Optimal. Leaf size=83 \[ -\frac {1}{16} (8 a+5 b) x+\frac {(8 a+11 b) \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {13 b \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d} \]
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Rubi [A]
time = 0.07, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3296, 1271,
1171, 393, 212} \begin {gather*} \frac {(8 a+11 b) \sinh (c+d x) \cosh (c+d x)}{16 d}-\frac {1}{16} x (8 a+5 b)+\frac {b \sinh (c+d x) \cosh ^5(c+d x)}{6 d}-\frac {13 b \sinh (c+d x) \cosh ^3(c+d x)}{24 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 393
Rule 1171
Rule 1271
Rule 3296
Rubi steps
\begin {align*} \int \sinh ^2(c+d x) \left (a+b \sinh ^4(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int \frac {x^2 \left (a-2 a x^2+(a+b) x^4\right )}{\left (1-x^2\right )^4} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}+\frac {\text {Subst}\left (\int \frac {-b+6 (a-b) x^2-6 (a+b) x^4}{\left (1-x^2\right )^3} \, dx,x,\tanh (c+d x)\right )}{6 d}\\ &=-\frac {13 b \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}-\frac {\text {Subst}\left (\int \frac {-9 b-24 (a+b) x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{24 d}\\ &=\frac {(8 a+11 b) \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {13 b \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}-\frac {(8 a+5 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{16 d}\\ &=-\frac {1}{16} (8 a+5 b) x+\frac {(8 a+11 b) \cosh (c+d x) \sinh (c+d x)}{16 d}-\frac {13 b \cosh ^3(c+d x) \sinh (c+d x)}{24 d}+\frac {b \cosh ^5(c+d x) \sinh (c+d x)}{6 d}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 63, normalized size = 0.76 \begin {gather*} \frac {-96 a c-60 b c-96 a d x-60 b d x+(48 a+45 b) \sinh (2 (c+d x))-9 b \sinh (4 (c+d x))+b \sinh (6 (c+d x))}{192 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.03, size = 61, normalized size = 0.73
method | result | size |
default | \(\frac {\left (\frac {15 b}{32}+\frac {a}{2}\right ) \sinh \left (2 d x +2 c \right )}{2 d}-\frac {a x}{2}-\frac {5 b x}{16}-\frac {3 b \sinh \left (4 d x +4 c \right )}{64 d}+\frac {b \sinh \left (6 d x +6 c \right )}{192 d}\) | \(61\) |
risch | \(-\frac {5 b x}{16}-\frac {a x}{2}+\frac {b \,{\mathrm e}^{6 d x +6 c}}{384 d}-\frac {3 \,{\mathrm e}^{4 d x +4 c} b}{128 d}+\frac {15 b \,{\mathrm e}^{2 d x +2 c}}{128 d}+\frac {{\mathrm e}^{2 d x +2 c} a}{8 d}-\frac {15 \,{\mathrm e}^{-2 d x -2 c} b}{128 d}-\frac {{\mathrm e}^{-2 d x -2 c} a}{8 d}+\frac {3 \,{\mathrm e}^{-4 d x -4 c} b}{128 d}-\frac {b \,{\mathrm e}^{-6 d x -6 c}}{384 d}\) | \(130\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 122, normalized size = 1.47 \begin {gather*} -\frac {1}{8} \, a {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{384} \, b {\left (\frac {{\left (9 \, e^{\left (-2 \, d x - 2 \, c\right )} - 45 \, e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )} e^{\left (6 \, d x + 6 \, c\right )}}{d} + \frac {120 \, {\left (d x + c\right )}}{d} + \frac {45 \, e^{\left (-2 \, d x - 2 \, c\right )} - 9 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )}}{d}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.42, size = 109, normalized size = 1.31 \begin {gather*} \frac {3 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{5} + 2 \, {\left (5 \, b \cosh \left (d x + c\right )^{3} - 9 \, b \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )^{3} - 6 \, {\left (8 \, a + 5 \, b\right )} d x + 3 \, {\left (b \cosh \left (d x + c\right )^{5} - 6 \, b \cosh \left (d x + c\right )^{3} + {\left (16 \, a + 15 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{96 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 206 vs.
\(2 (76) = 152\).
time = 0.46, size = 206, normalized size = 2.48 \begin {gather*} \begin {cases} \frac {a x \sinh ^{2}{\left (c + d x \right )}}{2} - \frac {a x \cosh ^{2}{\left (c + d x \right )}}{2} + \frac {a \sinh {\left (c + d x \right )} \cosh {\left (c + d x \right )}}{2 d} + \frac {5 b x \sinh ^{6}{\left (c + d x \right )}}{16} - \frac {15 b x \sinh ^{4}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{16} + \frac {15 b x \sinh ^{2}{\left (c + d x \right )} \cosh ^{4}{\left (c + d x \right )}}{16} - \frac {5 b x \cosh ^{6}{\left (c + d x \right )}}{16} + \frac {11 b \sinh ^{5}{\left (c + d x \right )} \cosh {\left (c + d x \right )}}{16 d} - \frac {5 b \sinh ^{3}{\left (c + d x \right )} \cosh ^{3}{\left (c + d x \right )}}{6 d} + \frac {5 b \sinh {\left (c + d x \right )} \cosh ^{5}{\left (c + d x \right )}}{16 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{4}{\left (c \right )}\right ) \sinh ^{2}{\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 = 0.42, size = 113, normalized size = 1.36 \begin {gather*} -\frac {1}{16} \, {\left (8 \, a + 5 \, b\right )} x + \frac {b e^{\left (6 \, d x + 6 \, c\right )}}{384 \, d} - \frac {3 \, b e^{\left (4 \, d x + 4 \, c\right )}}{128 \, d} + \frac {{\left (16 \, a + 15 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )}}{128 \, d} - \frac {{\left (16 \, a + 15 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{128 \, d} + \frac {3 \, b e^{\left (-4 \, d x - 4 \, c\right )}}{128 \, d} - \frac {b e^{\left (-6 \, d x - 6 \, c\right )}}{384 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.16, size = 64, normalized size = 0.77 \begin {gather*} \frac {12\,a\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )+\frac {45\,b\,\mathrm {sinh}\left (2\,c+2\,d\,x\right )}{4}-\frac {9\,b\,\mathrm {sinh}\left (4\,c+4\,d\,x\right )}{4}+\frac {b\,\mathrm {sinh}\left (6\,c+6\,d\,x\right )}{4}-24\,a\,d\,x-15\,b\,d\,x}{48\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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