Optimal. Leaf size=67 \[ \frac {a^3 \sinh (c+d x)}{d}+\frac {a^2 b \sinh ^3(c+d x)}{d}+\frac {3 a b^2 \sinh ^5(c+d x)}{5 d}+\frac {b^3 \sinh ^7(c+d x)}{7 d} \]
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Rubi [A]
time = 0.03, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {3269, 200}
\begin {gather*} \frac {a^3 \sinh (c+d x)}{d}+\frac {a^2 b \sinh ^3(c+d x)}{d}+\frac {3 a b^2 \sinh ^5(c+d x)}{5 d}+\frac {b^3 \sinh ^7(c+d x)}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 200
Rule 3269
Rubi steps
\begin {align*} \int \cosh (c+d x) \left (a+b \sinh ^2(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \left (a+b x^2\right )^3 \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \left (a^3+3 a^2 b x^2+3 a b^2 x^4+b^3 x^6\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^3 \sinh (c+d x)}{d}+\frac {a^2 b \sinh ^3(c+d x)}{d}+\frac {3 a b^2 \sinh ^5(c+d x)}{5 d}+\frac {b^3 \sinh ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 67, normalized size = 1.00 \begin {gather*} \frac {a^3 \sinh (c+d x)}{d}+\frac {a^2 b \sinh ^3(c+d x)}{d}+\frac {3 a b^2 \sinh ^5(c+d x)}{5 d}+\frac {b^3 \sinh ^7(c+d x)}{7 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.82, size = 56, normalized size = 0.84
method | result | size |
derivativedivides | \(\frac {\frac {b^{3} \left (\sinh ^{7}\left (d x +c \right )\right )}{7}+\frac {3 a \,b^{2} \left (\sinh ^{5}\left (d x +c \right )\right )}{5}+a^{2} b \left (\sinh ^{3}\left (d x +c \right )\right )+a^{3} \sinh \left (d x +c \right )}{d}\) | \(56\) |
default | \(\frac {\frac {b^{3} \left (\sinh ^{7}\left (d x +c \right )\right )}{7}+\frac {3 a \,b^{2} \left (\sinh ^{5}\left (d x +c \right )\right )}{5}+a^{2} b \left (\sinh ^{3}\left (d x +c \right )\right )+a^{3} \sinh \left (d x +c \right )}{d}\) | \(56\) |
risch | \(\frac {b^{3} {\mathrm e}^{7 d x +7 c}}{896 d}+\frac {3 a \,b^{2} {\mathrm e}^{5 d x +5 c}}{160 d}-\frac {{\mathrm e}^{5 d x +5 c} b^{3}}{128 d}+\frac {{\mathrm e}^{3 d x +3 c} a^{2} b}{8 d}-\frac {3 a \,b^{2} {\mathrm e}^{3 d x +3 c}}{32 d}+\frac {3 \,{\mathrm e}^{3 d x +3 c} b^{3}}{128 d}+\frac {a^{3} {\mathrm e}^{d x +c}}{2 d}-\frac {3 b \,{\mathrm e}^{d x +c} a^{2}}{8 d}+\frac {3 a \,{\mathrm e}^{d x +c} b^{2}}{16 d}-\frac {5 b^{3} {\mathrm e}^{d x +c}}{128 d}-\frac {{\mathrm e}^{-d x -c} a^{3}}{2 d}+\frac {3 \,{\mathrm e}^{-d x -c} a^{2} b}{8 d}-\frac {3 a \,{\mathrm e}^{-d x -c} b^{2}}{16 d}+\frac {5 b^{3} {\mathrm e}^{-d x -c}}{128 d}-\frac {{\mathrm e}^{-3 d x -3 c} a^{2} b}{8 d}+\frac {3 a \,b^{2} {\mathrm e}^{-3 d x -3 c}}{32 d}-\frac {3 b^{3} {\mathrm e}^{-3 d x -3 c}}{128 d}-\frac {3 a \,b^{2} {\mathrm e}^{-5 d x -5 c}}{160 d}+\frac {{\mathrm e}^{-5 d x -5 c} b^{3}}{128 d}-\frac {b^{3} {\mathrm e}^{-7 d x -7 c}}{896 d}\) | \(340\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 63, normalized size = 0.94 \begin {gather*} \frac {b^{3} \sinh \left (d x + c\right )^{7}}{7 \, d} + \frac {3 \, a b^{2} \sinh \left (d x + c\right )^{5}}{5 \, d} + \frac {a^{2} b \sinh \left (d x + c\right )^{3}}{d} + \frac {a^{3} \sinh \left (d x + c\right )}{d} \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. 209 vs.
\(2 (63) = 126\).
time = 0.51, size = 209, normalized size = 3.12 \begin {gather*} \frac {5 \, b^{3} \sinh \left (d x + c\right )^{7} + 7 \, {\left (15 \, b^{3} \cosh \left (d x + c\right )^{2} + 12 \, a b^{2} - 5 \, b^{3}\right )} \sinh \left (d x + c\right )^{5} + 35 \, {\left (5 \, b^{3} \cosh \left (d x + c\right )^{4} + 16 \, a^{2} b - 12 \, a b^{2} + 3 \, b^{3} + 2 \, {\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{3} + 35 \, {\left (b^{3} \cosh \left (d x + c\right )^{6} + {\left (12 \, a b^{2} - 5 \, b^{3}\right )} \cosh \left (d x + c\right )^{4} + 64 \, a^{3} - 48 \, a^{2} b + 24 \, a b^{2} - 5 \, b^{3} + 3 \, {\left (16 \, a^{2} b - 12 \, a b^{2} + 3 \, b^{3}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )}{2240 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.63, size = 75, normalized size = 1.12 \begin {gather*} \begin {cases} \frac {a^{3} \sinh {\left (c + d x \right )}}{d} + \frac {a^{2} b \sinh ^{3}{\left (c + d x \right )}}{d} + \frac {3 a b^{2} \sinh ^{5}{\left (c + d x \right )}}{5 d} + \frac {b^{3} \sinh ^{7}{\left (c + d x \right )}}{7 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\left (c \right )}\right )^{3} \cosh {\left (c \right )} & \text {otherwise} \end {cases} \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. 222 vs.
\(2 (63) = 126\).
time = 0.43, size = 222, normalized size = 3.31 \begin {gather*} \frac {b^{3} e^{\left (7 \, d x + 7 \, c\right )}}{896 \, d} - \frac {b^{3} e^{\left (-7 \, d x - 7 \, c\right )}}{896 \, d} + \frac {{\left (12 \, a b^{2} - 5 \, b^{3}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {{\left (16 \, a^{2} b - 12 \, a b^{2} + 3 \, b^{3}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{128 \, d} + \frac {{\left (64 \, a^{3} - 48 \, a^{2} b + 24 \, a b^{2} - 5 \, b^{3}\right )} e^{\left (d x + c\right )}}{128 \, d} - \frac {{\left (64 \, a^{3} - 48 \, a^{2} b + 24 \, a b^{2} - 5 \, b^{3}\right )} e^{\left (-d x - c\right )}}{128 \, d} - \frac {{\left (16 \, a^{2} b - 12 \, a b^{2} + 3 \, b^{3}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{128 \, d} - \frac {{\left (12 \, a b^{2} - 5 \, b^{3}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.15, size = 58, normalized size = 0.87 \begin {gather*} \frac {\mathrm {sinh}\left (c+d\,x\right )\,\left (35\,a^3+35\,a^2\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+21\,a\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^4+5\,b^3\,{\mathrm {sinh}\left (c+d\,x\right )}^6\right )}{35\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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