Optimal. Leaf size=33 \[ \frac {(a+b) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {1}{2} x (a-b) \]
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Rubi [A] time = 0.04, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3675, 385, 206} \[ \frac {(a+b) \sinh (c+d x) \cosh (c+d x)}{2 d}+\frac {1}{2} x (a-b) \]
Antiderivative was successfully verified.
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Rule 206
Rule 385
Rule 3675
Rubi steps
\begin {align*} \int \cosh ^2(c+d x) \left (a+b \tanh ^2(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b x^2}{\left (1-x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {(a+b) \cosh (c+d x) \sinh (c+d x)}{2 d}+\frac {(a-b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 d}\\ &=\frac {1}{2} (a-b) x+\frac {(a+b) \cosh (c+d x) \sinh (c+d x)}{2 d}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 32, normalized size = 0.97 \[ \frac {2 (a-b) (c+d x)+(a+b) \sinh (2 (c+d x))}{4 d} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 30, normalized size = 0.91 \[ \frac {{\left (a - b\right )} d x + {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.14, size = 81, normalized size = 2.45 \[ \frac {4 \, {\left (a - b\right )} d x - {\left (2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a e^{\left (2 \, d x + 4 \, c\right )} + b e^{\left (2 \, d x + 4 \, c\right )}\right )} e^{\left (-2 \, c\right )}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.21, size = 54, normalized size = 1.64 \[ \frac {a \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}+\frac {d x}{2}+\frac {c}{2}\right )+b \left (\frac {\cosh \left (d x +c \right ) \sinh \left (d x +c \right )}{2}-\frac {d x}{2}-\frac {c}{2}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.32, size = 69, normalized size = 2.09 \[ \frac {1}{8} \, a {\left (4 \, x + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} - \frac {1}{8} \, b {\left (4 \, x - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.15, size = 27, normalized size = 0.82 \[ x\,\left (\frac {a}{2}-\frac {b}{2}\right )+\frac {\mathrm {sinh}\left (2\,c+2\,d\,x\right )\,\left (a+b\right )}{4\,d} \]
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 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right ) \cosh ^{2}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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