Optimal. Leaf size=51 \[ -\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b (2 a+b) \text {sech}(c+d x)}{d}+\frac {b^2 \text {sech}^3(c+d x)}{3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.06, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {3664, 390, 207} \[ -\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b (2 a+b) \text {sech}(c+d x)}{d}+\frac {b^2 \text {sech}^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 207
Rule 390
Rule 3664
Rubi steps
\begin {align*} \int \text {csch}(c+d x) \left (a+b \tanh ^2(c+d x)\right )^2 \, dx &=\frac {\operatorname {Subst}\left (\int \frac {\left (a+b-b x^2\right )^2}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=\frac {\operatorname {Subst}\left (\int \left (-b (2 a+b)+b^2 x^2+\frac {a^2}{-1+x^2}\right ) \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=-\frac {b (2 a+b) \text {sech}(c+d x)}{d}+\frac {b^2 \text {sech}^3(c+d x)}{3 d}+\frac {a^2 \operatorname {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\text {sech}(c+d x)\right )}{d}\\ &=-\frac {a^2 \tanh ^{-1}(\cosh (c+d x))}{d}-\frac {b (2 a+b) \text {sech}(c+d x)}{d}+\frac {b^2 \text {sech}^3(c+d x)}{3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.17, size = 50, normalized size = 0.98 \[ \frac {3 a^2 \log \left (\tanh \left (\frac {1}{2} (c+d x)\right )\right )-3 b (2 a+b) \text {sech}(c+d x)+b^2 \text {sech}^3(c+d x)}{3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.90, size = 890, normalized size = 17.45 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [B] time = 0.23, size = 126, normalized size = 2.47 \[ -\frac {3 \, a^{2} \log \left (e^{\left (d x + c\right )} + 1\right ) - 3 \, a^{2} \log \left ({\left | e^{\left (d x + c\right )} - 1 \right |}\right ) + \frac {2 \, {\left (6 \, a b e^{\left (5 \, d x + 5 \, c\right )} + 3 \, b^{2} e^{\left (5 \, d x + 5 \, c\right )} + 12 \, a b e^{\left (3 \, d x + 3 \, c\right )} + 2 \, b^{2} e^{\left (3 \, d x + 3 \, c\right )} + 6 \, a b e^{\left (d x + c\right )} + 3 \, b^{2} e^{\left (d x + c\right )}\right )}}{{\left (e^{\left (2 \, d x + 2 \, c\right )} + 1\right )}^{3}}}{3 \, d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.24, size = 63, normalized size = 1.24 \[ \frac {-2 a^{2} \arctanh \left ({\mathrm e}^{d x +c}\right )-\frac {2 a b}{\cosh \left (d x +c \right )}+b^{2} \left (-\frac {\sinh ^{2}\left (d x +c \right )}{\cosh \left (d x +c \right )^{3}}-\frac {2}{3 \cosh \left (d x +c \right )^{3}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.35, size = 196, normalized size = 3.84 \[ -\frac {2}{3} \, b^{2} {\left (\frac {3 \, e^{\left (-d x - c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {2 \, e^{\left (-3 \, d x - 3 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}} + \frac {3 \, e^{\left (-5 \, d x - 5 \, c\right )}}{d {\left (3 \, e^{\left (-2 \, d x - 2 \, c\right )} + 3 \, e^{\left (-4 \, d x - 4 \, c\right )} + e^{\left (-6 \, d x - 6 \, c\right )} + 1\right )}}\right )} + \frac {a^{2} \log \left (\tanh \left (\frac {1}{2} \, d x + \frac {1}{2} \, c\right )\right )}{d} - \frac {4 \, a b}{d {\left (e^{\left (d x + c\right )} + e^{\left (-d x - c\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 0.15, size = 160, normalized size = 3.14 \[ \frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (2\,{\mathrm {e}}^{2\,c+2\,d\,x}+{\mathrm {e}}^{4\,c+4\,d\,x}+1\right )}-\frac {8\,b^2\,{\mathrm {e}}^{c+d\,x}}{3\,d\,\left (3\,{\mathrm {e}}^{2\,c+2\,d\,x}+3\,{\mathrm {e}}^{4\,c+4\,d\,x}+{\mathrm {e}}^{6\,c+6\,d\,x}+1\right )}-\frac {2\,{\mathrm {e}}^{c+d\,x}\,\left (b^2+2\,a\,b\right )}{d\,\left ({\mathrm {e}}^{2\,c+2\,d\,x}+1\right )}-\frac {2\,\mathrm {atan}\left (\frac {a^2\,{\mathrm {e}}^{d\,x}\,{\mathrm {e}}^c\,\sqrt {-d^2}}{d\,\sqrt {a^4}}\right )\,\sqrt {a^4}}{\sqrt {-d^2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \left (a + b \tanh ^{2}{\left (c + d x \right )}\right )^{2} \operatorname {csch}{\left (c + d x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________