Optimal. Leaf size=79 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {a-b} d}+\frac {\tanh (c+d x)}{2 a d \left (a-(a-b) \tanh ^2(c+d x)\right )} \]
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Rubi [A]
time = 0.05, antiderivative size = 79, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3270, 205, 214}
\begin {gather*} \frac {\tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} d \sqrt {a-b}}+\frac {\tanh (c+d x)}{2 a d \left (a-(a-b) \tanh ^2(c+d x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 205
Rule 214
Rule 3270
Rubi steps
\begin {align*} \int \frac {\cosh ^2(c+d x)}{\left (a+b \sinh ^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{\left (a-(a-b) x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\tanh (c+d x)}{2 a d \left (a-(a-b) \tanh ^2(c+d x)\right )}+\frac {\text {Subst}\left (\int \frac {1}{a+(-a+b) x^2} \, dx,x,\tanh (c+d x)\right )}{2 a d}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{2 a^{3/2} \sqrt {a-b} d}+\frac {\tanh (c+d x)}{2 a d \left (a-(a-b) \tanh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A]
time = 0.15, size = 78, normalized size = 0.99 \begin {gather*} \frac {\frac {\tanh ^{-1}\left (\frac {\sqrt {a-b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a-b}}+\frac {\sqrt {a} \sinh (2 (c+d x))}{2 a-b+b \cosh (2 (c+d x))}}{2 a^{3/2} 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. \(251\) vs.
\(2(67)=134\).
time = 1.49, size = 252, normalized size = 3.19
method | result | size |
risch | \(-\frac {2 a \,{\mathrm e}^{2 d x +2 c}-b \,{\mathrm e}^{2 d x +2 c}+b}{b d a \left (b \,{\mathrm e}^{4 d x +4 c}+4 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+b \right )}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}-2 a^{2}+2 a b}{b \sqrt {a^{2}-a b}}\right )}{4 \sqrt {a^{2}-a b}\, d a}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {2 a \sqrt {a^{2}-a b}-b \sqrt {a^{2}-a b}+2 a^{2}-2 a b}{b \sqrt {a^{2}-a b}}\right )}{4 \sqrt {a^{2}-a b}\, d a}\) | \(236\) |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}}{d}\) | \(252\) |
default | \(\frac {-\frac {2 \left (-\frac {\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}-\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (\sqrt {-b \left (a -b \right )}-b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}+a -2 b \right ) a}}-\frac {\left (\sqrt {-b \left (a -b \right )}+b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}\right )}{2 a \sqrt {-b \left (a -b \right )}\, \sqrt {\left (2 \sqrt {-b \left (a -b \right )}-a +2 b \right ) a}}}{d}\) | \(252\) |
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. 582 vs.
\(2 (68) = 136\).
time = 0.41, size = 1421, normalized size = 17.99 \begin {gather*} \left [-\frac {4 \, a^{2} b - 4 \, a b^{2} + 4 \, {\left (2 \, a^{3} - 3 \, a^{2} b + a b^{2}\right )} \cosh \left (d x + c\right )^{2} + 8 \, {\left (2 \, a^{3} - 3 \, a^{2} b + a b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + 4 \, {\left (2 \, a^{3} - 3 \, a^{2} b + a b^{2}\right )} \sinh \left (d x + c\right )^{2} - {\left (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} + 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 )\right )} \sqrt {a^{2} - a b} \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 (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 {a^{2} - a b}}{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 ({\left (a^{3} b^{2} - a^{2} b^{3}\right )} d \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{3} b^{2} - a^{2} b^{3}\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{3} b^{2} - a^{2} b^{3}\right )} d \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a^{4} b - 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{3} b^{2} - a^{2} b^{3}\right )} d \cosh \left (d x + c\right )^{2} + {\left (2 \, a^{4} b - 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} d\right )} \sinh \left (d x + c\right )^{2} + {\left (a^{3} b^{2} - a^{2} b^{3}\right )} d + 4 \, {\left ({\left (a^{3} b^{2} - a^{2} b^{3}\right )} d \cosh \left (d x + c\right )^{3} + {\left (2 \, a^{4} b - 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}}, -\frac {2 \, a^{2} b - 2 \, a b^{2} + 2 \, {\left (2 \, a^{3} - 3 \, a^{2} b + a b^{2}\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (2 \, a^{3} - 3 \, a^{2} b + a b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + 2 \, {\left (2 \, a^{3} - 3 \, a^{2} b + a b^{2}\right )} \sinh \left (d x + c\right )^{2} + {\left (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} + 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 )\right )} \sqrt {-a^{2} + a b} \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 {-a^{2} + a b}}{2 \, {\left (a^{2} - a b\right )}}\right )}{2 \, {\left ({\left (a^{3} b^{2} - a^{2} b^{3}\right )} d \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{3} b^{2} - a^{2} b^{3}\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{3} b^{2} - a^{2} b^{3}\right )} d \sinh \left (d x + c\right )^{4} + 2 \, {\left (2 \, a^{4} b - 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{3} b^{2} - a^{2} b^{3}\right )} d \cosh \left (d x + c\right )^{2} + {\left (2 \, a^{4} b - 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} d\right )} \sinh \left (d x + c\right )^{2} + {\left (a^{3} b^{2} - a^{2} b^{3}\right )} d + 4 \, {\left ({\left (a^{3} b^{2} - a^{2} b^{3}\right )} d \cosh \left (d x + c\right )^{3} + {\left (2 \, a^{4} b - 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\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 = 1.36, size = 126, normalized size = 1.59 \begin {gather*} \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} a} - \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 (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 )} a b}}{2 \, 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 {cosh}\left (c+d\,x\right )}^2}{{\left (b\,{\mathrm {sinh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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