Optimal. Leaf size=51 \[ \frac {(a+b)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {1}{2} x (a-3 b) (a+b)+\frac {b^2 \tanh (c+d x)}{d} \]
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Rubi [A] time = 0.08, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3675, 390, 385, 206} \[ \frac {(a+b)^2 \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {1}{2} x (a-3 b) (a+b)+\frac {b^2 \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 385
Rule 390
Rule 3675
Rubi steps
\begin {align*} \int \cosh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b x^2\right )^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (b^2+\frac {a^2-b^2+2 b (a+b) x^2}{\left (1-x^2\right )^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {b^2 \tanh (c+d x)}{d}+\frac {\operatorname {Subst}\left (\int \frac {a^2-b^2+2 b (a+b) x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {(a+b)^2 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac {b^2 \tanh (c+d x)}{d}+\frac {((a-3 b) (a+b)) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {1}{2} (a-3 b) (a+b) x+\frac {(a+b)^2 \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac {b^2 \tanh (c+d x)}{d}\\ \end {align*}
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Mathematica [A] time = 0.41, size = 54, normalized size = 1.06 \[ \frac {(a-3 b) (a+b) (c+d x)}{2 d}+\frac {(a+b)^2 \sinh (2 (c+d x))}{4 d}+\frac {b^2 \tanh (c+d x)}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 105, normalized size = 2.06 \[ \frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{3} + 4 \, {\left ({\left (a^{2} - 2 \, a b - 3 \, b^{2}\right )} d x - 2 \, b^{2}\right )} \cosh \left (d x + c\right ) + {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, a b + 9 \, b^{2}\right )} \sinh \left (d x + c\right )}{8 \, d \cosh \left (d x + c\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 170, normalized size = 3.33 \[ \frac {4 \, {\left (a^{2} - 2 \, a b - 3 \, b^{2}\right )} d x + {\left (a^{2} e^{\left (2 \, d x + 8 \, c\right )} + 2 \, a b e^{\left (2 \, d x + 8 \, c\right )} + b^{2} e^{\left (2 \, d x + 8 \, c\right )}\right )} e^{\left (-6 \, c\right )} - \frac {{\left (a^{2} e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} + 14 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-2 \, c\right )}}{e^{\left (2 \, d x\right )} + e^{\left (4 \, d x + 2 \, c\right )}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.21, size = 96, normalized size = 1.88 \[ \frac {a^{2} \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+2 a b \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )+b^{2} \left (\frac {\sinh ^{3}\left (d x +c \right )}{2 \cosh \left (d x +c \right )}-\frac {3 d x}{2}-\frac {3 c}{2}+\frac {3 \tanh \left (d x +c \right )}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 140, normalized size = 2.75 \[ \frac {1}{8} \, a^{2} {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{4} \, a b {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{8} \, b^{2} {\left (\frac {12 \, {\left (d x + c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d} - \frac {17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 1}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )}\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.29, size = 77, normalized size = 1.51 \[ \frac {{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a+b\right )}^2}{8\,d}-\frac {2\,b^2}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}\,{\left (a+b\right )}^2}{8\,d}-x\,\left (-\frac {a^2}{2}+a\,b+\frac {3\,b^2}{2}\right ) \]
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 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2} \cosh ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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