Optimal. Leaf size=119 \[ \frac {a \left (a^2+3 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )^2}+\frac {\text {sech}^2(c+d x) (a \sinh (c+d x)+b)}{2 d \left (a^2+b^2\right )}+\frac {b^3 \log (a+b \sinh (c+d x))}{d \left (a^2+b^2\right )^2}-\frac {b^3 \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.14, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {2668, 741, 801, 635, 203, 260} \[ \frac {b^3 \log (a+b \sinh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {a \left (a^2+3 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 d \left (a^2+b^2\right )^2}-\frac {b^3 \log (\cosh (c+d x))}{d \left (a^2+b^2\right )^2}+\frac {\text {sech}^2(c+d x) (a \sinh (c+d x)+b)}{2 d \left (a^2+b^2\right )} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 203
Rule 260
Rule 635
Rule 741
Rule 801
Rule 2668
Rubi steps
\begin {align*} \int \frac {\text {sech}^3(c+d x)}{a+b \sinh (c+d x)} \, dx &=\frac {b^3 \operatorname {Subst}\left (\int \frac {1}{(a+x) \left (-b^2-x^2\right )^2} \, dx,x,b \sinh (c+d x)\right )}{d}\\ &=\frac {\text {sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {b \operatorname {Subst}\left (\int \frac {a^2+2 b^2+a x}{(a+x) \left (-b^2-x^2\right )} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right ) d}\\ &=\frac {\text {sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {b \operatorname {Subst}\left (\int \left (-\frac {2 b^2}{\left (a^2+b^2\right ) (a+x)}+\frac {-a^3-3 a b^2+2 b^2 x}{\left (a^2+b^2\right ) \left (b^2+x^2\right )}\right ) \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right ) d}\\ &=\frac {b^3 \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {\text {sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {b \operatorname {Subst}\left (\int \frac {-a^3-3 a b^2+2 b^2 x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=\frac {b^3 \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {\text {sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}-\frac {b^3 \operatorname {Subst}\left (\int \frac {x}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{\left (a^2+b^2\right )^2 d}+\frac {\left (a b \left (a^2+3 b^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2+x^2} \, dx,x,b \sinh (c+d x)\right )}{2 \left (a^2+b^2\right )^2 d}\\ &=\frac {a \left (a^2+3 b^2\right ) \tan ^{-1}(\sinh (c+d x))}{2 \left (a^2+b^2\right )^2 d}-\frac {b^3 \log (\cosh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {b^3 \log (a+b \sinh (c+d x))}{\left (a^2+b^2\right )^2 d}+\frac {\text {sech}^2(c+d x) (b+a \sinh (c+d x))}{2 \left (a^2+b^2\right ) d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.18, size = 104, normalized size = 0.87 \[ \frac {b \left (a^2+b^2\right ) \text {sech}^2(c+d x)+2 a \left (a^2+3 b^2\right ) \tan ^{-1}\left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )+a \left (a^2+b^2\right ) \tanh (c+d x) \text {sech}(c+d x)+2 b^3 (\log (a+b \sinh (c+d x))-\log (\cosh (c+d x)))}{2 d \left (a^2+b^2\right )^2} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.50, size = 893, normalized size = 7.50 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.59, size = 282, normalized size = 2.37 \[ \frac {\frac {4 \, b^{4} \log \left ({\left | b {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a \right |}\right )}{a^{4} b + 2 \, a^{2} b^{3} + b^{5}} - \frac {2 \, b^{3} \log \left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {{\left (\pi + 2 \, \arctan \left (\frac {1}{2} \, {\left (e^{\left (2 \, d x + 2 \, c\right )} - 1\right )} e^{\left (-d x - c\right )}\right )\right )} {\left (a^{3} + 3 \, a b^{2}\right )}}{a^{4} + 2 \, a^{2} b^{2} + b^{4}} + \frac {2 \, {\left (b^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 2 \, a^{3} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 2 \, a