\ & = -\frac {b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac {b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^3 \left (10 a^2+5 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^4 (5 a+b) \tanh ^7(c+d x)}{7 d}-\frac {b^5 \tanh ^9(c+d x)}{9 d}+\frac {(a+b)^5 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = (a+b)^5 x-\frac {b \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right ) \tanh (c+d x)}{d}-\frac {b^2 \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^3(c+d x)}{3 d}-\frac {b^3 \left (10 a^2+5 a b+b^2\right ) \tanh ^5(c+d x)}{5 d}-\frac {b^4 (5 a+b) \tanh ^7(c+d x)}{7 d}-\frac {b^5 \tanh ^9(c+d x)}{9 d} \\ \end{align*}
Time = 2.72 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.06 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx=\frac {\tanh (c+d x) \left (\frac {315 (a+b)^5 \text {arctanh}\left (\sqrt {\tanh ^2(c+d x)}\right )}{\sqrt {\tanh ^2(c+d x)}}-b \left (315 \left (5 a^4+10 a^3 b+10 a^2 b^2+5 a b^3+b^4\right )+105 b \left (10 a^3+10 a^2 b+5 a b^2+b^3\right ) \tanh ^2(c+d x)+63 b^2 \left (10 a^2+5 a b+b^2\right ) \tanh ^4(c+d x)+45 b^3 (5 a+b) \tanh ^6(c+d x)+35 b^4 \tanh ^8(c+d x)\right )\right )}{315 d} \]
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Time = 0.21 (sec) , antiderivative size = 262, normalized size of antiderivative = 1.64
method | result | size |
parallelrisch | \(-\frac {3150 a^{3} b^{2} \tanh \left (d x +c \right )+1575 a \,b^{4} \tanh \left (d x +c \right )+525 a \,b^{4} \tanh \left (d x +c \right )^{3}+3150 a^{2} b^{3} \tanh \left (d x +c \right )+225 a \,b^{4} \tanh \left (d x +c \right )^{7}+630 a^{2} b^{3} \tanh \left (d x +c \right )^{5}+315 a \,b^{4} \tanh \left (d x +c \right )^{5}+1050 a^{3} b^{2} \tanh \left (d x +c \right )^{3}-3150 a^{2} b^{3} d x -315 a^{5} d x -315 b^{5} d x +63 b^{5} \tanh \left (d x +c \right )^{5}+105 b^{5} \tanh \left (d x +c \right )^{3}+45 b^{5} \tanh \left (d x +c \right )^{7}+315 b^{5} \tanh \left (d x +c \right )+35 b^{5} \tanh \left (d x +c \right )^{9}-1575 a^{4} b d x -3150 a^{3} b^{2} d x -1575 a \,b^{4} d x +1050 a^{2} b^{3} \tanh \left (d x +c \right )^{3}+1575 a^{4} b \tanh \left (d x +c \right )}{315 d}\) | \(262\) |
derivativedivides | \(\frac {-5 a^{4} b \tanh \left (d x +c \right )-10 a^{3} b^{2} \tanh \left (d x +c \right )-10 a^{2} b^{3} \tanh \left (d x +c \right )-5 a \,b^{4} \tanh \left (d x +c \right )-\frac {5 a \,b^{4} \tanh \left (d x +c \right )^{7}}{7}-b^{5} \tanh \left (d x +c \right )-\frac {b^{5} \tanh \left (d x +c \right )^{9}}{9}-2 a^{2} b^{3} \tanh \left (d x +c \right )^{5}-a \,b^{4} \tanh \left (d x +c \right )^{5}-\frac {10 a^{3} b^{2} \tanh \left (d x +c \right )^{3}}{3}-\frac {10 a^{2} b^{3} \tanh \left (d x +c \right )^{3}}{3}-\frac {5 a \,b^{4} \tanh \left (d x +c \right )^{3}}{3}-\frac {\left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}-\frac {b^{5} \tanh \left (d x +c \right )^{5}}{5}-\frac {b^{5} \tanh \left (d x +c \right )^{3}}{3}-\frac {b^{5} \tanh \left (d x +c \right )^{7}}{7}+\frac {\left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )}{2}}{d}\) | \(303\) |
