Optimal. Leaf size=38 \[ \frac {\tanh ^3(a+b x)}{3 b}-\frac {2 \tanh (a+b x)}{b}-\frac {\coth (a+b x)}{b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {2620, 270} \[ \frac {\tanh ^3(a+b x)}{3 b}-\frac {2 \tanh (a+b x)}{b}-\frac {\coth (a+b x)}{b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 270
Rule 2620
Rubi steps
\begin {align*} \int \text {csch}^2(a+b x) \text {sech}^4(a+b x) \, dx &=\frac {i \operatorname {Subst}\left (\int \frac {\left (1+x^2\right )^2}{x^2} \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=\frac {i \operatorname {Subst}\left (\int \left (2+\frac {1}{x^2}+x^2\right ) \, dx,x,i \tanh (a+b x)\right )}{b}\\ &=-\frac {\coth (a+b x)}{b}-\frac {2 \tanh (a+b x)}{b}+\frac {\tanh ^3(a+b x)}{3 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 46, normalized size = 1.21 \[ -\frac {5 \tanh (a+b x)}{3 b}-\frac {\coth (a+b x)}{b}-\frac {\tanh (a+b x) \text {sech}^2(a+b x)}{3 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.40, size = 230, normalized size = 6.05 \[ -\frac {16 \, {\left (3 \, \cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{3 \, {\left (b \cosh \left (b x + a\right )^{7} + 7 \, b \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{6} + b \sinh \left (b x + a\right )^{7} + 2 \, b \cosh \left (b x + a\right )^{5} + {\left (21 \, b \cosh \left (b x + a\right )^{2} + 2 \, b\right )} \sinh \left (b x + a\right )^{5} + 5 \, {\left (7 \, b \cosh \left (b x + a\right )^{3} + 2 \, b \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{4} + 5 \, {\left (7 \, b \cosh \left (b x + a\right )^{4} + 4 \, b \cosh \left (b x + a\right )^{2}\right )} \sinh \left (b x + a\right )^{3} + {\left (21 \, b \cosh \left (b x + a\right )^{5} + 20 \, b \cosh \left (b x + a\right )^{3}\right )} \sinh \left (b x + a\right )^{2} - 3 \, b \cosh \left (b x + a\right ) + {\left (7 \, b \cosh \left (b x + a\right )^{6} + 10 \, b \cosh \left (b x + a\right )^{4} - b\right )} \sinh \left (b x + a\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.15, size = 60, normalized size = 1.58 \[ -\frac {2 \, {\left (\frac {3}{e^{\left (2 \, b x + 2 \, a\right )} - 1} - \frac {3 \, e^{\left (4 \, b x + 4 \, a\right )} + 12 \, e^{\left (2 \, b x + 2 \, a\right )} + 5}{{\left (e^{\left (2 \, b x + 2 \, a\right )} + 1\right )}^{3}}\right )}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.30, size = 44, normalized size = 1.16 \[ \frac {-\frac {1}{\sinh \left (b x +a \right ) \cosh \left (b x +a \right )^{3}}-4 \left (\frac {2}{3}+\frac {\mathrm {sech}\left (b x +a \right )^{2}}{3}\right ) \tanh \left (b x +a \right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.31, size = 94, normalized size = 2.47 \[ -\frac {32 \, e^{\left (-2 \, b x - 2 \, a\right )}}{3 \, b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )} + 1\right )}} - \frac {16}{3 \, b {\left (2 \, e^{\left (-2 \, b x - 2 \, a\right )} - 2 \, e^{\left (-6 \, b x - 6 \, a\right )} - e^{\left (-8 \, b x - 8 \, a\right )} + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.52, size = 152, normalized size = 4.00 \[ \frac {\frac {2}{3\,b}+\frac {4\,{\mathrm {e}}^{2\,a+2\,b\,x}}{b}+\frac {2\,{\mathrm {e}}^{4\,a+4\,b\,x}}{3\,b}}{3\,{\mathrm {e}}^{2\,a+2\,b\,x}+3\,{\mathrm {e}}^{4\,a+4\,b\,x}+{\mathrm {e}}^{6\,a+6\,b\,x}+1}+\frac {\frac {2}{b}+\frac {2\,{\mathrm {e}}^{2\,a+2\,b\,x}}{3\,b}}{2\,{\mathrm {e}}^{2\,a+2\,b\,x}+{\mathrm {e}}^{4\,a+4\,b\,x}+1}-\frac {2}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )}+\frac {2}{3\,b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}+1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}^{2}{\left (a + b x \right )} \operatorname {sech}^{4}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________