Optimal. Leaf size=111 \[ a^4 x+\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tanh (c+d x)}{d}-\frac {b^2 \left (6 a^2+8 a b+3 b^2\right ) \tanh ^3(c+d x)}{3 d}+\frac {b^3 (4 a+3 b) \tanh ^5(c+d x)}{5 d}-\frac {b^4 \tanh ^7(c+d x)}{7 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 111, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4213, 398, 212}
\begin {gather*} a^4 x-\frac {b^2 \left (6 a^2+8 a b+3 b^2\right ) \tanh ^3(c+d x)}{3 d}+\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tanh (c+d x)}{d}+\frac {b^3 (4 a+3 b) \tanh ^5(c+d x)}{5 d}-\frac {b^4 \tanh ^7(c+d x)}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 398
Rule 4213
Rubi steps
\begin {align*} \int \left (a+b \text {sech}^2(c+d x)\right )^4 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b-b x^2\right )^4}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (b (2 a+b) \left (2 a^2+2 a b+b^2\right )-b^2 \left (6 a^2+8 a b+3 b^2\right ) x^2+b^3 (4 a+3 b) x^4-b^4 x^6+\frac {a^4}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tanh (c+d x)}{d}-\frac {b^2 \left (6 a^2+8 a b+3 b^2\right ) \tanh ^3(c+d x)}{3 d}+\frac {b^3 (4 a+3 b) \tanh ^5(c+d x)}{5 d}-\frac {b^4 \tanh ^7(c+d x)}{7 d}+\frac {a^4 \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=a^4 x+\frac {b (2 a+b) \left (2 a^2+2 a b+b^2\right ) \tanh (c+d x)}{d}-\frac {b^2 \left (6 a^2+8 a b+3 b^2\right ) \tanh ^3(c+d x)}{3 d}+\frac {b^3 (4 a+3 b) \tanh ^5(c+d x)}{5 d}-\frac {b^4 \tanh ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(455\) vs. \(2(111)=222\).
time = 1.11, size = 455, normalized size = 4.10 \begin {gather*} \frac {\text {sech}(c) \text {sech}^7(c+d x) \left (3675 a^4 d x \cosh (d x)+3675 a^4 d x \cosh (2 c+d x)+2205 a^4 d x \cosh (2 c+3 d x)+2205 a^4 d x \cosh (4 c+3 d x)+735 a^4 d x \cosh (4 c+5 d x)+735 a^4 d x \cosh (6 c+5 d x)+105 a^4 d x \cosh (6 c+7 d x)+105 a^4 d x \cosh (8 c+7 d x)+16800 a^3 b \sinh (d x)+18480 a^2 b^2 \sinh (d x)+11200 a b^3 \sinh (d x)+3360 b^4 \sinh (d x)-12600 a^3 b \sinh (2 c+d x)-10920 a^2 b^2 \sinh (2 c+d x)-4480 a b^3 \sinh (2 c+d x)+12600 a^3 b \sinh (2 c+3 d x)+15120 a^2 b^2 \sinh (2 c+3 d x)+9408 a b^3 \sinh (2 c+3 d x)+2016 b^4 \sinh (2 c+3 d x)-5040 a^3 b \sinh (4 c+3 d x)-2520 a^2 b^2 \sinh (4 c+3 d x)+5040 a^3 b \sinh (4 c+5 d x)+5880 a^2 b^2 \sinh (4 c+5 d x)+3136 a b^3 \sinh (4 c+5 d x)+672 b^4 \sinh (4 c+5 d x)-840 a^3 b \sinh (6 c+5 d x)+840 a^3 b \sinh (6 c+7 d x)+840 a^2 b^2 \sinh (6 c+7 d x)+448 a b^3 \sinh (6 c+7 d x)+96 b^4 \sinh (6 c+7 d x)\right )}{13440 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. \(309\) vs.