default | \(\frac {-5 a^{4} b \tanh \left (d x +c \right )-10 a^{3} b^{2} \tanh \left (d x +c \right )-10 a^{2} b^{3} \tanh \left (d x +c \right )-5 a \,b^{4} \tanh \left (d x +c \right )-\frac {5 a \,b^{4} \tanh \left (d x +c \right )^{7}}{7}-b^{5} \tanh \left (d x +c \right )-\frac {b^{5} \tanh \left (d x +c \right )^{9}}{9}-2 a^{2} b^{3} \tanh \left (d x +c \right )^{5}-a \,b^{4} \tanh \left (d x +c \right )^{5}-\frac {10 a^{3} b^{2} \tanh \left (d x +c \right )^{3}}{3}-\frac {10 a^{2} b^{3} \tanh \left (d x +c \right )^{3}}{3}-\frac {5 a \,b^{4} \tanh \left (d x +c \right )^{3}}{3}-\frac {\left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right ) \ln \left (\tanh \left (d x +c \right )-1\right )}{2}-\frac {b^{5} \tanh \left (d x +c \right )^{5}}{5}-\frac {b^{5} \tanh \left (d x +c \right )^{3}}{3}-\frac {b^{5} \tanh \left (d x +c \right )^{7}}{7}+\frac {\left (a^{5}+5 a^{4} b +10 a^{3} b^{2}+10 a^{2} b^{3}+5 a \,b^{4}+b^{5}\right ) \ln \left (\tanh \left (d x +c \right )+1\right )}{2}}{d}\) | \(303\) |
parts | \(a^{5} x +\frac {b^{5} \left (-\frac {\tanh \left (d x +c \right )^{9}}{9}-\frac {\tanh \left (d x +c \right )^{7}}{7}-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {5 a \,b^{4} \left (-\frac {\tanh \left (d x +c \right )^{7}}{7}-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {10 a^{2} b^{3} \left (-\frac {\tanh \left (d x +c \right )^{5}}{5}-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {5 a^{4} b \left (-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}+\frac {10 a^{3} b^{2} \left (-\frac {\tanh \left (d x +c \right )^{3}}{3}-\tanh \left (d x +c \right )-\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2}+\frac {\ln \left (\tanh \left (d x +c \right )+1\right )}{2}\right )}{d}\) | \(309\) |
risch | \(a^{5} x +5 b \,a^{4} x +10 a^{3} b^{2} x +10 b^{3} a^{2} x +5 a \,b^{4} x +b^{5} x +\frac {2 b \left (31500 a^{3} b \,{\mathrm e}^{2 d x +2 c}+34020 a^{2} b^{2} {\mathrm e}^{2 d x +2 c}+17460 a \,b^{3} {\mathrm e}^{2 d x +2 c}+107100 a^{3} b \,{\mathrm e}^{4 d x +4 c}+63540 a \,b^{3} {\mathrm e}^{4 d x +4 c}+157500 a \,b^{3} {\mathrm e}^{10 d x +10 c}+175140 a \,b^{3} {\mathrm e}^{8 d x +8 c}+131460 a \,b^{3} {\mathrm e}^{6 d x +6 c}+90300 a \,b^{3} {\mathrm e}^{12 d x +12 c}+31500 a \,b^{3} {\mathrm e}^{14 d x +14 c}+6300 a \,b^{3} {\mathrm e}^{16 d x +16 c}+9450 a^{2} b^{2} {\mathrm e}^{16 d x +16 c}+6300 a^{3} b \,{\mathrm e}^{16 d x +16 c}+1575 a^{4}+563 b^{4}+4830 a^{2} b^{2}+4200 a^{3} b +2640 a \,b^{3}+6300 \,{\mathrm e}^{14 d x +14 c} b^{4}+1575 a^{4} {\mathrm e}^{16 d x +16 c}+1575 \,{\mathrm e}^{16 d x +16 c} b^{4}+31500 b^{4} {\mathrm e}^{10 d x +10 c}+12600 a^{4} {\mathrm e}^{2 d x +2 c}+3492 b^{4} {\mathrm e}^{2 d x +2 c}+21000 b^{4} {\mathrm e}^{12 d x +12 c}+39438 b^{4} {\mathrm e}^{8 d x +8 c}+26292 b^{4} {\mathrm e}^{6 d x +6 c}+161700 \,{\mathrm e}^{12 d x +12 c} a^{2} b^{2}+245700 \,{\mathrm e}^{10 d x +10 c} a^{3} b +325080 \,{\mathrm e}^{8 d x +8 c} a^{2} b^{2}+44100 a^{4} {\mathrm e}^{4 d x +4 c}+13968 b^{4} {\mathrm e}^{4 d x +4 c}+117180 \,{\mathrm e}^{4 d x +4 c} a^{2} b^{2}+283500 a^{2} b^{2} {\mathrm e}^{10 d x +10 c}+283500 \,{\mathrm e}^{8 d x +8 c} a^{3} b +216300 b \,a^{3} {\mathrm e}^{6 d x +6 c}+136500 \,{\mathrm e}^{12 d x +12 c} a^{3} b +44100 \,{\mathrm e}^{14 d x +14 c} a^{3} b +56700 \,{\mathrm e}^{14 d x +14 c} a^{2} b^{2}+110250 \,{\mathrm e}^{8 d x +8 c} a^{4}+88200 \,{\mathrm e}^{6 d x +6 c} a^{4}+44100 \,{\mathrm e}^{12 d x +12 c} a^{4}+88200 \,{\mathrm e}^{10 d x +10 c} a^{4}+12600 \,{\mathrm e}^{14 d x +14 c} a^{4}+244020 a^{2} b^{2} {\mathrm e}^{6 d x +6 c}\right )}{315 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{9}}\) | \(694\) |
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Leaf count of result is larger than twice the leaf count of optimal. 2133 vs. \(2 (152) = 304\).
Time = 0.30 (sec) , antiderivative size = 2133, normalized size of antiderivative = 13.33 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx=\text {Too large to display} \]
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Leaf count of result is larger than twice the leaf count of optimal. 308 vs. \(2 (148) = 296\).
Time = 0.32 (sec) , antiderivative size = 308, normalized size of antiderivative = 1.92 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx=\begin {cases} a^{5} x + 5 a^{4} b x - \frac {5 a^{4} b \tanh {\left (c + d x \right )}}{d} + 10 a^{3} b^{2} x - \frac {10 a^{3} b^{2} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {10 a^{3} b^{2} \tanh {\left (c + d x \right )}}{d} + 10 a^{2} b^{3} x - \frac {2 a^{2} b^{3} \tanh ^{5}{\left (c + d x \right )}}{d} - \frac {10 a^{2} b^{3} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {10 a^{2} b^{3} \tanh {\left (c + d x \right )}}{d} + 5 a b^{4} x - \frac {5 a b^{4} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {a b^{4} \tanh ^{5}{\left (c + d x \right )}}{d} - \frac {5 a b^{4} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {5 a b^{4} \tanh {\left (c + d x \right )}}{d} + b^{5} x - \frac {b^{5} \tanh ^{9}{\left (c + d x \right )}}{9 d} - \frac {b^{5} \tanh ^{7}{\left (c + d x \right )}}{7 d} - \frac {b^{5} \tanh ^{5}{\left (c + d x \right )}}{5 d} - \frac {b^{5} \tanh ^{3}{\left (c + d x \right )}}{3 d} - \frac {b^{5} \tanh {\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \tanh ^{2}{\left (c \right )}\right )^{5} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 624 vs. \(2 (152) = 304\).