b^{2} {\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )} + 4 \, a^{2} b + 8 \, b^{3}\right )}}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} {\left ({\left (e^{\left (d x + c\right )} - e^{\left (-d x - c\right )}\right )}^{2} + 4\right )}}}{4 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.00, size = 468, normalized size = 3.93 \[ \frac {b^{3} \ln \left (\left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a -2 \tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) b -a \right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}-\frac {\left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {\left (\tanh ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{2} b}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right ) b^{3}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a^{3}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}+\frac {\tanh \left (\frac {d x}{2}+\frac {c}{2}\right ) a \,b^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right ) \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {b^{3} \ln \left (\tanh ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {\arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a^{3}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )}+\frac {3 \arctan \left (\tanh \left (\frac {d x}{2}+\frac {c}{2}\right )\right ) a \,b^{2}}{d \left (a^{4}+2 a^{2} b^{2}+b^{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.54, size = 216, normalized size = 1.82 \[ \frac {b^{3} \log \left (-2 \, a e^{\left (-d x - c\right )} + b e^{\left (-2 \, d x - 2 \, c\right )} - b\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {b^{3} \log \left (e^{\left (-2 \, d x - 2 \, c\right )} + 1\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} - \frac {{\left (a^{3} + 3 \, a b^{2}\right )} \arctan \left (e^{\left (-d x - c\right )}\right )}{{\left (a^{4} + 2 \, a^{2} b^{2} + b^{4}\right )} d} + \frac {a e^{\left (-d x - c\right )} + 2 \, b e^{\left (-2 \, d x - 2 \, c\right )} - a e^{\left (-3 \, d x - 3 \, c\right )}}{{\left (a^{2} + b^{2} + 2 \, {\left (a^{2} + b^{2}\right )} e^{\left (-2 \, d x - 2 \, c\right )} + {\left (a^{2} + b^{2}\right )} e^{\left (-4 \, d x - 4 \, c\right )}\right )} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 2.20, size = 381, normalized size = 3.20 \[ \frac {\frac {2\,\left (a^2\,b+b^3\right )}{d\,{\left (a^2+b^2\right )}^2}+\frac {{\mathrm {e}}^{c+d\,x}\,\left (a^3+a\,b^2\right )}{d\,{\left (a^2+b^2\right )}^2}}{{\mathrm {e}}^{2\,c+2\,d\,x}+1}-\frac {\frac {2\,b}{d\,\left (a^2+b^2\right )}+\frac {2\,a\,{\mathrm {e}}^{c+d\,x}}{d\,\left (a^2+b^2\right )}}{2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1}-\frac {\ln \left ({\mathrm {e}}^{c+d\,x}+1{}\mathrm {i}\right )\,\left (2\,b+a\,1{}\mathrm {i}\right )}{2\,\left (-d\,a^2+2{}\mathrm {i}\,d\,a\,b+d\,b^2\right )}-\frac {\ln \left (1+{\mathrm {e}}^{c+d\,x}\,1{}\mathrm {i}\right )\,\left (a+b\,2{}\mathrm {i}\right )}{2\,\left (-1{}\mathrm {i}\,d\,a^2+2\,d\,a\,b+1{}\mathrm {i}\,d\,b^2\right )}+\frac {b^3\,\ln \left (2\,a^7\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c-16\,b^7-9\,a^2\,b^5-6\,a^4\,b^3-a^6\,b+16\,b^7\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+a^6\,b\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+18\,a^3\,b^4\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+12\,a^5\,b^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c+9\,a^2\,b^5\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+6\,a^4\,b^3\,{\mathrm {e}}^{2\,c}\,{\mathrm {e}}^{2\,d\,x}+32\,a\,b^6\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\right )}{d\,a^4+2\,d\,a^2\,b^2+d\,b^4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {sech}^{3}{\left (c + d x \right )}}{a + b \sinh {\left (c + d x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________