\(2(105)=210\).
time = 1.86, size = 310, normalized size = 2.79
method | result | size |
risch | \(x \,a^{4}-\frac {8 b \left (105 a^{3} {\mathrm e}^{12 d x +12 c}+630 a^{3} {\mathrm e}^{10 d x +10 c}+315 a^{2} b \,{\mathrm e}^{10 d x +10 c}+1575 a^{3} {\mathrm e}^{8 d x +8 c}+1365 a^{2} b \,{\mathrm e}^{8 d x +8 c}+560 a \,b^{2} {\mathrm e}^{8 d x +8 c}+2100 a^{3} {\mathrm e}^{6 d x +6 c}+2310 a^{2} b \,{\mathrm e}^{6 d x +6 c}+1400 a \,b^{2} {\mathrm e}^{6 d x +6 c}+420 b^{3} {\mathrm e}^{6 d x +6 c}+1575 a^{3} {\mathrm e}^{4 d x +4 c}+1890 a^{2} b \,{\mathrm e}^{4 d x +4 c}+1176 a \,b^{2} {\mathrm e}^{4 d x +4 c}+252 b^{3} {\mathrm e}^{4 d x +4 c}+630 a^{3} {\mathrm e}^{2 d x +2 c}+735 a^{2} b \,{\mathrm e}^{2 d x +2 c}+392 a \,b^{2} {\mathrm e}^{2 d x +2 c}+84 b^{3} {\mathrm e}^{2 d x +2 c}+105 a^{3}+105 a^{2} b +56 a \,b^{2}+12 b^{3}\right )}{105 d \left (1+{\mathrm e}^{2 d x +2 c}\right )^{7}}\) | \(310\) |
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. 703 vs.
\(2 (105) = 210\).
time = 0.29, size = 703, normalized size = 6.33 \begin {gather*} a^{4} x + \frac {32}{35} \, b^{4} {\left (\frac {7 \, e^{\left (-2 \, d x - 2 \, c\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 )}} + \frac {21 \, e^{\left (-4 \, d x - 4 \, c\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 )}} + \frac {35 \, e^{\left (-6 \, d x - 6 \, c\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 )}} + \frac {1}{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 {64}{15} \, a b^{3} {\left (\frac {5 \, e^{\left (-2 \, d x - 2 \, c\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 )}} + \frac {10 \, e^{\left (-4 \, d x - 4 \, c\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 )}} + \frac {1}{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 )} + 8 \, a^{2} b^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\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 )}} + \frac {1}{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 )} + \frac {8 \, a^{3} b}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\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. 941 vs.
\(2 (105) = 210\).
time = 0.39, size = 941, normalized size = 8.48 \begin {gather*} \frac {{\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )^{7} + 7 \, {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + 4 \, {\left (105 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \sinh \left (d x + c\right )^{7} + 7 \, {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )^{5} + 28 \, {\left (75 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4} + 3 \, {\left (105 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{5} + 35 \, {\left ({\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )^{3} + {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 21 \, {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )^{3} + 28 \, {\left (5 \, {\left (105 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \cosh \left (d x + c\right )^{4} + 135 \, a^{3} b + 225 \, a^{2} b^{2} + 168 \, a b^{3} + 36 \, b^{4} + 10 \, {\left (75 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 7 \, {\left (3 \, {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )^{5} + 10 \, {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )^{3} + 9 \, {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \, {\left (105 \, a^{4} d x - 420 \, a^{3} b - 420 \, a^{2} b^{2} - 224 \, a b^{3} - 48 \, b^{4}\right )} \cosh \left (d x + c\right ) + 28 \, {\left ({\left (105 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \cosh \left (d x + c\right )^{6} + 5 \, {\left (75 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \cosh \left (d x + c\right )^{4} + 75 \, a^{3} b + 135 \, a^{2} b^{2} + 120 \, a b^{3} + 60 \, b^{4} + 9 \, {\left (45 \, a^{3} b + 75 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{105 \, {\left (d \cosh \left (d x + c\right )^{7} + 7 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{6} + 7 \, d \cosh \left (d x + c\right )^{5} + 35 \, {\left (d \cosh \left (d x + c\right )^{3} + d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{4} + 21 \, d \cosh \left (d x + c\right )^{3} + 7 \, {\left (3 \, d \cosh \left (d x + c\right )^{5} + 10 \, d \cosh \left (d x + c\right )^{3} + 9 \, d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{2} + 35 \, d \cosh \left (d x + c\right )\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*} \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{4}\, dx \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. 334 vs.
