Optimal. Leaf size=124 \[ -\frac {\coth ^2(c+d x)}{2 a^2 d}+\frac {\log (\cosh (c+d x))}{(a+b)^2 d}+\frac {(a-2 b) \log (\tanh (c+d x))}{a^3 d}+\frac {b^2 (3 a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^3 (a+b)^2 d}-\frac {b^2}{2 a^2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.14, antiderivative size = 124, 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 = {3751, 457, 90}
\begin {gather*} \frac {b^2 (3 a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^3 d (a+b)^2}+\frac {(a-2 b) \log (\tanh (c+d x))}{a^3 d}-\frac {b^2}{2 a^2 d (a+b) \left (a+b \tanh ^2(c+d x)\right )}-\frac {\coth ^2(c+d x)}{2 a^2 d}+\frac {\log (\cosh (c+d x))}{d (a+b)^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 90
Rule 457
Rule 3751
Rubi steps
\begin {align*} \int \frac {\coth ^3(c+d x)}{\left (a+b \tanh ^2(c+d x)\right )^2} \, dx &=\frac {\text {Subst}\left (\int \frac {1}{x^3 \left (1-x^2\right ) \left (a+b x^2\right )^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {1}{(1-x) x^2 (a+b x)^2} \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=\frac {\text {Subst}\left (\int \left (-\frac {1}{(a+b)^2 (-1+x)}+\frac {1}{a^2 x^2}+\frac {a-2 b}{a^3 x}+\frac {b^3}{a^2 (a+b) (a+b x)^2}+\frac {b^3 (3 a+2 b)}{a^3 (a+b)^2 (a+b x)}\right ) \, dx,x,\tanh ^2(c+d x)\right )}{2 d}\\ &=-\frac {\coth ^2(c+d x)}{2 a^2 d}+\frac {\log (\cosh (c+d x))}{(a+b)^2 d}+\frac {(a-2 b) \log (\tanh (c+d x))}{a^3 d}+\frac {b^2 (3 a+2 b) \log \left (a+b \tanh ^2(c+d x)\right )}{2 a^3 (a+b)^2 d}-\frac {b^2}{2 a^2 (a+b) d \left (a+b \tanh ^2(c+d x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.61, size = 93, normalized size = 0.75 \begin {gather*} \frac {-\frac {\coth ^2(c+d x)}{a^2}+\frac {b^3}{a^3 (a+b) \left (b+a \coth ^2(c+d x)\right )}+\frac {b^2 (3 a+2 b) \log \left (b+a \coth ^2(c+d x)\right )}{a^3 (a+b)^2}+\frac {2 \log (\sinh (c+d x))}{(a+b)^2}}{2 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 3.16, size = 226, normalized size = 1.82
method | result | size |
derivativedivides | \(\frac {-\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{2}}+\frac {b^{2} \left (\frac {2 b \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (3 a +2 b \right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )}{2}\right )}{\left (a +b \right )^{2} a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{\left (a +b \right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{\left (a +b \right )^{2}}-\frac {1}{8 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-8 b +4 a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}}{d}\) | \(226\) |
default | \(\frac {-\frac {\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 a^{2}}+\frac {b^{2} \left (\frac {2 b \left (a +b \right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a}+\frac {\left (3 a +2 b \right ) \ln \left (a \left (\tanh ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 a \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+4 b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+a \right )}{2}\right )}{\left (a +b \right )^{2} a^{3}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{\left (a +b \right )^{2}}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{\left (a +b \right )^{2}}-\frac {1}{8 a^{2} \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}+\frac {\left (-8 b +4 a \right ) \ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 a^{3}}}{d}\) | \(226\) |
