Optimal. Leaf size=75 \[ \frac {\sqrt {b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a^2 d}-\frac {x (a+2 b)}{2 a^2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 a d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.11, antiderivative size = 75, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {4132, 471, 522, 206, 208} \[ \frac {\sqrt {b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a^2 d}-\frac {x (a+2 b)}{2 a^2}+\frac {\sinh (c+d x) \cosh (c+d x)}{2 a d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 208
Rule 471
Rule 522
Rule 4132
Rubi steps
\begin {align*} \int \frac {\sinh ^2(c+d x)}{a+b \text {sech}^2(c+d x)} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^2}{\left (1-x^2\right )^2 \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{d}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 a d}-\frac {\operatorname {Subst}\left (\int \frac {a+b+b x^2}{\left (1-x^2\right ) \left (a+b-b x^2\right )} \, dx,x,\tanh (c+d x)\right )}{2 a d}\\ &=\frac {\cosh (c+d x) \sinh (c+d x)}{2 a d}+\frac {(b (a+b)) \operatorname {Subst}\left (\int \frac {1}{a+b-b x^2} \, dx,x,\tanh (c+d x)\right )}{a^2 d}-\frac {(a+2 b) \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\tanh (c+d x)\right )}{2 a^2 d}\\ &=-\frac {(a+2 b) x}{2 a^2}+\frac {\sqrt {b} \sqrt {a+b} \tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{a^2 d}+\frac {\cosh (c+d x) \sinh (c+d x)}{2 a d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [B] time = 1.00, size = 236, normalized size = 3.15 \[ \frac {\text {sech}^2(c+d x) (a \cosh (2 (c+d x))+a+2 b) \left (\frac {\frac {\left (a^2+8 a b+8 b^2\right ) (\cosh (2 c)-\sinh (2 c)) \tanh ^{-1}\left (\frac {(\cosh (2 c)-\sinh (2 c)) \text {sech}(d x) ((a+2 b) \sinh (d x)-a \sinh (2 c+d x))}{2 \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}\right )}{d \sqrt {a+b} \sqrt {b (\cosh (c)-\sinh (c))^4}}-4 x (a+2 b)+\frac {2 a \sinh (2 c) \cosh (2 d x)}{d}+\frac {2 a \cosh (2 c) \sinh (2 d x)}{d}}{a^2}-\frac {\tanh ^{-1}\left (\frac {\sqrt {b} \tanh (c+d x)}{\sqrt {a+b}}\right )}{\sqrt {b} d \sqrt {a+b}}\right )}{16 \left (a+b \text {sech}^2(c+d x)\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.48, size = 805, normalized size = 10.73 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 1.60, size = 132, normalized size = 1.76 \[ -\frac {\frac {4 \, {\left (d x + c\right )} {\left (a + 2 \, b\right )}}{a^{2}} - \frac {e^{\left (2 \, d x + 2 \, c\right )}}{a} - \frac {{\left (2 \, a e^{\left (2 \, d x + 2 \, c\right )} + 4 \, b e^{\left (2 \, d x + 2 \, c\right )} - a\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{a^{2}} - \frac {8 \, {\left (a b + b^{2}\right )} \arctan \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b}{2 \, \sqrt {-a b - b^{2}}}\right )}{\sqrt {-a b - b^{2}} a^{2}}}{8 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.32, size = 383, normalized size = 5.11 \[ \frac {1}{2 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}+\frac {1}{2 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 d a}+\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right ) b}{d \,a^{2}}-\frac {1}{2 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {1}{2 d a \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{2 d a}-\frac {\ln \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right ) b}{d \,a^{2}}-\frac {\sqrt {b}\, \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d a \sqrt {a +b}}+\frac {\sqrt {b}\, \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d a \sqrt {a +b}}-\frac {b^{\frac {3}{2}} \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d \,a^{2} \sqrt {a +b}}+\frac {b^{\frac {3}{2}} \ln \left (\sqrt {a +b}\, \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )+2 \sqrt {b}\, \tanh \left (\frac {d x}{2}+\frac {c}{2}\right )+\sqrt {a +b}\right )}{2 d \,a^{2} \sqrt {a +b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.42, size = 352, normalized size = 4.69 \[ -\frac {b \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{4 \, \sqrt {{\left (a + b\right )} b} a d} - \frac {d x + c}{2 \, a d} + \frac {e^{\left (2 \, d x + 2 \, c\right )}}{8 \, a d} - \frac {e^{\left (-2 \, d x - 2 \, c\right )}}{8 \, a d} - \frac {b \log \left (a e^{\left (4 \, d x + 4 \, c\right )} + 2 \, {\left (a + 2 \, b\right )} e^{\left (2 \, d x + 2 \, c\right )} + a\right )}{4 \, a^{2} d} + \frac {b \log \left (2 \, {\left (a + 2 \, b\right )} e^{\left (-2 \, d x - 2 \, c\right )} + a e^{\left (-4 \, d x - 4 \, c\right )} + a\right )}{4 \, a^{2} d} + \frac {{\left (a b + 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (2 \, d x + 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, \sqrt {{\left (a + b\right )} b} a^{2} d} - \frac {{\left (a b + 2 \, b^{2}\right )} \log \left (\frac {a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b - 2 \, \sqrt {{\left (a + b\right )} b}}{a e^{\left (-2 \, d x - 2 \, c\right )} + a + 2 \, b + 2 \, \sqrt {{\left (a + b\right )} b}}\right )}{8 \, \sqrt {{\left (a + b\right )} b} a^{2} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.03, size = 276, normalized size = 3.68 \[ \frac {{\mathrm {e}}^{2\,c+2\,d\,x}}{8\,a\,d}-\frac {{\mathrm {e}}^{-2\,c-2\,d\,x}}{8\,a\,d}-\frac {x\,\left (a+2\,b\right )}{2\,a^2}-\frac {\sqrt {b}\,\ln \left (2\,a\,b+a^2+a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}-2\,a\,\sqrt {b}\,\sqrt {a+b}-8\,b^{3/2}\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\sqrt {a+b}+8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}-4\,a\,\sqrt {b}\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\sqrt {a+b}\right )\,\sqrt {a+b}}{2\,a^2\,d}+\frac {\sqrt {b}\,\ln \left (2\,a\,b+a^2+a^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+8\,b^2\,{\mathrm {e}}^{2\,c+2\,d\,x}+2\,a\,\sqrt {b}\,\sqrt {a+b}+8\,b^{3/2}\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\sqrt {a+b}+8\,a\,b\,{\mathrm {e}}^{2\,c+2\,d\,x}+4\,a\,\sqrt {b}\,{\mathrm {e}}^{2\,c+2\,d\,x}\,\sqrt {a+b}\right )\,\sqrt {a+b}}{2\,a^2\,d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sinh ^{2}{\left (c + d x \right )}}{a + b \operatorname {sech}^{2}{\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________