Time = 0.21 (sec) , antiderivative size = 624, normalized size of antiderivative = 3.90 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx=\frac {1}{315} \, b^{5} {\left (315 \, x + \frac {315 \, c}{d} - \frac {2 \, {\left (3492 \, e^{\left (-2 \, d x - 2 \, c\right )} + 13968 \, e^{\left (-4 \, d x - 4 \, c\right )} + 26292 \, e^{\left (-6 \, d x - 6 \, c\right )} + 39438 \, e^{\left (-8 \, d x - 8 \, c\right )} + 31500 \, e^{\left (-10 \, d x - 10 \, c\right )} + 21000 \, e^{\left (-12 \, d x - 12 \, c\right )} + 6300 \, e^{\left (-14 \, d x - 14 \, c\right )} + 1575 \, e^{\left (-16 \, d x - 16 \, c\right )} + 563\right )}}{d {\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} + 36 \, e^{\left (-4 \, d x - 4 \, c\right )} + 84 \, e^{\left (-6 \, d x - 6 \, c\right )} + 126 \, e^{\left (-8 \, d x - 8 \, c\right )} + 126 \, e^{\left (-10 \, d x - 10 \, c\right )} + 84 \, e^{\left (-12 \, d x - 12 \, c\right )} + 36 \, e^{\left (-14 \, d x - 14 \, c\right )} + 9 \, e^{\left (-16 \, d x - 16 \, c\right )} + e^{\left (-18 \, d x - 18 \, c\right )} + 1\right )}}\right )} + \frac {1}{21} \, a b^{4} {\left (105 \, x + \frac {105 \, c}{d} - \frac {8 \, {\left (203 \, e^{\left (-2 \, d x - 2 \, c\right )} + 609 \, e^{\left (-4 \, d x - 4 \, c\right )} + 770 \, e^{\left (-6 \, d x - 6 \, c\right )} + 770 \, e^{\left (-8 \, d x - 8 \, c\right )} + 315 \, e^{\left (-10 \, d x - 10 \, c\right )} + 105 \, e^{\left (-12 \, d x - 12 \, c\right )} + 44\right )}}{d {\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 21 \, e^{\left (-4 \, d x - 4 \, c\right )} + 35 \, e^{\left (-6 \, d x - 6 \, c\right )} + 35 \, e^{\left (-8 \, d x - 8 \, c\right )} + 21 \, e^{\left (-10 \, d x - 10 \, c\right )} + 7 \, e^{\left (-12 \, d x - 12 \, c\right )} + e^{\left (-14 \, d x - 14 \, c\right )} + 1\right )}}\right )} + \frac {2}{3} \, a^{2} b^{3} {\left (15 \, x + \frac {15 \, c}{d} - \frac {2 \, {\left (70 \, e^{\left (-2 \, d x - 2 \, c\right )} + 140 \, e^{\left (-4 \, d x - 4 \, c\right )} + 90 \, e^{\left (-6 \, d x - 6 \, c\right )} + 45 \, e^{\left (-8 \, d x - 8 \, c\right )} + 23\right )}}{d {\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} + 10 \, e^{\left (-4 \, d x - 4 \, c\right )} + 10 \, e^{\left (-6 \, d x - 6 \, c\right )} + 5 \, e^{\left (-8 \, d x - 8 \, c\right )} + e^{\left (-10 \, d x - 10 \, c\right )} + 1\right )}}\right )} + \frac {10}{3} \, a^{3} b^{2} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + 5 \, a^{4} b {\left (x + \frac {c}{d} - \frac {2}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}}\right )} + a^{5} x \]
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Leaf count of result is larger than twice the leaf count of optimal. 721 vs. \(2 (152) = 304\).
