Optimal. Leaf size=100 \[ -\frac {\sqrt {b} (3 a-2 b) \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{2 a^2 d (a-b)^{3/2}}+\frac {x}{a^2}+\frac {b \coth (c+d x)}{2 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )} \]
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Rubi [A] time = 0.13, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.357, Rules used = {4128, 414, 522, 206, 205} \[ -\frac {\sqrt {b} (3 a-2 b) \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{2 a^2 d (a-b)^{3/2}}+\frac {x}{a^2}+\frac {b \coth (c+d x)}{2 a d (a-b) \left (a+b \coth ^2(c+d x)-b\right )} \]
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 414
Rule 522
Rule 4128
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \text {csch}^2(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\coth (c+d x)\right )}{d}\\ &=\frac {b \coth (c+d x)}{2 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {-2 a+b+b x^2}{\left (1-x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\coth (c+d x)\right )}{2 a (a-b) d}\\ &=\frac {b \coth (c+d x)}{2 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\coth (c+d x)\right )}{a^2 d}+\frac {((3 a-2 b) b) \operatorname {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\coth (c+d x)\right )}{2 a^2 (a-b) d}\\ &=\frac {x}{a^2}-\frac {(3 a-2 b) \sqrt {b} \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )}{2 a^2 (a-b)^{3/2} d}+\frac {b \coth (c+d x)}{2 a (a-b) d \left (a-b+b \coth ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.71, size = 199, normalized size = 1.99 \[ \frac {\text {csch}^4(c+d x) (a \cosh (2 (c+d x))-a+2 b) \left (a b \sqrt {a-b} \sinh (2 (c+d x))+(a-2 b) \left (\sqrt {b} (3 a-2 b) \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )-2 (a-b)^{3/2} (c+d x)\right )+a \cosh (2 (c+d x)) \left (2 (a-b)^{3/2} (c+d x)+\sqrt {b} (2 b-3 a) \tan ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {b}}\right )\right )\right )}{8 a^2 d (a-b)^{3/2} \left (a+b \text {csch}^2(c+d x)\right )^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.48, size = 1747, normalized size = 17.47 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.44, size = 166, normalized size = 1.66 \[ -\frac {\frac {{\left (3 \, a b - 2 \, b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} - a + 2 \, b}{2 \, \sqrt {a b - b^{2}}}\right )}{{\left (a^{3} - a^{2} b\right )} \sqrt {a b - b^{2}}} - \frac {2 \, {\left (a b e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b^{2} e^{\left (2 \, d x + 2 \, c\right )} - a b\right )}}{{\left (a^{3} - a^{2} b\right )} {\left (a e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} + a\right )}} - \frac {2 \, {\left (d x + c\right )}}{a^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.29, size = 778, normalized size = 7.78 \[ -\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d \,a^{2}}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \,a^{2}}+\frac {b \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a \left (b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +b \right ) \left (a -b \right )}+\frac {b \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d a \left (b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +b \right ) \left (a -b \right )}+\frac {3 b \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{2 d a \left (a -b \right ) \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}+\frac {3 b \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{2 d \left (a -b \right ) \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}-\frac {3 b \arctanh \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{2 d a \left (a -b \right ) \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}+\frac {3 b \arctanh \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{2 d \left (a -b \right ) \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}-\frac {b^{2} \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{d \,a^{2} \left (a -b \right ) \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}-\frac {b^{2} \arctan \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}\right )}{d a \left (a -b \right ) \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}+2 a -b \right ) b}}+\frac {b^{2} \arctanh \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{d \,a^{2} \left (a -b \right ) \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}-\frac {b^{2} \arctanh \left (\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b}{\sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}}\right )}{d a \left (a -b \right ) \sqrt {a \left (a -b \right )}\, \sqrt {\left (2 \sqrt {a \left (a -b \right )}-2 a +b \right ) b}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{{\left (a+\frac {b}{{\mathrm {sinh}\left (c+d\,x\right )}^2}\right )}^2} \,d x \]
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 \[ \int \frac {1}{\left (a + b \operatorname {csch}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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