Integrand size = 21, antiderivative size = 99 \[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {3 b \left (4 (a+b)^2+(2 a+b)^2\right ) \arctan (\sinh (c+d x))}{8 d}+\frac {(a+b)^3 \sinh (c+d x)}{d}+\frac {3 b^2 (4 a+3 b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}-\frac {b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 d} \]
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Time = 0.09 (sec) , antiderivative size = 99, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {3757, 398, 1171, 393, 209} \[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=-\frac {3 b \left (4 (a+b)^2+(2 a+b)^2\right ) \arctan (\sinh (c+d x))}{8 d}+\frac {3 b^2 (4 a+3 b) \tanh (c+d x) \text {sech}(c+d x)}{8 d}+\frac {(a+b)^3 \sinh (c+d x)}{d}-\frac {b^3 \tanh (c+d x) \text {sech}^3(c+d x)}{4 d} \]
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Rule 209
Rule 393
Rule 398
Rule 1171
Rule 3757
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (a+(a+b) x^2\right )^3}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d} \\ & = \frac {\text {Subst}\left (\int \left ((a+b)^3-\frac {b \left (3 a^2+3 a b+b^2\right )+3 b (a+b) (2 a+b) x^2+3 b (a+b)^2 x^4}{\left (1+x^2\right )^3}\right ) \, dx,x,\sinh (c+d x)\right )}{d} \\ & = \frac {(a+b)^3 \sinh (c+d x)}{d}-\frac {\text {Subst}\left (\int \frac {b \left (3 a^2+3 a b+b^2\right )+3 b (a+b) (2 a+b) x^2+3 b (a+b)^2 x^4}{\left (1+x^2\right )^3} \, dx,x,\sinh (c+d x)\right )}{d} \\ & = \frac {(a+b)^3 \sinh (c+d x)}{d}-\frac {b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 d}+\frac {\text {Subst}\left (\int \frac {-3 b (2 a+b)^2-12 b (a+b)^2 x^2}{\left (1+x^2\right )^2} \, dx,x,\sinh (c+d x)\right )}{4 d} \\ & = \frac {(a+b)^3 \sinh (c+d x)}{d}+\frac {3 b^2 (4 a+3 b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}-\frac {b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 d}-\frac {\left (3 b \left (4 (a+b)^2+(2 a+b)^2\right )\right ) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sinh (c+d x)\right )}{8 d} \\ & = -\frac {3 b \left (4 (a+b)^2+(2 a+b)^2\right ) \arctan (\sinh (c+d x))}{8 d}+\frac {(a+b)^3 \sinh (c+d x)}{d}+\frac {3 b^2 (4 a+3 b) \text {sech}(c+d x) \tanh (c+d x)}{8 d}-\frac {b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{4 d} \\ \end{align*}
Time = 0.44 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.90 \[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {-3 b \left (8 a^2+12 a b+5 b^2\right ) \arctan (\sinh (c+d x))+8 (a+b)^3 \sinh (c+d x)+3 b^2 (4 a+3 b) \text {sech}(c+d x) \tanh (c+d x)-2 b^3 \text {sech}^3(c+d x) \tanh (c+d x)}{8 d} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(192\) vs. \(2(93)=186\).
Time = 2.81 (sec) , antiderivative size = 193, normalized size of antiderivative = 1.95
method | result | size |
derivativedivides | \(\frac {\sinh \left (d x +c \right ) a^{3}+3 a^{2} b \left (\sinh \left (d x +c \right )-2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}-3 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{5}}{\cosh \left (d x +c \right )^{4}}+\frac {5 \sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{4}}+\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{4}}-5 \left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )-\frac {15 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )}{d}\) | \(193\) |
default | \(\frac {\sinh \left (d x +c \right ) a^{3}+3 a^{2} b \left (\sinh \left (d x +c \right )-2 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+3 a \,b^{2} \left (\frac {\sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{2}}+\frac {3 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{2}}-\frac {3 \,\operatorname {sech}\left (d x +c \right ) \tanh \left (d x +c \right )}{2}-3 \arctan \left ({\mathrm e}^{d x +c}\right )\right )+b^{3} \left (\frac {\sinh \left (d x +c \right )^{5}}{\cosh \left (d x +c \right )^{4}}+\frac {5 \sinh \left (d x +c \right )^{3}}{\cosh \left (d x +c \right )^{4}}+\frac {5 \sinh \left (d x +c \right )}{\cosh \left (d x +c \right )^{4}}-5 \left (\frac {\operatorname {sech}\left (d x +c \right )^{3}}{4}+\frac {3 \,\operatorname {sech}\left (d x +c \right )}{8}\right ) \tanh \left (d x +c \right )-\frac {15 \arctan \left ({\mathrm e}^{d x +c}\right )}{4}\right )}{d}\) | \(193\) |
risch | \(\frac {{\mathrm e}^{d x +c} a^{3}}{2 d}+\frac {3 \,{\mathrm e}^{d x +c} a^{2} b}{2 d}+\frac {3 \,{\mathrm e}^{d x +c} a \,b^{2}}{2 d}+\frac {b^{3} {\mathrm e}^{d x +c}}{2 d}-\frac {{\mathrm e}^{-d x -c} a^{3}}{2 d}-\frac {3 \,{\mathrm e}^{-d x -c} a^{2} b}{2 d}-\frac {3 \,{\mathrm e}^{-d x -c} a \,b^{2}}{2 d}-\frac {{\mathrm e}^{-d x -c} b^{3}}{2 d}+\frac {b^{2} {\mathrm e}^{d x +c} \left (12 a \,{\mathrm e}^{6 d x +6 c}+9 b \,{\mathrm e}^{6 d x +6 c}+12 a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}-12 \,{\mathrm e}^{2 d x +2 c} a -b \,{\mathrm e}^{2 d x +2 c}-12 a -9 b \right )}{4 d \left ({\mathrm e}^{2 d x +2 c}+1\right )^{4}}+\frac {3 i b \ln \left ({\mathrm e}^{d x +c}-i\right ) a^{2}}{d}+\frac {9 i b^{2} \ln \left ({\mathrm e}^{d x +c}-i\right ) a}{2 d}+\frac {15 i b^{3} \ln \left ({\mathrm e}^{d x +c}-i\right )}{8 d}-\frac {3 i b \ln \left ({\mathrm e}^{d x +c}+i\right ) a^{2}}{d}-\frac {9 i b^{2} \ln \left ({\mathrm e}^{d x +c}+i\right ) a}{2 d}-\frac {15 i b^{3} \ln \left ({\mathrm e}^{d x +c}+i\right )}{8 d}\) | \(353\) |
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Leaf count of result is larger than twice the leaf count of optimal. 2411 vs. \(2 (93) = 186\).
