Optimal. Leaf size=76 \[ \frac {(a+b)^2}{2 a^2 b d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\log (\cosh (c+d x))}{b^2 d}+\frac {\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \log \left (b+a \cosh ^2(c+d x)\right )}{2 d} \]
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Rubi [A]
time = 0.08, antiderivative size = 76, 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 = {4223, 457, 90}
\begin {gather*} \frac {\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \log \left (a \cosh ^2(c+d x)+b\right )}{2 d}+\frac {(a+b)^2}{2 a^2 b d \left (a \cosh ^2(c+d x)+b\right )}+\frac {\log (\cosh (c+d x))}{b^2 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 90
Rule 457
Rule 4223
Rubi steps
\begin {align*} \int \frac {\tanh ^5(c+d x)}{\left (a+b \text {sech}^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {\left (1-x^2\right )^2}{x \left (b+a x^2\right )^2} \, dx,x,\cosh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {(1-x)^2}{x (b+a x)^2} \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{b^2 x}-\frac {(a+b)^2}{a b (b+a x)^2}+\frac {-a^2+b^2}{a b^2 (b+a x)}\right ) \, dx,x,\cosh ^2(c+d x)\right )}{2 d}\\ &=\frac {(a+b)^2}{2 a^2 b d \left (b+a \cosh ^2(c+d x)\right )}+\frac {\log (\cosh (c+d x))}{b^2 d}+\frac {\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \log \left (b+a \cosh ^2(c+d x)\right )}{2 d}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 109, normalized size = 1.43 \begin {gather*} \frac {(a+2 b+a \cosh (2 c+2 d x))^2 \left (\frac {(a+b)^2}{a^2 b \left (b+a \cosh ^2(c+d x)\right )}+\frac {2 \log (\cosh (c+d x))}{b^2}+\left (\frac {1}{a^2}-\frac {1}{b^2}\right ) \log \left (b+a \cosh ^2(c+d x)\right )\right ) \text {sech}^4(c+d x)}{8 d \left (a+b \text {sech}^2(c+d x)\right )^2} \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. \(208\) vs.
\(2(72)=144\).
time = 2.36, size = 209, normalized size = 2.75
method | result | size |
risch | \(-\frac {x}{a^{2}}-\frac {2 c}{a^{2} d}+\frac {2 \left (a^{2}+2 a b +b^{2}\right ) {\mathrm e}^{2 d x +2 c}}{a^{2} b d \left (a \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}+4 b \,{\mathrm e}^{2 d x +2 c}+a \right )}+\frac {\ln \left (1+{\mathrm e}^{2 d x +2 c}\right )}{b^{2} d}-\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 b^{2} d}+\frac {\ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (2 b +a \right ) {\mathrm e}^{2 d x +2 c}}{a}+1\right )}{2 a^{2} d}\) | \(184\) |
derivativedivides | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {\left (a +b \right ) \left (\frac {2 a b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (a -b \right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{2}\right )}{a^{2} b^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}+\frac {\ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}}{d}\) | \(209\) |
default | \(\frac {-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{a^{2}}-\frac {\left (a +b \right ) \left (\frac {2 a b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b}+\frac {\left (a -b \right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+b \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a +b \right )}{2}\right )}{a^{2} b^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{a^{2}}+\frac {\ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{b^{2}}}{d}\) | \(209\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 154 vs.
\(2 (72) = 144\).
time = 0.48, size = 154, normalized size = 2.03 \begin {gather*} \frac {2 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{{\left (a^{3} b e^{\left (-4 \, d x - 4 \, c\right )} + a^{3} b + 2 \, {\left (a^{3} b + 2 \, a^{2} b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )}\right )} d} + \frac {d x + c}{a^{2} d} + \frac {\log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{b^{2} d} - \frac {{\left (a^{2} - b^{2}\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 )}{2 \, a^{2} b^{2} d} \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. 853 vs.
\(2 (72) = 144\).
time = 0.46, size = 853, normalized size = 11.22 \begin {gather*} -\frac {2 \, a b^{2} d x \cosh \left (d x + c\right )^{4} + 8 \, a b^{2} d x \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + 2 \, a b^{2} d x \sinh \left (d x + c\right )^{4} + 2 \, a b^{2} d x - 4 \, {\left (a^{2} b + 2 \, a b^{2} + b^{3} - {\left (a b^{2} + 2 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )^{2} + 4 \, {\left (3 \, a b^{2} d x \cosh \left (d x + c\right )^{2} - a^{2} b - 2 \, a b^{2} - b^{3} + {\left (a b^{2} + 2 \, b^{3}\right )} d x\right )} \sinh \left (d x + c\right )^{2} + {\left ({\left (a^{3} - a b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{3} - a b^{2}\right )} \sinh \left (d x + c\right )^{4} + a^{3} - a b^{2} + 2 \, {\left (a^{3} + 2 \, a^{2} b - a b^{2} - 2 \, b^{3}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (a^{3} + 2 \, a^{2} b - a b^{2} - 2 \, b^{3} + 3 \, {\left (a^{3} - a b^{2}\right )} \cosh \left (d x + c\right )^{2}\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a^{3} - a b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{3} + 2 \, a^{2} b - a b^{2} - 2 \, b^{3}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, {\left (a \cosh \left (d x + c\right )^{2} + a \sinh \left (d x + c\right )^{2} + a + 2 \, b\right )}}{\cosh \left (d x + c\right )^{2} - 2 \, \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + \sinh \left (d x + c\right )^{2}}\right ) - 2 \, {\left (a^{3} \cosh \left (d x + c\right )^{4} + 4 \, a^{3} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} \sinh \left (d x + c\right )^{4} + a^{3} + 2 \, {\left (a^{3} + 2 \, a^{2} b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} \cosh \left (d x + c\right )^{2} + a^{3} + 2 \, a^{2} b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} \cosh \left (d x + c\right )^{3} + {\left (a^{3} + 2 \, a^{2} b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )} \log \left (\frac {2 \, \cosh \left (d x + c\right )}{\cosh \left (d x + c\right ) - \sinh \left (d x + c\right )}\right ) + 8 \, {\left (a b^{2} d x \cosh \left (d x + c\right )^{3} - {\left (a^{2} b + 2 \, a b^{2} + b^{3} - {\left (a b^{2} + 2 \, b^{3}\right )} d x\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )}{2 \, {\left (a^{3} b^{2} d \cosh \left (d x + c\right )^{4} + 4 \, a^{3} b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + a^{3} b^{2} d \sinh \left (d x + c\right )^{4} + a^{3} b^{2} d + 2 \, {\left (a^{3} b^{2} + 2 \, a^{2} b^{3}\right )} d \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{3} b^{2} d \cosh \left (d x + c\right )^{2} + {\left (a^{3} b^{2} + 2 \, a^{2} b^{3}\right )} d\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left (a^{3} b^{2} d \cosh \left (d x + c\right )^{3} + {\left (a^{3} b^{2} + 2 \, a^{2} b^{3}\right )} d \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right )\right )}} \end {gather*}
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 \begin {gather*} \int \frac {\tanh ^{5}{\left (c + d x \right )}}{\left (a + b \operatorname {sech}^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}
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 \begin {gather*} \text {Exception raised: TypeError} \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 )}^4\,{\mathrm {tanh}\left (c+d\,x\right )}^5}{{\left (a\,{\mathrm {cosh}\left (c+d\,x\right )}^2+b\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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