Optimal. Leaf size=139 \[ -\frac {(4 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} (a-b)^{5/2} d}+\frac {\cosh (c+d x) \sinh (c+d x)}{4 (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac {(2 a+b) \cosh (c+d x) \sinh (c+d x)}{8 a (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )} \]
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Rubi [A]
time = 0.11, antiderivative size = 139, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3252, 12, 3260,
214} \begin {gather*} -\frac {(4 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} d (a-b)^{5/2}}+\frac {(2 a+b) \sinh (c+d x) \cosh (c+d x)}{8 a d (a-b)^2 \left (a+b \sinh ^2(c+d x)\right )}+\frac {\sinh (c+d x) \cosh (c+d x)}{4 d (a-b) \left (a+b \sinh ^2(c+d x)\right )^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 214
Rule 3252
Rule 3260
Rubi steps
\begin {align*} \int \frac {\sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^3} \, dx &=\frac {\cosh (c+d x) \sinh (c+d x)}{4 (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}-\frac {\int \frac {a-2 a \sinh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx}{4 a (a-b)}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{4 (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac {(2 a+b) \cosh (c+d x) \sinh (c+d x)}{8 a (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )}-\frac {\int \frac {a (4 a-b)}{a+b \sinh ^2(c+d x)} \, dx}{8 a^2 (a-b)^2}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{4 (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac {(2 a+b) \cosh (c+d x) \sinh (c+d x)}{8 a (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )}-\frac {(4 a-b) \int \frac {1}{a+b \sinh ^2(c+d x)} \, dx}{8 a (a-b)^2}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{4 (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac {(2 a+b) \cosh (c+d x) \sinh (c+d x)}{8 a (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )}-\frac {(4 a-b) \text {Subst}\left (\int \frac {1}{a-(a-b) x^2} \, dx,x,\tanh (c+d x)\right )}{8 a (a-b)^2 d}\\ &=-\frac {(4 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{8 a^{3/2} (a-b)^{5/2} d}+\frac {\cosh (c+d x) \sinh (c+d x)}{4 (a-b) d \left (a+b \sinh ^2(c+d x)\right )^2}+\frac {(2 a+b) \cosh (c+d x) \sinh (c+d x)}{8 a (a-b)^2 d \left (a+b \sinh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.90, size = 121, normalized size = 0.87 \begin {gather*} \frac {-\frac {(4 a-b) \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{a^{3/2} (a-b)^{5/2}}+\frac {\left (8 a^2-4 a b-b^2+b (2 a+b) \cosh (2 (c+d x))\right ) \sinh (2 (c+d x))}{a (a-b)^2 (2 a-b+b \cosh (2 (c+d x)))^2}}{8 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(400\) vs.
\(2(125)=250\).
time = 1.32, size = 401, normalized size = 2.88
method | result | size |
derivativedivides | \(\frac {-\frac {8 \left (-\frac {\left (4 a -b \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (4 a^{2}-9 a b -4 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (4 a^{2}-9 a b -4 b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (4 a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {\left (4 a -b \right ) \left (\frac {\left (-\sqrt {-b \left (a -b \right )}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 \sqrt {-b \left (a -b \right )}\, a \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (-\sqrt {-b \left (a -b \right )}+b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(401\) |
default | \(\frac {-\frac {8 \left (-\frac {\left (4 a -b \right ) \left (\tanh ^{7}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (4 a^{2}-9 a b -4 b^{2}\right ) \left (\tanh ^{5}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 a \left (a^{2}-2 a b +b^{2}\right )}+\frac {\left (4 a^{2}-9 a b -4 b^{2}\right ) \left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{32 a \left (a^{2}-2 a b +b^{2}\right )}-\frac {\left (4 a -b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{32 \left (a^{2}-2 a b +b^{2}\right )}\right )}{\left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )^{2}}-\frac {\left (4 a -b \right ) \left (\frac {\left (-\sqrt {-b \left (a -b \right )}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 \sqrt {-b \left (a -b \right )}\, a \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (-\sqrt {-b \left (a -b \right )}+b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{4 \left (a^{2}-2 a b +b^{2}\right )}}{d}\) | \(401\) |
risch | \(-\frac {4 a \,b^{2} {\mathrm e}^{6 d x +6 c}-b^{3} {\mathrm e}^{6 d x +6 c}+16 a^{3} {\mathrm e}^{4 d x +4 c}-8 a^{2} b \,{\mathrm e}^{4 d x +4 c}-2 a \,b^{2} {\mathrm e}^{4 d x +4 c}+3 b^{3} {\mathrm e}^{4 d x +4 c}+16 a^{2} b \,{\mathrm e}^{2 d x +2 c}-4 a \,b^{2} {\mathrm e}^{2 d x +2 c}-3 b^{3} {\mathrm e}^{2 d x +2 c}+2 a \,b^{2}+b^{3}}{4 b \left (a -b \right )^{2} d \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )^{2} a}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{4 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right ) b}{16 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d a}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{4 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right ) b}{16 \sqrt {a^{2}-a b}\, \left (a -b \right )^{2} d a}\) | \(540\) |
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 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 2632 vs.
\(2 (125) = 250\).
time = 0.46, size = 5519, normalized size = 39.71 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 277 vs.
\(2 (125) = 250\).
time = 1.85, size = 277, normalized size = 1.99 \begin {gather*} -\frac {\frac {{\left (4 \, a - b\right )} \arctan \left (\frac {b e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a - b}{2 \, \sqrt {-a^{2} + a b}}\right )}{{\left (a^{3} - 2 \, a^{2} b + a b^{2}\right )} \sqrt {-a^{2} + a b}} + \frac {2 \, {\left (4 \, a b^{2} e^{\left (6 \, d x + 6 \, c\right )} - b^{3} e^{\left (6 \, d x + 6 \, c\right )} + 16 \, a^{3} e^{\left (4 \, d x + 4 \, c\right )} - 8 \, a^{2} b e^{\left (4 \, d x + 4 \, c\right )} - 2 \, a b^{2} e^{\left (4 \, d x + 4 \, c\right )} + 3 \, b^{3} e^{\left (4 \, d x + 4 \, c\right )} + 16 \, a^{2} b e^{\left (2 \, d x + 2 \, c\right )} - 4 \, a b^{2} e^{\left (2 \, d x + 2 \, c\right )} - 3 \, b^{3} e^{\left (2 \, d x + 2 \, c\right )} + 2 \, a b^{2} + b^{3}\right )}}{{\left (a^{3} b - 2 \, a^{2} b^{2} + a b^{3}\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}}}{8 \, d} \end {gather*}
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 \begin {gather*} \int \frac {{\mathrm {sinh}\left (c+d\,x\right )}^2}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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