Optimal. Leaf size=81 \[ -\frac {(2 a-3 b) x}{2 b^2}+\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^2 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d} \]
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Rubi [A]
time = 0.09, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {3270, 425, 536,
212, 214} \begin {gather*} \frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^2 d}-\frac {x (2 a-3 b)}{2 b^2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 b d} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 214
Rule 425
Rule 536
Rule 3270
Rubi steps
\begin {align*} \int \frac {\cosh ^4(c+d x)}{a+b \sinh ^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a-(a-b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d}+\frac {\text {Subst}\left (\int \frac {-a+2 b+(-a+b) x^2}{\left (1-x^2\right ) \left (a+(-a+b) x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 b d}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d}-\frac {(2 a-3 b) \text {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 b^2 d}+\frac {(a-b)^2 \text {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{b^2 d}\\ &=-\frac {(2 a-3 b) x}{2 b^2}+\frac {(a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^2 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 b d}\\ \end {align*}
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Mathematica [A]
time = 0.12, size = 80, normalized size = 0.99 \begin {gather*} \frac {4 (a-b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )+\sqrt {a} (-2 (2 a-3 b) (c+d x)+b \sinh (2 (c+d x)))}{4 \sqrt {a} b^2 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. \(315\) vs.
\(2(69)=138\).
time = 1.64, size = 316, normalized size = 3.90
method | result | size |
risch | \(-\frac {a x}{b^{2}}+\frac {3 x}{2 b}+\frac {{\mathrm e}^{2 d x +2 c}}{8 b d}-\frac {{\mathrm e}^{-2 d x -2 c}}{8 b d}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{2 d \,b^{2}}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}-\frac {2 \sqrt {a \left (a -b \right )}-2 a +b}{b}\right )}{2 a d b}-\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{2 d \,b^{2}}+\frac {\sqrt {a \left (a -b \right )}\, \ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 \sqrt {a \left (a -b \right )}+2 a -b}{b}\right )}{2 a d b}\) | \(253\) |
derivativedivides | \(\frac {\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (2 a -3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{2}}+\frac {2 \left (a^{2}-2 a b +b^{2}\right ) a \left (\frac {\left (-\sqrt {-b \left (a -b \right )}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (-\sqrt {-b \left (a -b \right )}+b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{b^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-2 a +3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{2}}}{d}\) | \(316\) |
default | \(\frac {\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\left (2 a -3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 b^{2}}+\frac {2 \left (a^{2}-2 a b +b^{2}\right ) a \left (\frac {\left (-\sqrt {-b \left (a -b \right )}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}-\frac {\left (-\sqrt {-b \left (a -b \right )}+b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{b^{2}}-\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 b \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\left (-2 a +3 b \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 b^{2}}}{d}\) | \(316\) |
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. 305 vs.
\(2 (69) = 138\).
time = 0.42, size = 875, normalized size = 10.80 \begin {gather*} \left [-\frac {4 \, {\left (2 \, a - 3 \, b\right )} d x \cosh \left (d x + c\right )^{2} - b \cosh \left (d x + c\right )^{4} - 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} - b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, {\left (2 \, a - 3 \, b\right )} d x - 3 \, b \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a - b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a - b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {\frac {a - b}{a}} \log \left (\frac {b^{2} \cosh \left (d x + c\right )^{4} + 4 \, b^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b^{2} \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a b - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b^{2} \cosh \left (d x + c\right )^{2} + 2 \, a b - b^{2}\right )} \sinh \left (d x + c\right )^{2} + 8 \, a^{2} - 8 \, a b + b^{2} + 4 \, {\left (b^{2} \cosh \left (d x + c\right )^{3} + {\left (2 \, a b - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \, {\left (a b \cosh \left (d x + c\right )^{2} + 2 \, a b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b \sinh \left (d x + c\right )^{2} + 2 \, a^{2} - a b\right )} \sqrt {\frac {a - b}{a}}}{b \cosh \left (d x + c\right )^{4} + 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, b \cosh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (b \cosh \left (d x + c\right )^{3} + {\left (2 \, a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + b}\right ) + 4 \, {\left (2 \, {\left (2 \, a - 3 \, b\right )} d x \cosh \left (d x + c\right ) - b \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right ) + b}{8 \, {\left (b^{2} d \cosh \left (d x + c\right )^{2} + 2 \, b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} d \sinh \left (d x + c\right )^{2}\right )}}, -\frac {4 \, {\left (2 \, a - 3 \, b\right )} d x \cosh \left (d x + c\right )^{2} - b \cosh \left (d x + c\right )^{4} - 4 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} - b \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, {\left (2 \, a - 3 \, b\right )} d x - 3 \, b \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 8 \, {\left ({\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a - b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a - b\right )} \sinh \left (d x + c\right )^{2}\right )} \sqrt {-\frac {a - b}{a}} \arctan \left (-\frac {{\left (b \cosh \left (d x + c\right )^{2} + 2 \, b \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b \sinh \left (d x + c\right )^{2} + 2 \, a - b\right )} \sqrt {-\frac {a - b}{a}}}{2 \, {\left (a - b\right )}}\right ) + 4 \, {\left (2 \, {\left (2 \, a - 3 \, b\right )} d x \cosh \left (d x + c\right ) - b \cosh \left (d x + c\right )^{3}\right )} \sinh \left (d x + c\right ) + b}{8 \, {\left (b^{2} d \cosh \left (d x + c\right )^{2} + 2 \, b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b^{2} d \sinh \left (d x + c\right )^{2}\right )}}\right ] \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 [A]
time = 1.69, size = 138, normalized size = 1.70 \begin {gather*} -\frac {\frac {4 \, {\left (d x + c\right )} {\left (2 \, a - 3 \, b\right )}}{b^{2}} - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{b} - \frac {{\left (4 \, a e^{\left (2 \, d x + 2 \, c\right )} - 6 \, b e^{\left (2 \, d x + 2 \, c\right )} - b\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{b^{2}} - \frac {8 \, {\left (a^{2} - 2 \, a b + b^{2}\right )} \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} b^{2}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.61, size = 300, normalized size = 3.70 \begin {gather*} \frac {{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,b\,d}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,b\,d}-\frac {x\,\left (2\,a-3\,b\right )}{2\,b^2}-\frac {\ln \left (\frac {4\,{\left (a-b\right )}^3\,\left (2\,a\,b-b^2+8\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a\,b^6}-\frac {8\,{\left (a-b\right )}^{7/2}\,\left (b+4\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {a}\,b^6}\right )\,{\left (a-b\right )}^{3/2}}{2\,\sqrt {a}\,b^2\,d}+\frac {\ln \left (\frac {4\,{\left (a-b\right )}^3\,\left (2\,a\,b-b^2+8\,a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a\,b^6}+\frac {8\,{\left (a-b\right )}^{7/2}\,\left (b+4\,a\,{\mathrm {e}}^{2\,c+2\,d\,x}-2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {a}\,b^6}\right )\,{\left (a-b\right )}^{3/2}}{2\,\sqrt {a}\,b^2\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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