Integrand size = 23, antiderivative size = 118 \[ \int \frac {\sinh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {\left (3 a^2-6 a b-b^2\right ) x}{8 (a+b)^3}+\frac {a^{3/2} \sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{(a+b)^3 d}-\frac {(5 a+b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d} \]
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Time = 0.12 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3744, 481, 541, 536, 212, 211} \[ \int \frac {\sinh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {a^{3/2} \sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{d (a+b)^3}+\frac {x \left (3 a^2-6 a b-b^2\right )}{8 (a+b)^3}+\frac {\sinh (c+d x) \cosh ^3(c+d x)}{4 d (a+b)}-\frac {(5 a+b) \sinh (c+d x) \cosh (c+d x)}{8 d (a+b)^2} \]
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Rule 211
Rule 212
Rule 481
Rule 536
Rule 541
Rule 3744
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^3 \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d} \\ & = \frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d}-\frac {\text {Subst}\left (\int \frac {a+(4 a+b) x^2}{\left (1-x^2\right )^2 \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{4 (a+b) d} \\ & = -\frac {(5 a+b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d}-\frac {\text {Subst}\left (\int \frac {-a (3 a-b)+b (5 a+b) x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^2 d} \\ & = -\frac {(5 a+b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d}+\frac {\left (a^2 b\right ) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{(a+b)^3 d}+\frac {\left (3 a^2-6 a b-b^2\right ) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{8 (a+b)^3 d} \\ & = \frac {\left (3 a^2-6 a b-b^2\right ) x}{8 (a+b)^3}+\frac {a^{3/2} \sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{(a+b)^3 d}-\frac {(5 a+b) \cosh (c+d x) \sinh (c+d x)}{8 (a+b)^2 d}+\frac {\cosh ^3(c+d x) \sinh (c+d x)}{4 (a+b) d} \\ \end{align*}
Time = 0.89 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.79 \[ \int \frac {\sinh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {4 \left (3 a^2-6 a b-b^2\right ) (c+d x)+32 a^{3/2} \sqrt {b} \arctan \left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )-8 a (a+b) \sinh (2 (c+d x))+(a+b)^2 \sinh (4 (c+d x))}{32 (a+b)^3 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(214\) vs. \(2(104)=208\).
Time = 11.14 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.82
method | result | size |
risch | \(\frac {3 a^{2} x}{8 \left (a +b \right )^{3}}-\frac {3 a x b}{4 \left (a +b \right )^{3}}-\frac {x \,b^{2}}{8 \left (a +b \right )^{3}}+\frac {{\mathrm e}^{4 d x +4 c}}{64 d \left (a +b \right )}-\frac {a \,{\mathrm e}^{2 d x +2 c}}{8 \left (a +b \right )^{2} d}+\frac {a \,{\mathrm e}^{-2 d x -2 c}}{8 \left (a^{2}+2 a b +b^{2}\right ) d}-\frac {{\mathrm e}^{-4 d x -4 c}}{64 d \left (a +b \right )}+\frac {\sqrt {-a b}\, a \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {-a b}+a -b}{a +b}\right )}{2 \left (a +b \right )^{3} d}-\frac {\sqrt {-a b}\, a \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {-a b}-a +b}{a +b}\right )}{2 \left (a +b \right )^{3} d}\) | \(215\) |
derivativedivides | \(\frac {\frac {8}{\left (32 a +32 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {32}{\left (64 a +64 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a -3 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3 a -b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}+6 a b +b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a +b \right )^{3}}-\frac {8}{\left (32 a +32 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {32}{\left (64 a +64 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {-a +3 b}{8 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {3 a -b}{8 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (3 a^{2}-6 a b -b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{3}}-\frac {2 a^{3} b \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{\left (a +b \right )^{3}}}{d}\) | \(434\) |
