Optimal. Leaf size=50 \[ \frac {(a+b) \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} d}-\frac {\tanh (c+d x)}{b d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {3756, 396, 211}
\begin {gather*} \frac {(a+b) \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} d}-\frac {\tanh (c+d x)}{b d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 396
Rule 3756
Rubi steps
\begin {align*} \int \frac {\text {sech}^4(c+d x)}{a+b \tanh ^2(c+d x)} \, dx &=\frac {\text {Subst}\left (\int \frac {1-x^2}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=-\frac {\tanh (c+d x)}{b d}+\frac {(a+b) \text {Subst}\left (\int \frac {1}{a+b x^2} \, dx,x,\tanh (c+d x)\right )}{b d}\\ &=\frac {(a+b) \tan ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} d}-\frac {\tanh (c+d x)}{b d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.11, size = 50, normalized size = 1.00 \begin {gather*} \frac {(a+b) \text {ArcTan}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a}}\right )}{\sqrt {a} b^{3/2} d}-\frac {\tanh (c+d x)}{b d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(196\) vs.
\(2(42)=84\).
time = 2.18, size = 197, normalized size = 3.94
method | result | size |
derivativedivides | \(\frac {\frac {2 a \left (a +b \right ) \left (-\frac {\left (a -\sqrt {b \left (a +b \right )}+b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (-a -\sqrt {b \left (a +b \right )}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{b}-\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\) | \(197\) |
default | \(\frac {\frac {2 a \left (a +b \right ) \left (-\frac {\left (a -\sqrt {b \left (a +b \right )}+b \right ) \arctanh \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}-a -2 b \right ) a}}+\frac {\left (-a -\sqrt {b \left (a +b \right )}-b \right ) \arctan \left (\frac {a \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{\sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{2 a \sqrt {b \left (a +b \right )}\, \sqrt {\left (2 \sqrt {b \left (a +b \right )}+a +2 b \right ) a}}\right )}{b}-\frac {2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )}{b \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}}{d}\) | \(197\) |
risch | \(\frac {2}{b d \left (1+{\mathrm e}^{2 d x +2 c}\right )}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a}{2 \sqrt {-a b}\, d b}-\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}-2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{2 \sqrt {-a b}\, d}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right ) a}{2 \sqrt {-a b}\, d b}+\frac {\ln \left ({\mathrm e}^{2 d x +2 c}+\frac {a \sqrt {-a b}-b \sqrt {-a b}+2 a b}{\left (a +b \right ) \sqrt {-a b}}\right )}{2 \sqrt {-a b}\, d}\) | \(255\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.52, size = 63, normalized size = 1.26 \begin {gather*} -\frac {{\left (a + b\right )} \arctan \left (\frac {{\left (a + b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} b d} - \frac {2}{{\left (b e^{\left (-2 \, d x - 2 \, c\right )} + b\right )} 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. 172 vs.
\(2 (42) = 84\).
time = 0.39, size = 649, normalized size = 12.98 \begin {gather*} \left [-\frac {{\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 {-a b} \log \left (\frac {{\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a^{2} + 2 \, a b + b^{2}\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{2} + a^{2} - b^{2}\right )} \sinh \left (d x + c\right )^{2} + a^{2} - 6 \, a b + b^{2} + 4 \, {\left ({\left (a^{2} + 2 \, a b + b^{2}\right )} \cosh \left (d x + c\right )^{3} + {\left (a^{2} - b^{2}\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) - 4 \, {\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 {-a b}}{{\left (a + b\right )} \cosh \left (d x + c\right )^{4} + 4 \, {\left (a + b\right )} \cosh \left (d x + c\right ) \sinh \left (d x + c\right )^{3} + {\left (a + b\right )} \sinh \left (d x + c\right )^{4} + 2 \, {\left (a - b\right )} \cosh \left (d x + c\right )^{2} + 2 \, {\left (3 \, {\left (a + b\right )} \cosh \left (d x + c\right )^{2} + a - b\right )} \sinh \left (d x + c\right )^{2} + 4 \, {\left ({\left (a + b\right )} \cosh \left (d x + c\right )^{3} + {\left (a - b\right )} \cosh \left (d x + c\right )\right )} \sinh \left (d x + c\right ) + a + b}\right ) - 4 \, a b}{2 \, {\left (a b^{2} d \cosh \left (d x + c\right )^{2} + 2 \, a b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b^{2} d \sinh \left (d x + c\right )^{2} + a b^{2} d\right )}}, \frac {{\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 {a b} \arctan \left (\frac {{\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 {a b}}{2 \, a b}\right ) + 2 \, a b}{a b^{2} d \cosh \left (d x + c\right )^{2} + 2 \, a b^{2} d \cosh \left (d x + c\right ) \sinh \left (d x + c\right ) + a b^{2} d \sinh \left (d x + c\right )^{2} + a b^{2} d}\right ] \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 {\operatorname {sech}^{4}{\left (c + d x \right )}}{a + b \tanh ^{2}{\left (c + d x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.59, size = 70, normalized size = 1.40 \begin {gather*} \frac {\frac {{\left (a + b\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + b e^{\left (2 \, d x + 2 \, c\right )} + a - b}{2 \, \sqrt {a b}}\right )}{\sqrt {a b} b} + \frac {2}{b {\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}}}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.61, size = 176, normalized size = 3.52 \begin {gather*} \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}}{b}-\frac {2\,\left (a\,d+b\,d+a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {-a}\,b^{3/2}\,d}\right )\,\left (a+b\right )}{2\,\sqrt {-a}\,b^{3/2}\,d}-\frac {\ln \left (\frac {2\,\left (a\,d+b\,d+a\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}-b\,d\,{\mathrm {e}}^{2\,c+2\,d\,x}\right )}{\sqrt {-a}\,b^{3/2}\,d}-\frac {4\,{\mathrm {e}}^{2\,c+2\,d\,x}}{b}\right )\,\left (a+b\right )}{2\,\sqrt {-a}\,b^{3/2}\,d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________