Optimal. Leaf size=70 \[ \frac {\sqrt {b} (a+b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {(a+b) \coth (c+d x)}{a^2 d}-\frac {\coth ^3(c+d x)}{3 a d} \]
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Rubi [A] time = 0.09, antiderivative size = 70, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3663, 453, 325, 205} \[ \frac {(a+b) \coth (c+d x)}{a^2 d}+\frac {\sqrt {b} (a+b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{5/2} d}-\frac {\coth ^3(c+d x)}{3 a d} \]
Antiderivative was successfully verified.
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Rule 205
Rule 325
Rule 453
Rule 3663
Rubi steps
\begin {align*} \int \frac {\text {csch}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1-x^2}{x^4 \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {\coth ^3(c+d x)}{3 a d}-\frac {(a+b) \operatorname {Subst}\left (\int \frac {1}{x^2 \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{a d}\\ &=\frac {(a+b) \coth (c+d x)}{a^2 d}-\frac {\coth ^3(c+d x)}{3 a d}+\frac {(b (a+b)) \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{a^2 d}\\ &=\frac {\sqrt {b} (a+b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{5/2} d}+\frac {(a+b) \coth (c+d x)}{a^2 d}-\frac {\coth ^3(c+d x)}{3 a d}\\ \end {align*}
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Mathematica [A] time = 0.32, size = 71, normalized size = 1.01 \[ \frac {3 \sqrt {b} (a+b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )+\sqrt {a} \coth (c+d x) \left (-a \text {csch}^2(c+d x)+2 a+3 b\right )}{3 a^{5/2} d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.62, size = 1628, normalized size = 23.26 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.36, size = 132, normalized size = 1.89 \[ \frac {\frac {3 \, {\left (a b e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right ) e^{\left (-2 \, c\right )}}{\sqrt {a b} a^{2}} + \frac {2 \, {\left (3 \, b e^{\left (4 \, d x + 4 \, c\right )} - 6 \, a e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a + 3 \, b\right )}}{a^{2} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.37, size = 750, normalized size = 10.71 \[ -\frac {\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{24 d a}+\frac {3 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d a}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{2 d \,a^{2}}-\frac {b \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{d \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {b \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{d a \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}-\frac {2 \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right ) b^{2}}{d \sqrt {b \left (a +b \right )}\, a \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}-\frac {b \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{d \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}-\frac {b \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{d a \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}-\frac {2 \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right ) b^{2}}{d \sqrt {b \left (a +b \right )}\, a \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}+\frac {b^{2} \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{d \,a^{2} \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}-\frac {b^{3} \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{d \,a^{2} \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}-\frac {b^{2} \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{d \,a^{2} \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}-\frac {b^{3} \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{d \,a^{2} \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}-\frac {1}{24 d a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}+\frac {3}{8 d a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b}{2 d \,a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.46, size = 134, normalized size = 1.91 \[ \frac {2 \, {\left (6 \, {\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, b e^{\left (-4 \, d x - 4 \, c\right )} - 2 \, a - 3 \, b\right )}}{3 \, {\left (3 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )} - 3 \, a^{2} e^{\left (-4 \, d x - 4 \, c\right )} + a^{2} e^{\left (-6 \, d x - 6 \, c\right )} - a^{2}\right )} d} - \frac {{\left (a b + b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.44, size = 254, normalized size = 3.63 \[ \frac {2\,b}{a^2\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )}-\frac {8}{3\,a\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}-3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}-1\right )}-\frac {4}{a\,d\,\left ({\mathrm {e}}^{4\,c+4\,d\,x}-2\,{\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {\sqrt {-b}\,\ln \left (-\frac {4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{a^2}-\frac {2\,\sqrt {-b}\,\left (a\,d+b\,d+a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^{5/2}\,d}\right )\,\left (a+b\right )}{2\,a^{5/2}\,d}-\frac {\sqrt {-b}\,\ln \left (\frac {2\,\sqrt {-b}\,\left (a\,d+b\,d+a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^{5/2}\,d}-\frac {4\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}}{a^2}\right )\,\left (a+b\right )}{2\,a^{5/2}\,d} \]
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 \[ \int \frac {\operatorname {csch}^{4}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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