Optimal. Leaf size=84 \[ \frac {\sinh (c+d x) \cosh (c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 \sqrt {a} d (a-b)^{3/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 84, 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 = {3173, 12, 3181, 208} \[ \frac {\sinh (c+d x) \cosh (c+d x)}{2 d (a-b) \left (a+b \sinh ^2(c+d x)\right )}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 \sqrt {a} d (a-b)^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 208
Rule 3173
Rule 3181
Rubi steps
\begin {align*} \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a-b) d \left (a+b \sinh ^2(c+d x)\right )}-\frac {\int \frac {a}{a+b \sinh ^2(c+d x)} \, dx}{2 a (a-b)}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a-b) d \left (a+b \sinh ^2(c+d x)\right )}-\frac {\int \frac {1}{a+b \sinh ^2(c+d x)} \, dx}{2 (a-b)}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a-b) d \left (a+b \sinh ^2(c+d x)\right )}-\frac {\operatorname {Subst}\left (\int \frac {1}{a-(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{2 (a-b) d}\\ &=-\frac {\tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 \sqrt {a} (a-b)^{3/2} d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a-b) d \left (a+b \sinh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.44, size = 81, normalized size = 0.96 \[ \frac {\frac {\sinh (2 (c+d x))}{(a-b) (2 a+b \cosh (2 (c+d x))-b)}-\frac {\tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a-b)^{3/2}}}{2 d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.53, size = 1523, normalized size = 18.13 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 2.12, size = 135, normalized size = 1.61 \[ -\frac {\frac {\arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{\sqrt {-a^{2} + a b} {\left (a - b\right )}} + \frac {2 \, {\left (2 \, a e^{\left (2 \, d x + 2 \, c\right )} - b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}}{{\left (a b - b^{2}\right )} {\left (b e^{\left (4 \, d x + 4 \, c\right )} + 4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + b\right )}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.07, size = 428, normalized size = 5.10 \[ \frac {\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \right ) \left (a -b \right )}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d \left (\left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a +4 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b +a \right ) \left (a -b \right )}+\frac {\arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 d \left (a -b \right ) \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}+\frac {\arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right ) b}{2 d \left (a -b \right ) \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 d \left (a -b \right ) \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}+\frac {\arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right ) b}{2 d \left (a -b \right ) \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}} \]
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 \[ \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \]
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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