Optimal. Leaf size=83 \[ -\frac {a^2}{2 b^2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac {a (a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 b^2 d (a+b)^2}+\frac {\log (\cosh (c+d x))}{d (a+b)^2} \]
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Rubi [A] time = 0.15, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3670, 446, 88} \[ -\frac {a^2}{2 b^2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac {a (a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 b^2 d (a+b)^2}+\frac {\log (\cosh (c+d x))}{d (a+b)^2} \]
Antiderivative was successfully verified.
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Rule 88
Rule 446
Rule 3670
Rubi steps
\begin {align*} \int \frac {\tanh ^5(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^5}{\left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{(1-x) (a+b x)^2} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{(a+b)^2 (-1+x)}+\frac {a^2}{b (a+b) (a+b x)^2}-\frac {a (a+2 b)}{b (a+b)^2 (a+b x)}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac {\log (\cosh (c+d x))}{(a+b)^2 d}-\frac {a (a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 b^2 (a+b)^2 d}-\frac {a^2}{2 b^2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}\\ \end {align*}
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Mathematica [A] time = 0.56, size = 69, normalized size = 0.83 \[ -\frac {\frac {a^2 (a+b)}{b^2 \left (a+b \tanh ^2(c+d x)\right )}+\frac {a (a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{b^2}-2 \log (\cosh (c+d x))}{2 d (a+b)^2} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.55, size = 1141, normalized size = 13.75 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.45, size = 194, normalized size = 2.34 \[ -\frac {\frac {{\left (a^{2} + 2 \, a b\right )} \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )}{a^{2} b^{2} + 2 \, a b^{3} + b^{4}} + \frac {2 \, {\left (d x + c\right )}}{a^{2} + 2 \, a b + b^{2}} + \frac {4 \, a^{2} e^{\left (2 \, d x + 2 \, c\right )}}{{\left (a e^{\left (4 \, d x + 4 \, c\right )} + b e^{\left (4 \, d x + 4 \, c\right )} + 2 \, a e^{\left (2 \, d x + 2 \, c\right )} - 2 \, b e^{\left (2 \, d x + 2 \, c\right )} + a + b\right )} {\left (a + b\right )}^{2} b} - \frac {2 \, \log \left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}{b^{2}}}{2 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.10, size = 156, normalized size = 1.88 \[ -\frac {\ln \left (\tanh \left (d x +c \right )-1\right )}{2 d \left (a +b \right )^{2}}-\frac {\ln \left (1+\tanh \left (d x +c \right )\right )}{2 d \left (a +b \right )^{2}}-\frac {a^{2} \ln \left (a +b \left (\tanh ^{2}\left (d x +c \right )\right )\right )}{2 d \left (a +b \right )^{2} b^{2}}-\frac {a \ln \left (a +b \left (\tanh ^{2}\left (d x +c \right )\right )\right )}{d \left (a +b \right )^{2} b}-\frac {a^{3}}{2 d \left (a +b \right )^{2} b^{2} \left (a +b \left (\tanh ^{2}\left (d x +c \right )\right )\right )}-\frac {a^{2}}{2 d \left (a +b \right )^{2} b \left (a +b \left (\tanh ^{2}\left (d x +c \right )\right )\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.44, size = 217, normalized size = 2.61 \[ -\frac {2 \, a^{2} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4} + 2 \, {\left (a^{3} b + a^{2} b^{2} - a b^{3} - b^{4}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{3} b + 3 \, a^{2} b^{2} + 3 \, a b^{3} + b^{4}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} - \frac {{\left (a^{2} + 2 \, a b\right )} \log \left (2 \, {\left (a - b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a + b\right )} e^{\left (-4 \, d x - 4 \, c\right )} + a + b\right )}{2 \, {\left (a^{2} b^{2} + 2 \, a b^{3} + b^{4}\right )} d} + \frac {d x + c}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} + \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{b^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.61, size = 170, normalized size = 2.05 \[ -\frac {a^2}{2\,\left (d\,a^2\,b^2+d\,a\,b^3\,{\mathrm {tanh}\left (c+d\,x\right )}^2+d\,a\,b^3+d\,b^4\,{\mathrm {tanh}\left (c+d\,x\right )}^2\right )}-\frac {\ln \left ({\mathrm {tanh}\left (c+d\,x\right )}^2-1\right )}{2\,\left (d\,a^2+2\,d\,a\,b+d\,b^2\right )}-\frac {a^2\,\ln \left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}{2\,\left (d\,a^2\,b^2+2\,d\,a\,b^3+d\,b^4\right )}-\frac {a\,b\,\ln \left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}{d\,a^2\,b^2+2\,d\,a\,b^3+d\,b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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