Optimal. Leaf size=74 \[ \frac {a^2 \sinh (c+d x)}{d}+\frac {b (2 a+b) \sinh ^5(c+d x)}{5 d}+\frac {a (a+2 b) \sinh ^3(c+d x)}{3 d}+\frac {b^2 \sinh ^7(c+d x)}{7 d} \]
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Rubi [A] time = 0.07, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {3190, 373} \[ \frac {a^2 \sinh (c+d x)}{d}+\frac {b (2 a+b) \sinh ^5(c+d x)}{5 d}+\frac {a (a+2 b) \sinh ^3(c+d x)}{3 d}+\frac {b^2 \sinh ^7(c+d x)}{7 d} \]
Antiderivative was successfully verified.
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Rule 373
Rule 3190
Rubi steps
\begin {align*} \int \cosh ^3(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \left (1+x^2\right ) \left (a+b x^2\right )^2 \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2+a (a+2 b) x^2+b (2 a+b) x^4+b^2 x^6\right ) \, dx,x,\sinh (c+d x)\right )}{d}\\ &=\frac {a^2 \sinh (c+d x)}{d}+\frac {a (a+2 b) \sinh ^3(c+d x)}{3 d}+\frac {b (2 a+b) \sinh ^5(c+d x)}{5 d}+\frac {b^2 \sinh ^7(c+d x)}{7 d}\\ \end {align*}
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Mathematica [A] time = 0.11, size = 64, normalized size = 0.86 \[ \frac {105 a^2 \sinh (c+d x)+21 b (2 a+b) \sinh ^5(c+d x)+35 a (a+2 b) \sinh ^3(c+d x)+15 b^2 \sinh ^7(c+d x)}{105 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.55, size = 188, normalized size = 2.54 \[ \frac {15 \, b^{2} \sinh \left (d x + c\right )^{7} + 21 \, {\left (15 \, b^{2} \cosh \left (d x + c\right )^{2} + 8 \, a b - b^{2}\right )} \sinh \left (d x + c\right )^{5} + 35 \, {\left (15 \, b^{2} \cosh \left (d x + c\right )^{4} + 6 \, {\left (8 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 16 \, a^{2} + 8 \, a b - 3 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 105 \, {\left (b^{2} \cosh \left (d x + c\right )^{6} + {\left (8 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{4} + {\left (16 \, a^{2} + 8 \, a b - 3 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + 48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )} \sinh \left (d x + c\right )}{6720 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 196, normalized size = 2.65 \[ \frac {b^{2} e^{\left (7 \, d x + 7 \, c\right )}}{896 \, d} - \frac {b^{2} e^{\left (-7 \, d x - 7 \, c\right )}}{896 \, d} + \frac {{\left (8 \, a b - b^{2}\right )} e^{\left (5 \, d x + 5 \, c\right )}}{640 \, d} + \frac {{\left (16 \, a^{2} + 8 \, a b - 3 \, b^{2}\right )} e^{\left (3 \, d x + 3 \, c\right )}}{384 \, d} + \frac {{\left (48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )} e^{\left (d x + c\right )}}{128 \, d} - \frac {{\left (48 \, a^{2} - 16 \, a b + 3 \, b^{2}\right )} e^{\left (-d x - c\right )}}{128 \, d} - \frac {{\left (16 \, a^{2} + 8 \, a b - 3 \, b^{2}\right )} e^{\left (-3 \, d x - 3 \, c\right )}}{384 \, d} - \frac {{\left (8 \, a b - b^{2}\right )} e^{\left (-5 \, d x - 5 \, c\right )}}{640 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 128, normalized size = 1.73 \[ \frac {b^{2} \left (\frac {\left (\sinh ^{3}\left (d x +c \right )\right ) \left (\cosh ^{4}\left (d x +c \right )\right )}{7}-\frac {3 \sinh \left (d x +c \right ) \left (\cosh ^{4}\left (d x +c \right )\right )}{35}+\frac {3 \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (d x +c \right )\right )}{3}\right ) \sinh \left (d x +c \right )}{35}\right )+2 a b \left (\frac {\sinh \left (d x +c \right ) \left (\cosh ^{4}\left (d x +c \right )\right )}{5}-\frac {\left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (d x +c \right )\right )}{3}\right ) \sinh \left (d x +c \right )}{5}\right )+a^{2} \left (\frac {2}{3}+\frac {\left (\cosh ^{2}\left (d x +c \right )\right )}{3}\right ) \sinh \left (d x +c \right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.33, size = 242, normalized size = 3.27 \[ -\frac {1}{4480} \, b^{2} {\left (\frac {{\left (7 \, e^{\left (-2 \, d x - 2 \, c\right )} + 35 \, e^{\left (-4 \, d x - 4 \, c\right )} - 105 \, e^{\left (-6 \, d x - 6 \, c\right )} - 5\right )} e^{\left (7 \, d x + 7 \, c\right )}}{d} + \frac {105 \, e^{\left (-d x - c\right )} - 35 \, e^{\left (-3 \, d x - 3 \, c\right )} - 7 \, e^{\left (-5 \, d x - 5 \, c\right )} + 5 \, e^{\left (-7 \, d x - 7 \, c\right )}}{d}\right )} + \frac {1}{240} \, a b {\left (\frac {{\left (5 \, e^{\left (-2 \, d x - 2 \, c\right )} - 30 \, e^{\left (-4 \, d x - 4 \, c\right )} + 3\right )} e^{\left (5 \, d x + 5 \, c\right )}}{d} + \frac {30 \, e^{\left (-d x - c\right )} - 5 \, e^{\left (-3 \, d x - 3 \, c\right )} - 3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d}\right )} + \frac {1}{24} \, a^{2} {\left (\frac {e^{\left (3 \, d x + 3 \, c\right )}}{d} + \frac {9 \, e^{\left (d x + c\right )}}{d} - \frac {9 \, e^{\left (-d x - c\right )}}{d} - \frac {e^{\left (-3 \, d x - 3 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.96, size = 80, normalized size = 1.08 \[ \frac {35\,a^2\,{\mathrm {sinh}\left (c+d\,x\right )}^3+105\,a^2\,\mathrm {sinh}\left (c+d\,x\right )+42\,a\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^5+70\,a\,b\,{\mathrm {sinh}\left (c+d\,x\right )}^3+15\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^7+21\,b^2\,{\mathrm {sinh}\left (c+d\,x\right )}^5}{105\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.27, size = 136, normalized size = 1.84 \[ \begin {cases} - \frac {2 a^{2} \sinh ^{3}{\left (c + d x \right )}}{3 d} + \frac {a^{2} \sinh {\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{d} - \frac {4 a b \sinh ^{5}{\left (c + d x \right )}}{15 d} + \frac {2 a b \sinh ^{3}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{3 d} - \frac {2 b^{2} \sinh ^{7}{\left (c + d x \right )}}{35 d} + \frac {b^{2} \sinh ^{5}{\left (c + d x \right )} \cosh ^{2}{\left (c + d x \right )}}{5 d} & \text {for}\: d \neq 0 \\x \left (a + b \sinh ^{2}{\relax (c )}\right )^{2} \cosh ^{3}{\relax (c )} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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