Time = 0.28 (sec) , antiderivative size = 2411, normalized size of antiderivative = 24.35 \[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\text {Too large to display} \]
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\[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{3} \cosh {\left (c + d x \right )}\, dx \]
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Leaf count of result is larger than twice the leaf count of optimal. 295 vs. \(2 (93) = 186\).
Time = 0.29 (sec) , antiderivative size = 295, normalized size of antiderivative = 2.98 \[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {1}{4} \, b^{3} {\left (\frac {15 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {2 \, e^{\left (-d x - c\right )}}{d} + \frac {17 \, e^{\left (-2 \, d x - 2 \, c\right )} + 13 \, e^{\left (-4 \, d x - 4 \, c\right )} + 7 \, e^{\left (-6 \, d x - 6 \, c\right )} - 7 \, e^{\left (-8 \, d x - 8 \, c\right )} + 2}{d {\left (e^{\left (-d x - c\right )} + 4 \, e^{\left (-3 \, d x - 3 \, c\right )} + 6 \, e^{\left (-5 \, d x - 5 \, c\right )} + 4 \, e^{\left (-7 \, d x - 7 \, c\right )} + e^{\left (-9 \, d x - 9 \, c\right )}\right )}}\right )} + \frac {3}{2} \, a b^{2} {\left (\frac {6 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} - \frac {e^{\left (-d x - c\right )}}{d} + \frac {4 \, e^{\left (-2 \, d x - 2 \, c\right )} - e^{\left (-4 \, d x - 4 \, c\right )} + 1}{d {\left (e^{\left (-d x - c\right )} + 2 \, e^{\left (-3 \, d x - 3 \, c\right )} + e^{\left (-5 \, d x - 5 \, c\right )}\right )}}\right )} + \frac {3}{2} \, a^{2} b {\left (\frac {4 \, \arctan \left (e^{\left (-d x - c\right )}\right )}{d} + \frac {e^{\left (d x + c\right )}}{d} - \frac {e^{\left (-d x - c\right )}}{d}\right )} + \frac {a^{3} \sinh \left (d x + c\right )}{d} \]
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Leaf count of result is larger than twice the leaf count of optimal. 272 vs. \(2 (93) = 186\).
Time = 0.41 (sec) , antiderivative size = 272, normalized size of antiderivative = 2.75 \[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {8 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 24 \, a^{2} b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 24 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 8 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} - 3 \, {\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (8 \, a^{2} b + 12 \, a b^{2} + 5 \, b^{3}\right )} + \frac {4 \, {\left (12 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 9 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{3} + 48 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 28 \, b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}\right )}}{{\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}^{2}}}{16 \, d} \]
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Time = 0.29 (sec) , antiderivative size = 355, normalized size of antiderivative = 3.59 \[ \int \cosh (c+d x) \left (a+b \tanh ^2(c+d x)\right )^3 \, dx=\frac {{\mathrm {e}}^{c+d\,x}\,{\left (a+b\right )}^3}{2\,d}-\frac {{\mathrm {e}}^{-c-d\,x}\,{\left (a+b\right )}^3}{2\,d}-\frac {3\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\left (5\,b^3\,\sqrt {d^2}+12\,a\,b^2\,\sqrt {d^2}+8\,a^2\,b\,\sqrt {d^2}\right )}{d\,\sqrt {64\,a^4\,b^2+192\,a^3\,b^3+224\,a^2\,b^4+120\,a\,b^5+25\,b^6}}\right )\,\sqrt {64\,a^4\,b^2+192\,a^3\,b^3+224\,a^2\,b^4+120\,a\,b^5+25\,b^6}}{4\,\sqrt {d^2}}+\frac {3\,{\mathrm {e}}^{c+d\,x}\,\left (3\,b^3+4\,a\,b^2\right )}{4\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {6\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {{\mathrm {e}}^{c+d\,x}\,\left (13\,b^3+12\,a\,b^2\right )}{2\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {4\,b^3\,{\mathrm {e}}^{c+d\,x}}{d\,\left (4\,{\mathrm {e}}^{2\,c+2\,d\,x}+6\,{\mathrm {e}}^{4\,c+4\,d\,x}+4\,{\mathrm {e}}^{6\,c+6\,d\,x}+{\mathrm {e}}^{8\,c+8\,d\,x}+1\right )} \]
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