risch | \(\frac {x}{a^{2}+2 a b +b^{2}}+\frac {4 b x}{a^{3}}+\frac {4 b c}{a^{3} d}-\frac {2 x}{a^{2}}-\frac {2 c}{a^{2} d}-\frac {6 b^{2} x}{a^{2} \left (a^{2}+2 a b +b^{2}\right )}-\frac {6 b^{2} c}{a^{2} d \left (a^{2}+2 a b +b^{2}\right )}-\frac {4 b^{3} x}{a^{3} \left (a^{2}+2 a b +b^{2}\right )}-\frac {4 b^{3} c}{a^{3} d \left (a^{2}+2 a b +b^{2}\right )}-\frac {2 \,{\mathrm e}^{2 d x +2 c} \left (a^{3} {\mathrm e}^{4 d x +4 c}+3 a^{2} b \,{\mathrm e}^{4 d x +4 c}+3 a \,b^{2} {\mathrm e}^{4 d x +4 c}+2 b^{3} {\mathrm e}^{4 d x +4 c}+2 a^{3} {\mathrm e}^{2 d x +2 c}+2 a^{2} b \,{\mathrm e}^{2 d x +2 c}-2 a \,b^{2} {\mathrm e}^{2 d x +2 c}-4 b^{3} {\mathrm e}^{2 d x +2 c}+a^{3}+3 a^{2} b +3 a \,b^{2}+2 b^{3}\right )}{a^{2} d \left ({\mathrm e}^{2 d x +2 c}-1\right )^{2} \left (a +b \right )^{2} \left (a \,{\mathrm e}^{4 d x +4 c}+b \,{\mathrm e}^{4 d x +4 c}+2 a \,{\mathrm e}^{2 d x +2 c}-2 b \,{\mathrm e}^{2 d x +2 c}+a +b \right )}-\frac {2 \ln \left ({\mathrm e}^{2 d x +2 c}-1\right ) b}{a^{3} d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}-1\right )}{a^{2} d}+\frac {3 b^{2} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{2 a^{2} d \left (a^{2}+2 a b +b^{2}\right )}+\frac {b^{3} \ln \left ({\mathrm e}^{4 d x +4 c}+\frac {2 \left (a -b \right ) {\mathrm e}^{2 d x +2 c}}{a +b}+1\right )}{a^{3} d \left (a^{2}+2 a b +b^{2}\right )}\) | \(518\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 402 vs.
\(2 (118) = 236\).
time = 0.28, size = 402, normalized size = 3.24 \begin {gather*} \frac {{\left (3 \, a b^{2} + 2 \, b^{3}\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^{5} + 2 \, a^{4} b + a^{3} b^{2}\right )} d} + \frac {d x + c}{{\left (a^{2} + 2 \, a b + b^{2}\right )} d} - \frac {2 \, {\left ({\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + 2 \, b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + 2 \, {\left (a^{3} + a^{2} b - a b^{2} - 2 \, b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} + {\left (a^{3} + 3 \, a^{2} b + 3 \, a b^{2} + 2 \, b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )}\right )}}{{\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3} - 4 \, {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} e^{\left (-2 \, d x - 2 \, c\right )} - 2 \, {\left (a^{5} - a^{4} b - 5 \, a^{3} b^{2} - 3 \, a^{2} b^{3}\right )} e^{\left (-4 \, d x - 4 \, c\right )} - 4 \, {\left (a^{4} b + 2 \, a^{3} b^{2} + a^{2} b^{3}\right )} e^{\left (-6 \, d x - 6 \, c\right )} + {\left (a^{5} + 3 \, a^{4} b + 3 \, a^{3} b^{2} + a^{2} b^{3}\right )} e^{\left (-8 \, d x - 8 \, c\right )}\right )} d} + \frac {{\left (a - 2 \, b\right )} \log \left (e^{\left (-d x - c\right )} + 1\right )}{a^{3} d} + \frac {{\left (a - 2 \, b\right )} \log \left (e^{\left (-d x - c\right )} - 1\right )}{a^{3} d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 3468 vs.
\(2 (118) = 236\).
time = 0.52, size = 3468, normalized size = 27.97 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\coth ^{3}{\left (c + d x \right )}}{\left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 323 vs.
\(2 (118) = 236\).
time = 0.52, size = 323, normalized size = 2.60 \begin {gather*} \frac {\frac {{\left (3 \, a b^{2} + 2 \, b^{3}\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^{5} + 2 \, a^{4} b + a^{3} b^{2}} - \frac {2 \, {\left (d x + c\right )}}{a^{2} + 2 \, a b + b^{2}} + \frac {2 \, {\left (a - 2 \, b\right )} \log \left ({\left | e^{\left (2 \, d x + 2 \, c\right )} - 1 \right |}\right )}{a^{3}} - \frac {4 \, {\left (\frac {{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + 2 \, a b^{3}\right )} e^{\left (6 \, d x + 6 \, c\right )}}{a + b} + \frac {2 \, {\left (a^{4} + a^{3} b - a^{2} b^{2} - 2 \, a b^{3}\right )} e^{\left (4 \, d x + 4 \, c\right )}}{a + b} + \frac {{\left (a^{4} + 3 \, a^{3} b + 3 \, a^{2} b^{2} + 2 \, a b^{3}\right )} e^{\left (2 \, d x + 2 \, c\right )}}{a + b}\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 )} a^{3} {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )}^{2}}}{2 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {coth}\left (c+d\,x\right )}^3}{{\left (b\,{\mathrm {tanh}\left (c+d\,x\right )}^2+a\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________