Optimal. Leaf size=62 \[ -\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a d (a+b)^{3/2}}-\frac {\coth (c+d x)}{d (a+b)}+\frac {x}{a} \]
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Rubi [A] time = 0.18, antiderivative size = 62, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {4141, 1975, 480, 522, 206, 208} \[ -\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a d (a+b)^{3/2}}-\frac {\coth (c+d x)}{d (a+b)}+\frac {x}{a} \]
Antiderivative was successfully verified.
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Rule 206
Rule 208
Rule 480
Rule 522
Rule 1975
Rule 4141
Rubi steps
\begin {align*} \int \frac {\coth ^2(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right ) \left (a+b \left (1-x^2\right )\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {1}{x^2 \left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {\coth (c+d x)}{(a+b) d}+\frac {\operatorname {Subst}\left (\int \frac {a+2 b-b x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{(a+b) d}\\ &=-\frac {\coth (c+d x)}{(a+b) d}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{a d}-\frac {b^2 \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{a (a+b) d}\\ &=\frac {x}{a}-\frac {b^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a (a+b)^{3/2} d}-\frac {\coth (c+d x)}{(a+b) d}\\ \end {align*}
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Mathematica [B] time = 1.16, size = 193, normalized size = 3.11 \[ \frac {\text {sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (b^2 (\sinh (2 c)-\cosh (2 c)) \tanh ^{-1}\left (\frac {(\cosh (2 c)-\sinh (2 c)) \text {sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right )+\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4} (d x (a+b)+a \text {csch}(c) \sinh (d x) \text {csch}(c+d x))\right )}{2 a d (a+b)^{3/2} \sqrt {b (\cosh (c)-\sinh (c))^4} \left (a+b \text {sech}^2(c+d x)\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.46, size = 749, normalized size = 12.08 \[ \left [\frac {2 \, {\left (a + b\right )} d x \cosh \left (d x + c\right )^{2} + 4 \, {\left (a + b\right )} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + 2 \, {\left (a + b\right )} d x \sinh \left (d x + c\right )^{2} - 2 \, {\left (a + b\right )} d x + {\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} - b\right )} \sqrt {\frac {b}{a + b}} \log \left (\frac {a^{2} \cosh \left (d x + c\right )^{4} + 4 \, a^{2} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{2} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{2} \cosh \left (d x + c\right )^{2} + a^{2} + 2 \, a b\right )} \sinh \left (d x + c\right )^{2} + a^{2} + 8 \, a b + 8 \, b^{2} + 4 \, {\left (a^{2} \cosh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + 4 \, {\left ({\left (a^{2} + a b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} + a b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} + a b\right )} \sinh \left (d x + c\right )^{2} + a^{2} + 3 \, a b + 2 \, b^{2}\right )} \sqrt {\frac {b}{a + b}}}{a \cosh \left (d x + c\right )^{4} + 4 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a \sinh \left (d x + c\right )^{4} + 2 \, {\left (a + 2 \, b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a \cosh \left (d x + c\right )^{2} + a + 2 \, b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a \cosh \left (d x + c\right )^{3} + {\left (a + 2 \, b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a}\right ) - 4 \, a}{2 \, {\left ({\left (a^{2} + a b\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} + a b\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} + a b\right )} d \sinh \left (d x + c\right )^{2} - {\left (a^{2} + a b\right )} d\right )}}, \frac {{\left (a + b\right )} d x \cosh \left (d x + c\right )^{2} + 2 \, {\left (a + b\right )} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a + b\right )} d x \sinh \left (d x + c\right )^{2} - {\left (a + b\right )} d x - {\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} - b\right )} \sqrt {-\frac {b}{a + b}} \arctan \left (\frac {{\left (a \cosh \left (d x + c\right )^{2} + 2 \, a \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a \sinh \left (d x + c\right )^{2} + a + 2 \, b\right )} \sqrt {-\frac {b}{a + b}}}{2 \, b}\right ) - 2 \, a}{{\left (a^{2} + a