Optimal. Leaf size=91 \[ \frac {3}{8} (a+b) \left (a^2-2 a b+5 b^2\right ) x+\frac {3 (a-3 b) (a+b)^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {(a+b)^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac {b^3 \tanh (c+d x)}{d} \]
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Rubi [A]
time = 0.09, antiderivative size = 91, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3756, 398,
1171, 393, 212} \begin {gather*} \frac {3}{8} x (a+b) \left (a^2-2 a b+5 b^2\right )+\frac {(a+b)^3 \sinh (c+d x) \cosh ^3(c+d x)}{4 d}+\frac {3 (a-3 b) (a+b)^2 \sinh (c+d x) \cosh (c+d x)}{8 d}-\frac {b^3 \tanh (c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 393
Rule 398
Rule 1171
Rule 3756
Rubi steps
\begin {align*} \int \cosh ^4(c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \frac {\left (a+b x^2\right )^3}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (-b^3+\frac {a^3+b^3+3 b \left (a^2-b^2\right ) x^2+3 b^2 (a+b) x^4}{\left (1-x^2\right )^3}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {b^3 \tanh (c+d x)}{d}+\frac {\text {Subst}\left (\int \frac {a^3+b^3+3 b \left (a^2-b^2\right ) x^2+3 b^2 (a+b) x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {(a+b)^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac {b^3 \tanh (c+d x)}{d}-\frac {\text {Subst}\left (\int \frac {-3 (a-b)^2 (a+b)+12 b^2 (a+b) x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{4 d}\\ &=\frac {3 (a-3 b) (a+b)^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {(a+b)^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac {b^3 \tanh (c+d x)}{d}+\frac {\left (3 (a+b) \left (a^2-2 a b+5 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 d}\\ &=\frac {3}{8} (a+b) \left (a^2-2 a b+5 b^2\right ) x+\frac {3 (a-3 b) (a+b)^2 \cosh (c+d x) \sinh (c+d x)}{8 d}+\frac {(a+b)^3 \cosh ^3(c+d x) \sinh (c+d x)}{4 d}-\frac {b^3 \tanh (c+d x)}{d}\\ \end {align*}
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Mathematica [A]
time = 0.50, size = 81, normalized size = 0.89 \begin {gather*} \frac {12 \left (a^3-a^2 b+3 a b^2+5 b^3\right ) (c+d x)+8 (a-2 b) (a+b)^2 \sinh (2 (c+d x))+(a+b)^3 \sinh (4 (c+d x))-32 b^3 \tanh (c+d x)}{32 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. \(292\) vs.
\(2(85)=170\).
time = 1.86, size = 293, normalized size = 3.22
method | result | size |
risch | \(\frac {3 a^{3} x}{8}-\frac {3 a^{2} b x}{8}+\frac {9 a \,b^{2} x}{8}+\frac {15 b^{3} x}{8}+\frac {{\mathrm e}^{4 d x +4 c} a^{3}}{64 d}+\frac {3 \,{\mathrm e}^{4 d x +4 c} a^{2} b}{64 d}+\frac {3 b^{2} {\mathrm e}^{4 d x +4 c} a}{64 d}+\frac {b^{3} {\mathrm e}^{4 d x +4 c}}{64 d}+\frac {{\mathrm e}^{2 d x +2 c} a^{3}}{8 d}-\frac {3 \,{\mathrm e}^{2 d x +2 c} a \,b^{2}}{8 d}-\frac {b^{3} {\mathrm e}^{2 d x +2 c}}{4 d}-\frac {{\mathrm e}^{-2 d x -2 c} a^{3}}{8 d}+\frac {3 \,{\mathrm e}^{-2 d x -2 c} a \,b^{2}}{8 d}+\frac {b^{3} {\mathrm e}^{-2 d x -2 c}}{4 d}-\frac {{\mathrm e}^{-4 d x -4 c} a^{3}}{64 d}-\frac {3 \,{\mathrm e}^{-4 d x -4 c} a^{2} b}{64 d}-\frac {3 b^{2} {\mathrm e}^{-4 d x -4 c} a}{64 d}-\frac {b^{3} {\mathrm e}^{-4 d x -4 c}}{64 d}+\frac {2 b^{3}}{d \left (1+{\mathrm e}^{2 d x +2 c}\right )}\) | \(293\) |
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. 267 vs.
\(2 (85) = 170\).
time = 0.27, size = 267, normalized size = 2.93 \begin {gather*} \frac {1}{64} \, a^{3} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {3}{64} \, a b^{2} {\left (24 \, x + \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} - \frac {8 \, e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {8 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {e^{\left (-4 \, d x - 4 \, c\right )}}{d}\right )} + \frac {1}{64} \, b^{3} {\left (\frac {120 \, {\left (d x + c\right )}}{d} + \frac {16 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )}}{d} - \frac {15 \, e^{\left (-2 \, d x - 2 \, c\right )} + 144 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1}{d {\left (e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}\right )}}\right )} - \frac {3}{64} \, a^{2} b {\left (\frac {8 \, {\left (d x + c\right )}}{d} - \frac {e^{\left (4 \, d x + 4 \, c\right )}}{d} + \frac {e^{\left (-4 \, d x - 4 \, c\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. 227 vs.
\(2 (85) = 170\).
time = 0.37, size = 227, normalized size = 2.49 \begin {gather*} \frac {{\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sinh \left (d x + c\right )^{5} + {\left (9 \, a^{3} + 3 \, a^{2} b - 21 \, a b^{2} - 15 \, b^{3} + 10 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 8 \, {\left (8 \, b^{3} + 3 \, {\left (a^{3} - a^{2} b + 3 \, a b^{2} + 5 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right ) + {\left (5 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \cosh \left (d x + c\right )^{4} + 8 \, a^{3} - 24 \, a b^{2} - 80 \, b^{3} + 9 \, {\left (3 \, a^{3} + a^{2} b - 7 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{64 \, d \cosh \left (d x + c\right )} \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. 282 vs.
\(2 (85) = 170\).
time = 0.62, size = 282, normalized size = 3.10 \begin {gather*} \frac {a^{3} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 24 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 24 \, {\left (a^{3} - a^{2} b + 3 \, a b^{2} + 5 \, b^{3}\right )} {\left (d x + c\right )} + \frac {128 \, b^{3}}{e^{\left (2 \, d x + 2 \, c\right )} + 1} - {\left (18 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 18 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} + 54 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 90 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 8 \, a^{3} e^{\left (2 \, d x + 2 \, c\right )} - 24 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 16 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.41, size = 133, normalized size = 1.46 \begin {gather*} x\,\left (\frac {3\,a^3}{8}-\frac {3\,a^2\,b}{8}+\frac {9\,a\,b^2}{8}+\frac {15\,b^3}{8}\right )+\frac {2\,b^3}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}\,{\left (a+b\right )}^3}{64\,d}+\frac {{\mathrm {e}}^{4\,c+4\,d\,x}\,{\left (a+b\right )}^3}{64\,d}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,{\left (a+b\right )}^2\,\left (a-2\,b\right )}{8\,d}+\frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a+b\right )}^2\,\left (a-2\,b\right )}{8\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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