\(2 (105) = 210\).
time = 0.42, size = 334, normalized size = 3.01 \begin {gather*} \frac {105 \, {\left (d x + c\right )} a^{4} - \frac {8 \, {\left (105 \, a^{3} b e^{\left (12 \, d x + 12 \, c\right )} + 630 \, a^{3} b e^{\left (10 \, d x + 10 \, c\right )} + 315 \, a^{2} b^{2} e^{\left (10 \, d x + 10 \, c\right )} + 1575 \, a^{3} b e^{\left (8 \, d x + 8 \, c\right )} + 1365 \, a^{2} b^{2} e^{\left (8 \, d x + 8 \, c\right )} + 560 \, a b^{3} e^{\left (8 \, d x + 8 \, c\right )} + 2100 \, a^{3} b e^{\left (6 \, d x + 6 \, c\right )} + 2310 \, a^{2} b^{2} e^{\left (6 \, d x + 6 \, c\right )} + 1400 \, a b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 420 \, b^{4} e^{\left (6 \, d x + 6 \, c\right )} + 1575 \, a^{3} b e^{\left (4 \, d x + 4 \, c\right )} + 1890 \, a^{2} b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 1176 \, a b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 252 \, b^{4} e^{\left (4 \, d x + 4 \, c\right )} + 630 \, a^{3} b e^{\left (2 \, d x + 2 \, c\right )} + 735 \, a^{2} b^{2} e^{\left (2 \, d x + 2 \, c\right )} + 392 \, a b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 84 \, b^{4} e^{\left (2 \, d x + 2 \, c\right )} + 105 \, a^{3} b + 105 \, a^{2} b^{2} + 56 \, a b^{3} + 12 \, b^{4}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{7}}}{105 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 1083, normalized size = 9.76 \begin {gather*} a^4\,x-\frac {\frac {8\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{21\,d}+\frac {16\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{21\,d}+\frac {16\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (5\,a^3\,b+9\,a^2\,b^2+8\,a\,b^3+4\,b^4\right )}{7\,d}+\frac {40\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{10\,c+10\,d\,x}}{7\,d}}{6\,{\mathrm {e}}^{2\,c+2\,d\,x}+15\,{\mathrm {e}}^{4\,c+4\,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}-\frac {\frac {8\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{7\,d}+\frac {8\,{\mathrm {e}}^{8\,c+8\,d\,x}\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{7\,d}+\frac {8\,a^3\,b}{7\,d}+\frac {32\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (5\,a^3\,b+9\,a^2\,b^2+8\,a\,b^3+4\,b^4\right )}{7\,d}+\frac {48\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {48\,{\mathrm {e}}^{10\,c+10\,d\,x}\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{12\,c+12\,d\,x}}{7\,d}}{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}-\frac {\frac {8\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{7\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\frac {8\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{105\,d}+\frac {16\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{35\,d}+\frac {32\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (5\,a^3\,b+9\,a^2\,b^2+8\,a\,b^3+4\,b^4\right )}{35\,d}+\frac {32\,{\mathrm {e}}^{6\,c+6\,d\,x}\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{8\,c+8\,d\,x}}{7\,d}}{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}-\frac {\frac {8\,\left (5\,a^3\,b+9\,a^2\,b^2+8\,a\,b^3+4\,b^4\right )}{35\,d}+\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{35\,d}+\frac {24\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{6\,c+6\,d\,x}}{7\,d}}{4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1}-\frac {\frac {8\,\left (15\,a^3\,b+24\,a^2\,b^2+16\,a\,b^3\right )}{105\,d}+\frac {16\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a^3\,b+a^2\,b^2\right )}{7\,d}+\frac {8\,a^3\,b\,{\mathrm {e}}^{4\,c+4\,d\,x}}{7\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}-\frac {8\,a^3\,b}{7\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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