Time = 0.36 (sec) , antiderivative size = 721, normalized size of antiderivative = 4.51 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx=\frac {315 \, {\left (a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}\right )} {\left (d x + c\right )} + \frac {2 \, {\left (1575 \, a^{4} b e^{\left (16 \, d x + 16 \, c\right )} + 6300 \, a^{3} b^{2} e^{\left (16 \, d x + 16 \, c\right )} + 9450 \, a^{2} b^{3} e^{\left (16 \, d x + 16 \, c\right )} + 6300 \, a b^{4} e^{\left (16 \, d x + 16 \, c\right )} + 1575 \, b^{5} e^{\left (16 \, d x + 16 \, c\right )} + 12600 \, a^{4} b e^{\left (14 \, d x + 14 \, c\right )} + 44100 \, a^{3} b^{2} e^{\left (14 \, d x + 14 \, c\right )} + 56700 \, a^{2} b^{3} e^{\left (14 \, d x + 14 \, c\right )} + 31500 \, a b^{4} e^{\left (14 \, d x + 14 \, c\right )} + 6300 \, b^{5} e^{\left (14 \, d x + 14 \, c\right )} + 44100 \, a^{4} b e^{\left (12 \, d x + 12 \, c\right )} + 136500 \, a^{3} b^{2} e^{\left (12 \, d x + 12 \, c\right )} + 161700 \, a^{2} b^{3} e^{\left (12 \, d x + 12 \, c\right )} + 90300 \, a b^{4} e^{\left (12 \, d x + 12 \, c\right )} + 21000 \, b^{5} e^{\left (12 \, d x + 12 \, c\right )} + 88200 \, a^{4} b e^{\left (10 \, d x + 10 \, c\right )} + 245700 \, a^{3} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 283500 \, a^{2} b^{3} e^{\left (10 \, d x + 10 \, c\right )} + 157500 \, a b^{4} e^{\left (10 \, d x + 10 \, c\right )} + 31500 \, b^{5} e^{\left (10 \, d x + 10 \, c\right )} + 110250 \, a^{4} b e^{\left (8 \, d x + 8 \, c\right )} + 283500 \, a^{3} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 325080 \, a^{2} b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 175140 \, a b^{4} e^{\left (8 \, d x + 8 \, c\right )} + 39438 \, b^{5} e^{\left (8 \, d x + 8 \, c\right )} + 88200 \, a^{4} b e^{\left (6 \, d x + 6 \, c\right )} + 216300 \, a^{3} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 244020 \, a^{2} b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 131460 \, a b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 26292 \, b^{5} e^{\left (6 \, d x + 6 \, c\right )} + 44100 \, a^{4} b e^{\left (4 \, d x + 4 \, c\right )} + 107100 \, a^{3} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 117180 \, a^{2} b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 63540 \, a b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 13968 \, b^{5} e^{\left (4 \, d x + 4 \, c\right )} + 12600 \, a^{4} b e^{\left (2 \, d x + 2 \, c\right )} + 31500 \, a^{3} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 34020 \, a^{2} b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 17460 \, a b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 3492 \, b^{5} e^{\left (2 \, d x + 2 \, c\right )} + 1575 \, a^{4} b + 4200 \, a^{3} b^{2} + 4830 \, a^{2} b^{3} + 2640 \, a b^{4} + 563 \, b^{5}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{9}}}{315 \, d} \]
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Time = 1.95 (sec) , antiderivative size = 188, normalized size of antiderivative = 1.18 \[ \int \left (a+b \tanh ^2(c+d x)\right )^5 \, dx=x\,\left (a^5+5\,a^4\,b+10\,a^3\,b^2+10\,a^2\,b^3+5\,a\,b^4+b^5\right )-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^3\,\left (10\,a^3\,b^2+10\,a^2\,b^3+5\,a\,b^4+b^5\right )}{3\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^5\,\left (10\,a^2\,b^3+5\,a\,b^4+b^5\right )}{5\,d}-\frac {{\mathrm {tanh}\left (c+d\,x\right )}^7\,\left (b^5+5\,a\,b^4\right )}{7\,d}-\frac {b^5\,{\mathrm {tanh}\left (c+d\,x\right )}^9}{9\,d}-\frac {b\,\mathrm {tanh}\left (c+d\,x\right )\,\left (5\,a^4+10\,a^3\,b+10\,a^2\,b^2+5\,a\,b^3+b^4\right )}{d} \]
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