Optimal. Leaf size=47 \[ \frac {\left (a^2-b^2\right ) \tanh (c+d x)}{d}-\frac {(a-b)^2 \tanh ^3(c+d x)}{3 d}+b^2 x \]
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Rubi [A] time = 0.06, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3191, 390, 206} \[ \frac {\left (a^2-b^2\right ) \tanh (c+d x)}{d}-\frac {(a-b)^2 \tanh ^3(c+d x)}{3 d}+b^2 x \]
Antiderivative was successfully verified.
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Rule 206
Rule 390
Rule 3191
Rubi steps
\begin {align*} \int \text {sech}^4(c+d x) \left (a+b \sinh ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a-(a-b) x^2\right )^2}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (a^2-b^2-(a-b)^2 x^2+\frac {b^2}{1-x^2}\right ) \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\left (a^2-b^2\right ) \tanh (c+d x)}{d}-\frac {(a-b)^2 \tanh ^3(c+d x)}{3 d}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=b^2 x+\frac {\left (a^2-b^2\right ) \tanh (c+d x)}{d}-\frac {(a-b)^2 \tanh ^3(c+d x)}{3 d}\\ \end {align*}
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Mathematica [A] time = 0.35, size = 57, normalized size = 1.21 \[ \frac {(a-b) \tanh (c+d x) \text {sech}^2(c+d x) ((a+2 b) \cosh (2 (c+d x))+2 a+b)+3 b^2 (c+d x)}{3 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 1.74, size = 200, normalized size = 4.26 \[ \frac {{\left (3 \, b^{2} d x - 2 \, a^{2} - 2 \, a b + 4 \, b^{2}\right )} \cosh \left (d x + c\right )^{3} + 3 \, {\left (3 \, b^{2} d x - 2 \, a^{2} - 2 \, a b + 4 \, b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} + a b - 2 \, b^{2}\right )} \sinh \left (d x + c\right )^{3} + 3 \, {\left (3 \, b^{2} d x - 2 \, a^{2} - 2 \, a b + 4 \, b^{2}\right )} \cosh \left (d x + c\right ) + 6 \, {\left ({\left (a^{2} + a b - 2 \, b^{2}\right )} \cosh \left (d x + c\right )^{2} + a^{2} - a b\right )} \sinh \left (d x + c\right )}{3 \, {\left (d \cosh \left (d x + c\right )^{3} + 3 \, d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{2} + 3 \, d \cosh \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.19, size = 98, normalized size = 2.09 \[ \frac {3 \, {\left (d x + c\right )} b^{2} - \frac {4 \, {\left (3 \, a b e^{\left (4 \, d x + 4 \, c\right )} - 3 \, b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} + a^{2} + a b - 2 \, b^{2}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.11, size = 96, normalized size = 2.04 \[ \frac {a^{2} \left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )+2 a b \left (-\frac {\sinh \left (d x +c \right )}{2 \cosh \left (d x +c \right )^{3}}+\frac {\left (\frac {2}{3}+\frac {\mathrm {sech}\left (d x +c \right )^{2}}{3}\right ) \tanh \left (d x +c \right )}{2}\right )+b^{2} \left (d x +c -\tanh \left (d x +c \right )-\frac {\left (\tanh ^{3}\left (d x +c \right )\right )}{3}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.41, size = 267, normalized size = 5.68 \[ \frac {1}{3} \, b^{2} {\left (3 \, x + \frac {3 \, c}{d} - \frac {4 \, {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + 2\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + \frac {4}{3} \, a^{2} {\left (\frac {3 \, e^{\left (-2 \, d x - 2 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + \frac {4}{3} \, a b {\left (\frac {3 \, e^{\left (-4 \, d x - 4 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {1}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.84, size = 194, normalized size = 4.13 \[ b^2\,x-\frac {\frac {4\,\left (a\,b-b^2\right )}{3\,d}-\frac {8\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b-a^2\right )}{3\,d}+\frac {4\,{\mathrm {e}}^{4\,c+4\,d\,x}\,\left (a\,b-b^2\right )}{3\,d}}{3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1}+\frac {\frac {4\,\left (a\,b-a^2\right )}{3\,d}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\left (a\,b-b^2\right )}{3\,d}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {4\,\left (a\,b-b^2\right )}{3\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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