Optimal. Leaf size=59 \[ -\frac {(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a b^{3/2} d}+\frac {x}{a}+\frac {\tanh (c+d x)}{b d} \]
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Rubi [A] time = 0.18, antiderivative size = 59, 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, 479, 522, 206, 208} \[ -\frac {(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a b^{3/2} d}+\frac {x}{a}+\frac {\tanh (c+d x)}{b d} \]
Antiderivative was successfully verified.
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Rule 206
Rule 208
Rule 479
Rule 522
Rule 1975
Rule 4141
Rubi steps
\begin {align*} \int \frac {\tanh ^4(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^4}{\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 {x^4}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\tanh (c+d x)}{b d}-\frac {\operatorname {Subst}\left (\int \frac {a+b+(-a-2 b) x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{b d}\\ &=\frac {\tanh (c+d x)}{b d}+\frac {\operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{a d}-\frac {(a+b)^2 \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{a b d}\\ &=\frac {x}{a}-\frac {(a+b)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a b^{3/2} d}+\frac {\tanh (c+d x)}{b d}\\ \end {align*}
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Mathematica [B] time = 1.09, size = 196, normalized size = 3.32 \[ \frac {\text {sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4} (a \text {sech}(c) \sinh (d x) \text {sech}(c+d x)+b d x)+(a+b)^2 (\sinh (2 c)-\cosh (2 c)) \tanh ^{-1}\left (\frac {(\cosh (2 c)-\sinh (2 c)) \text {sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right )\right )}{2 a b d \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4} \left (a+b \text {sech}^2(c+d x)\right )} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.45, size = 683, normalized size = 11.58 \[ \left [\frac {2 \, b d x \cosh \left (d x + c\right )^{2} + 4 \, b d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + 2 \, b d x \sinh \left (d x + c\right )^{2} + 2 \, b d x + {\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} + a + b\right )} \sqrt {\frac {a + b}{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 (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} + a b + 2 \, b^{2}\right )} \sqrt {\frac {a + b}{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 (a b d \cosh \left (d x + c\right )^{2} + 2 \, a b d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b d \sinh \left (d x + c\right )^{2} + a b d\right )}}, \frac {b d x \cosh \left (d x + c\right )^{2} + 2 \, b d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + b d x \sinh \left (d x + c\right )^{2} + b d x - {\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} + a + b\right )} \sqrt {-\frac {a + b}{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 {a + b}{b}}}{2 \, {\left (a + b\right )}}\right ) - 2 \, a}{a b d \cosh \left (d x + c\right )^{2} + 2 \, a b d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b d \sinh \left (d x + c\right )^{2} + a b 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 = 0.35, size = 386, normalized size = 6.54 \[ -\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{d a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d a}+\frac {a \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d \,b^{\frac {3}{2}} \sqrt {a +b}}+\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{d \sqrt {b}\, \sqrt {a +b}}+\frac {\sqrt {b}\, \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d a \sqrt {a +b}}-\frac {a \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d \,b^{\frac {3}{2}} \sqrt {a +b}}-\frac {\ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{d \sqrt {b}\, \sqrt {a +b}}-\frac {\sqrt {b}\, \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d a \sqrt {a +b}}+\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{d b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.64, size = 637, normalized size = 10.80 \[ -\frac {{\left (a + 2 \, b\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 \, \sqrt {{\left (a + b\right )} b} b d} + \frac {{\left (a + 2 \, b\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 \, \sqrt {{\left (a + b\right )} b} b d} + \frac {3 \, a \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 )}{16 \, \sqrt {{\left (a + b\right )} b} b d} + \frac {{\left (a + 2 \, b\right )} \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 )}{8 \, a b d} + \frac {\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 \, b d} - \frac {{\left (a + 2 \, b\right )} \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 )}{8 \, a b d} - \frac {\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 \, b d} - \frac {3 \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{4 \, b d} + \frac {3 \, \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{4 \, b d} - \frac {{\left (a^{2} + 8 \, a b + 8 \, 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 )}{32 \, \sqrt {{\left (a + b\right )} b} a b d} + \frac {{\left (a^{2} + 8 \, a b + 8 \, 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 )}{32 \, \sqrt {{\left (a + b\right )} b} a b d} - \frac {5}{8 \, {\left (b e^{\left (2 \, d x + 2 \, c\right )} + b\right )} d} + \frac {11}{8 \, {\left (b e^{\left (-2 \, d x - 2 \, c\right )} + b\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.87, size = 183, normalized size = 3.10 \[ \frac {x}{a}-\frac {2}{b\,d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}+\frac {\ln \left (\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a+b\right )}^2}{a^2\,b}-\frac {2\,{\left (a+b\right )}^{3/2}\,\left (a+a\,{\mathrm {e}}^{2\,c+2\,d\,x}+2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^2\,b^{3/2}}\right )\,{\left (a+b\right )}^{3/2}}{2\,a\,b^{3/2}\,d}-\frac {\ln \left (\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}\,{\left (a+b\right )}^2}{a^2\,b}+\frac {2\,{\left (a+b\right )}^{3/2}\,\left (a+a\,{\mathrm {e}}^{2\,c+2\,d\,x}+2\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{a^2\,b^{3/2}}\right )\,{\left (a+b\right )}^{3/2}}{2\,a\,b^{3/2}\,d} \]
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 \[ \int \frac {\tanh ^{4}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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