Optimal. Leaf size=77 \[ \frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d (a+b)^2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 d (a+b)}+\frac {x (a+3 b)}{2 (a+b)^2} \]
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Rubi [A] time = 0.10, antiderivative size = 77, 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 = {3675, 414, 522, 206, 205} \[ \frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} d (a+b)^2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 d (a+b)}+\frac {x (a+3 b)}{2 (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 205
Rule 206
Rule 414
Rule 522
Rule 3675
Rubi steps
\begin {align*} \int \frac {\cosh ^2(c+d x)}{a+b \tanh ^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{\left (1-x^2\right )^2 \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d}+\frac {\operatorname {Subst}\left (\int \frac {a+2 b+b x^2}{\left (1-x^2\right ) \left (a+b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 (a+b) d}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d}+\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{(a+b)^2 d}+\frac {(a+3 b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 (a+b)^2 d}\\ &=\frac {(a+3 b) x}{2 (a+b)^2}+\frac {b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} (a+b)^2 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 (a+b) d}\\ \end {align*}
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Mathematica [A] time = 0.16, size = 77, normalized size = 1.00 \[ \frac {4 b^{3/2} \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )+2 \sqrt {a} (a+3 b) (c+d x)+\sqrt {a} (a+b) \sinh (2 (c+d x))}{4 \sqrt {a} d (a+b)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 948, normalized size = 12.31 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.56, size = 172, normalized size = 2.23 \[ \frac {\frac {4 \, {\left (a + 3 \, b\right )} d x}{a^{2} + 2 \, a b + b^{2}} + \frac {8 \, b^{2} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{{\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b}} - \frac {{\left (2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 6 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} e^{\left (-2 \, d x\right )}}{a^{2} e^{\left (2 \, c\right )} + 2 \, a b e^{\left (2 \, c\right )} + b^{2} e^{\left (2 \, c\right )}} + \frac {e^{\left (2 \, d x + 8 \, c\right )}}{a e^{\left (6 \, c\right )} + b e^{\left (6 \, c\right )}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.43, size = 608, normalized size = 7.90 \[ \frac {1}{d \left (2 b +2 a \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {2}{d \left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) a}{2 d \left (a +b \right )^{2}}-\frac {3 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b}{2 d \left (a +b \right )^{2}}-\frac {1}{d \left (2 b +2 a \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {2}{d \left (4 a +4 b \right ) \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) a}{2 d \left (a +b \right )^{2}}+\frac {3 \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b}{2 d \left (a +b \right )^{2}}-\frac {a \,b^{2} \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 )}{d \left (a +b \right )^{2} \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {b^{2} \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 )}{d \left (a +b \right )^{2} \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}-\frac {b^{3} \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 )}{d \left (a +b \right )^{2} \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}-\frac {a \,b^{2} \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 )}{d \left (a +b \right )^{2} \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}-\frac {b^{2} \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 )}{d \left (a +b \right )^{2} \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}-\frac {b^{3} \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 )}{d \left (a +b \right )^{2} \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.62, size = 316, normalized size = 4.10 \[ \frac {b \log \left ({\left (a + b\right )} e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a - b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} - \frac {b \log \left (2 \, {\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} d} - \frac {{\left (a b - b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b} d} + \frac {{\left (a b - b^{2}\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \sqrt {a b} d} - \frac {b \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{2 \, \sqrt {a b} {\left (a + b\right )} d} + \frac {d x + c}{2 \, {\left (a + b\right )} d} + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{8 \, {\left (a + b\right )} d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, {\left (a + b\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.93, size = 880, normalized size = 11.43 \[ \frac {{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,d\,\left (a+b\right )}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,d\,\left (a+b\right )}+\frac {x\,\left (a+3\,b\right )}{2\,{\left (a+b\right )}^2}+\frac {\mathrm {atan}\left (\left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (\frac {2\,\left (2\,b^3\,\sqrt {a^5\,d^2+4\,a^4\,b\,d^2+6\,a^3\,b^2\,d^2+4\,a^2\,b^3\,d^2+a\,b^4\,d^2}+2\,a\,b^2\,\sqrt {a^5\,d^2+4\,a^4\,b\,d^2+6\,a^3\,b^2\,d^2+4\,a^2\,b^3\,d^2+a\,b^4\,d^2}\right )}{d\,{\left (a+b\right )}^5\,\sqrt {b^3}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )\,\sqrt {a^5\,d^2+4\,a^4\,b\,d^2+6\,a^3\,b^2\,d^2+4\,a^2\,b^3\,d^2+a\,b^4\,d^2}}-\frac {\left (a-b\right )\,\left (2\,a\,d\,{\left (b^3\right )}^{3/2}+b\,d\,{\left (b^3\right )}^{3/2}-a^4\,d\,\sqrt {b^3}-2\,a^3\,b\,d\,\sqrt {b^3}\right )}{b^2\,{\left (a+b\right )}^3\,\sqrt {a\,d^2\,{\left (a+b\right )}^4}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )\,\sqrt {a^5\,d^2+4\,a^4\,b\,d^2+6\,a^3\,b^2\,d^2+4\,a^2\,b^3\,d^2+a\,b^4\,d^2}}\right )+\frac {\left (a-b\right )\,\left (4\,a\,d\,{\left (b^3\right )}^{3/2}+b\,d\,{\left (b^3\right )}^{3/2}+a^4\,d\,\sqrt {b^3}+4\,a^3\,b\,d\,\sqrt {b^3}+6\,a^2\,b^2\,d\,\sqrt {b^3}\right )}{b^2\,{\left (a+b\right )}^3\,\sqrt {a\,d^2\,{\left (a+b\right )}^4}\,\left (a^3+3\,a^2\,b+3\,a\,b^2+b^3\right )\,\sqrt {a^5\,d^2+4\,a^4\,b\,d^2+6\,a^3\,b^2\,d^2+4\,a^2\,b^3\,d^2+a\,b^4\,d^2}}\right )\,\left (\frac {a^4\,\sqrt {a^5\,d^2+4\,a^4\,b\,d^2+6\,a^3\,b^2\,d^2+4\,a^2\,b^3\,d^2+a\,b^4\,d^2}}{2}+\frac {b^4\,\sqrt {a^5\,d^2+4\,a^4\,b\,d^2+6\,a^3\,b^2\,d^2+4\,a^2\,b^3\,d^2+a\,b^4\,d^2}}{2}+3\,a^2\,b^2\,\sqrt {a^5\,d^2+4\,a^4\,b\,d^2+6\,a^3\,b^2\,d^2+4\,a^2\,b^3\,d^2+a\,b^4\,d^2}+2\,a\,b^3\,\sqrt {a^5\,d^2+4\,a^4\,b\,d^2+6\,a^3\,b^2\,d^2+4\,a^2\,b^3\,d^2+a\,b^4\,d^2}+2\,a^3\,b\,\sqrt {a^5\,d^2+4\,a^4\,b\,d^2+6\,a^3\,b^2\,d^2+4\,a^2\,b^3\,d^2+a\,b^4\,d^2}\right )\right )\,\sqrt {b^3}}{\sqrt {a^5\,d^2+4\,a^4\,b\,d^2+6\,a^3\,b^2\,d^2+4\,a^2\,b^3\,d^2+a\,b^4\,d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\cosh ^{2}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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