Optimal. Leaf size=54 \[ \frac {(a+3 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {(a+b) \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {b \text {sech}(c+d x)}{d} \]
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Rubi [A] time = 0.07, antiderivative size = 54, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {4133, 456, 453, 206} \[ \frac {(a+3 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {(a+b) \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {b \text {sech}(c+d x)}{d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 453
Rule 456
Rule 4133
Rubi steps
\begin {align*} \int \text {csch}^3(c+d x) \left (a+b \text {sech}^2(c+d x)\right ) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {b+a x^2}{x^2 \left (1-x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=-\frac {(a+b) \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {\operatorname {Subst}\left (\int \frac {-2 b-(a+b) x^2}{x^2 \left (1-x^2\right )} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=-\frac {(a+b) \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {b \text {sech}(c+d x)}{d}+\frac {(a+3 b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (c+d x)\right )}{2 d}\\ &=\frac {(a+3 b) \tanh ^{-1}(\cosh (c+d x))}{2 d}-\frac {(a+b) \coth (c+d x) \text {csch}(c+d x)}{2 d}-\frac {b \text {sech}(c+d x)}{d}\\ \end {align*}
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Mathematica [B] time = 0.05, size = 131, normalized size = 2.43 \[ -\frac {a \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {a \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d}-\frac {b \text {csch}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {b \text {sech}^2\left (\frac {1}{2} (c+d x)\right )}{8 d}-\frac {b \text {sech}(c+d x)}{d}-\frac {3 b \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.43, size = 924, normalized size = 17.11 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.15, size = 142, normalized size = 2.63 \[ \frac {{\left (a + 3 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} + 2\right ) - {\left (a + 3 \, b\right )} \log \left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )} - 2\right ) - \frac {4 \, {\left (a {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} + 3 \, b {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{2} - 8 \, b\right )}}{{\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}^{3} - 4 \, e^{\left (d x + c\right )} - 4 \, e^{\left (-d x - c\right )}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.38, size = 70, normalized size = 1.30 \[ \frac {a \left (-\frac {\mathrm {csch}\left (d x +c \right ) \coth \left (d x +c \right )}{2}+\arctanh \left ({\mathrm e}^{d x +c}\right )\right )+b \left (-\frac {1}{2 \sinh \left (d x +c \right )^{2} \cosh \left (d x +c \right )}-\frac {3}{2 \cosh \left (d x +c \right )}+3 \arctanh \left ({\mathrm e}^{d x +c}\right )\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.34, size = 198, normalized size = 3.67 \[ \frac {1}{2} \, b {\left (\frac {3 \, \log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {3 \, \log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (3 \, e^{\left (-d x - c\right )} - 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + 3 \, e^{\left (-5 \, d x - 5 \, c\right )}\right )}}{d {\left (e^{\left (-2 \, d x - 2 \, c\right )} + e^{\left (-4 \, d x - 4 \, c\right )} - e^{\left (-6 \, d x - 6 \, c\right )} - 1\right )}}\right )} + \frac {1}{2} \, a {\left (\frac {\log \left (e^{\left (-d x - c\right )} + 1\right )}{d} - \frac {\log \left (e^{\left (-d x - c\right )} - 1\right )}{d} + \frac {2 \, {\left (e^{\left (-d x - c\right )} + e^{\left (-3 \, d x - 3 \, c\right )}\right )}}{d {\left (2 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} - 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.17, size = 160, normalized size = 2.96 \[ \frac {\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (a\,\sqrt {-d^2}+3\,b\,\sqrt {-d^2}\right )}{d\,\sqrt {a^2+6\,a\,b+9\,b^2}}\right )\,\sqrt {a^2+6\,a\,b+9\,b^2}}{\sqrt {-d^2}}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (a+b\right )}{d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,b\,{\mathrm {e}}^{c+d\,x}}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right ) \operatorname {csch}^{3}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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