default | \(\frac {\frac {8}{\left (32 a +32 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{4}}+\frac {32}{\left (64 a +64 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {a -3 b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {3 a -b}{8 \left (a +b \right )^{2} \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (-3 a^{2}+6 a b +b^{2}\right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{8 \left (a +b \right )^{3}}-\frac {8}{\left (32 a +32 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{4}}+\frac {32}{\left (64 a +64 b \right ) \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{3}}-\frac {-a +3 b}{8 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )^{2}}-\frac {3 a -b}{8 \left (a +b \right )^{2} \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}+\frac {\left (3 a^{2}-6 a b -b^{2}\right ) \ln \left (1+\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{8 \left (a +b \right )^{3}}-\frac {2 a^{3} b \left (\frac {\left (a +\sqrt {\left (a +b \right ) b}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}+a +2 b \right ) a}}-\frac {\left (-a +\sqrt {\left (a +b \right ) b}-b \right ) \operatorname {arctanh}\left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{2 a \sqrt {\left (a +b \right ) b}\, \sqrt {\left (2 \sqrt {\left (a +b \right ) b}-a -2 b \right ) a}}\right )}{\left (a +b \right )^{3}}}{d}\) | \(434\) |
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Leaf count of result is larger than twice the leaf count of optimal. 860 vs. \(2 (104) = 208\).
Time = 0.30 (sec) , antiderivative size = 2024, normalized size of antiderivative = 17.15 \[ \int \frac {\sinh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sinh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int \frac {\sinh ^{4}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 514 vs. \(2 (104) = 208\).
Time = 0.34 (sec) , antiderivative size = 514, normalized size of antiderivative = 4.36 \[ \int \frac {\sinh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=-\frac {{\left (a b - b^{2}\right )} {\left (d x + c\right )}}{2 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} d} + \frac {{\left (8 \, b e^{\left (-2 \, d x - 2 \, c\right )} + a + b\right )} e^{\left (4 \, d x + 4 \, c\right )}}{64 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} - \frac {b \log \left ({\left (a + b\right )} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a - b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac {b \log \left (2 \, {\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{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 )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b} d} - \frac {{\left (a^{2} b - 6 \, a b^{2} + b^{3}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{8 \, {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + b^{3}\right )} \sqrt {a b} 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 )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b} d} - \frac {3 \, b \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{8 \, \sqrt {a b} {\left (a + b\right )} d} - \frac {8 \, b e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )}}{64 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac {3 \, {\left (d x + c\right )}}{8 \, {\left (a + b\right )} d} - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{8 \, {\left (a + b\right )} d} + \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, {\left (a + b\right )} d} \]
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\[ \int \frac {\sinh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\int { \frac {\sinh \left (d x + c\right )^{4}}{b \tanh \left (d x + c\right )^{2} + a} \,d x } \]
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Time = 2.32 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.12 \[ \int \frac {\sinh ^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx=\frac {{\mathrm {e}}^{4\,c+4\,d\,x}}{64\,d\,\left (a+b\right )}-\frac {{\mathrm {e}}^{-4\,c-4\,d\,x}}{64\,d\,\left (a+b\right )}-\frac {x\,\left (-3\,a^2+6\,a\,b+b^2\right )}{8\,{\left (a+b\right )}^3}+\frac {a\,{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d\,{\left (a+b\right )}^2}-\frac {a\,{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d\,{\left (a+b\right )}^2}+\frac {{\left (-a\right )}^{3/2}\,\sqrt {b}\,\ln \left ({\left (-a\right )}^{3/2}\,b^{3/2}\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )-2\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\left (-a\right )}^{5/2}\,\sqrt {b}\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\right )}{2\,d\,{\left (a+b\right )}^3}-\frac {{\left (-a\right )}^{3/2}\,\sqrt {b}\,\ln \left (2\,a^2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\left (-a\right )}^{3/2}\,b^{3/2}\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )+{\left (-a\right )}^{5/2}\,\sqrt {b}\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )\right )}{2\,d\,{\left (a+b\right )}^3} \]
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