b\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{2} + a b\right )} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + {\left (a^{2} + a b\right )} d \sinh \left (d x + c\right )^{2} - {\left (a^{2} + a b\right )} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.46, size = 189, normalized size = 3.05 \[ -\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 d \left (a +b \right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d a}-\frac {1}{2 d \left (a +b \right ) \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}+\frac {b^{\frac {3}{2}} \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d a \left (a +b \right )^{\frac {3}{2}}}-\frac {b^{\frac {3}{2}} \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d a \left (a +b \right )^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.48, size = 429, normalized size = 6.92 \[ \frac {b \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a + 2 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{4 \, {\left (a^{2} + a b\right )} d} - \frac {b \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{4 \, {\left (a^{2} + a b\right )} d} - \frac {{\left (a b + 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, {\left (a^{2} + a b\right )} \sqrt {{\left (a + b\right )} b} d} + \frac {{\left (a b + 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, {\left (a^{2} + a b\right )} \sqrt {{\left (a + b\right )} b} d} - \frac {b \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{4 \, \sqrt {{\left (a + b\right )} b} {\left (a + b\right )} d} + \frac {\log \left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}{2 \, {\left (a + b\right )} d} - \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} - 1\right )}{2 \, {\left (a + b\right )} d} - \frac {1}{2 \, {\left ({\left (a + b\right )} e^{\left (2 \, d x + 2 \, c\right )} - a - b\right )} d} + \frac {3}{2 \, {\left ({\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} - a - b\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 3.39, size = 977, normalized size = 15.76 \[ \frac {x}{a}-\frac {2}{\left ({\mathrm {e}}^{2\,c+2\,d\,x}-1\right )\,\left (a\,d+b\,d\right )}+\frac {\mathrm {atan}\left (\frac {\left ({\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}\,\left (\frac {8\,\left (a+2\,b\right )\,\left (4\,d\,a^4\,b^2+16\,d\,a^3\,b^3+20\,d\,a^2\,b^4+8\,d\,a\,b^5\right )}{a^6\,\left (a+b\right )\,\left (a^3+2\,a^2\,b+a\,b^2\right )\,\sqrt {-a^2\,d^2\,{\left (a+b\right )}^3}\,\sqrt {-a^5\,d^2-3\,a^4\,b\,d^2-3\,a^3\,b^2\,d^2-a^2\,b^3\,d^2}}+\frac {2\,\sqrt {b^3}\,\left (a^2+8\,a\,b+8\,b^2\right )\,\left (a^2\,\sqrt {b^3}\,\sqrt {-a^5\,d^2-3\,a^4\,b\,d^2-3\,a^3\,b^2\,d^2-a^2\,b^3\,d^2}+8\,b^2\,\sqrt {b^3}\,\sqrt {-a^5\,d^2-3\,a^4\,b\,d^2-3\,a^3\,b^2\,d^2-a^2\,b^3\,d^2}+8\,a\,b\,\sqrt {b^3}\,\sqrt {-a^5\,d^2-3\,a^4\,b\,d^2-3\,a^3\,b^2\,d^2-a^2\,b^3\,d^2}\right )}{a^7\,b^2\,d\,{\left (a+b\right )}^3\,\left (a^3+2\,a^2\,b+a\,b^2\right )\,\sqrt {-a^5\,d^2-3\,a^4\,b\,d^2-3\,a^3\,b^2\,d^2-a^2\,b^3\,d^2}}\right )+\frac {8\,\left (a+2\,b\right )\,\left (2\,d\,a^4\,b^2+4\,d\,a^3\,b^3+2\,d\,a^2\,b^4\right )}{a^6\,\left (a+b\right )\,\left (a^3+2\,a^2\,b+a\,b^2\right )\,\sqrt {-a^2\,d^2\,{\left (a+b\right )}^3}\,\sqrt {-a^5\,d^2-3\,a^4\,b\,d^2-3\,a^3\,b^2\,d^2-a^2\,b^3\,d^2}}+\frac {2\,\left (a^2\,\sqrt {b^3}\,\sqrt {-a^5\,d^2-3\,a^4\,b\,d^2-3\,a^3\,b^2\,d^2-a^2\,b^3\,d^2}+2\,a\,b\,\sqrt {b^3}\,\sqrt {-a^5\,d^2-3\,a^4\,b\,d^2-3\,a^3\,b^2\,d^2-a^2\,b^3\,d^2}\right )\,\sqrt {b^3}\,\left (a^2+8\,a\,b+8\,b^2\right )}{a^7\,b^2\,d\,{\left (a+b\right )}^3\,\left (a^3+2\,a^2\,b+a\,b^2\right )\,\sqrt {-a^5\,d^2-3\,a^4\,b\,d^2-3\,a^3\,b^2\,d^2-a^2\,b^3\,d^2}}\right )\,\left (a^7\,\sqrt {-a^5\,d^2-3\,a^4\,b\,d^2-3\,a^3\,b^2\,d^2-a^2\,b^3\,d^2}+a^4\,b^3\,\sqrt {-a^5\,d^2-3\,a^4\,b\,d^2-3\,a^3\,b^2\,d^2-a^2\,b^3\,d^2}+3\,a^5\,b^2\,\sqrt {-a^5\,d^2-3\,a^4\,b\,d^2-3\,a^3\,b^2\,d^2-a^2\,b^3\,d^2}+3\,a^6\,b\,\sqrt {-a^5\,d^2-3\,a^4\,b\,d^2-3\,a^3\,b^2\,d^2-a^2\,b^3\,d^2}\right )}{4\,\sqrt {b^3}}\right )\,\sqrt {b^3}}{\sqrt {-a^5\,d^2-3\,a^4\,b\,d^2-3\,a^3\,b^2\,d^2-a^2\,b^3\,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 \[ \int \frac {\coth ^{2}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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