Optimal. Leaf size=37 \[ -\frac {\text {csch}^{n+2}(a+b x)}{b (n+2)}-\frac {\text {csch}^n(a+b x)}{b n} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.04, antiderivative size = 37, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {2621, 14} \[ -\frac {\text {csch}^n(a+b x)}{b n}-\frac {\text {csch}^{n+2}(a+b x)}{b (n+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 14
Rule 2621
Rubi steps
\begin {align*} \int \cosh ^3(a+b x) \text {csch}^{3+n}(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int x^{-1+n} \left (-1-x^2\right ) \, dx,x,\text {csch}(a+b x)\right )}{b}\\ &=\frac {\operatorname {Subst}\left (\int \left (-x^{-1+n}-x^{1+n}\right ) \, dx,x,\text {csch}(a+b x)\right )}{b}\\ &=-\frac {\text {csch}^n(a+b x)}{b n}-\frac {\text {csch}^{2+n}(a+b x)}{b (2+n)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.07, size = 34, normalized size = 0.92 \[ -\frac {\text {csch}^n(a+b x) \left (n \text {csch}^2(a+b x)+n+2\right )}{b n (n+2)} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.45, size = 216, normalized size = 5.84 \[ \frac {{\left ({\left (n + 2\right )} \cosh \left (b x + a\right )^{2} + {\left (n + 2\right )} \sinh \left (b x + a\right )^{2} + n - 2\right )} \cosh \left (n \log \left (\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )\right ) + {\left ({\left (n + 2\right )} \cosh \left (b x + a\right )^{2} + {\left (n + 2\right )} \sinh \left (b x + a\right )^{2} + n - 2\right )} \sinh \left (n \log \left (\frac {2 \, {\left (\cosh \left (b x + a\right ) + \sinh \left (b x + a\right )\right )}}{\cosh \left (b x + a\right )^{2} + 2 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right ) + \sinh \left (b x + a\right )^{2} - 1}\right )\right )}{b n^{2} - {\left (b n^{2} + 2 \, b n\right )} \cosh \left (b x + a\right )^{2} - {\left (b n^{2} + 2 \, b n\right )} \sinh \left (b x + a\right )^{2} + 2 \, b n} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \operatorname {csch}\left (b x + a\right )^{n} \coth \left (b x + a\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [C] time = 0.50, size = 499, normalized size = 13.49 \[ -\frac {\left (n \,{\mathrm e}^{4 b x +4 a}+2 \,{\mathrm e}^{4 b x +4 a}+2 \,{\mathrm e}^{2 b x +2 a} n -4 \,{\mathrm e}^{2 b x +2 a}+n +2\right ) {\mathrm e}^{\frac {n \left (-i \mathrm {csgn}\left (\frac {i}{\left (1+{\mathrm e}^{b x +a}\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right )^{3} \pi +i \mathrm {csgn}\left (\frac {i}{\left (1+{\mathrm e}^{b x +a}\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right )^{2} \mathrm {csgn}\left (\frac {i}{1+{\mathrm e}^{b x +a}}\right ) \pi +i \mathrm {csgn}\left (\frac {i}{\left (1+{\mathrm e}^{b x +a}\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right )^{2} \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{b x +a}-1}\right ) \pi -i \mathrm {csgn}\left (\frac {i}{\left (1+{\mathrm e}^{b x +a}\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right ) \mathrm {csgn}\left (\frac {i}{1+{\mathrm e}^{b x +a}}\right ) \mathrm {csgn}\left (\frac {i}{{\mathrm e}^{b x +a}-1}\right ) \pi +i \mathrm {csgn}\left (\frac {i}{\left (1+{\mathrm e}^{b x +a}\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{\left (1+{\mathrm e}^{b x +a}\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right )^{2} \pi -i \mathrm {csgn}\left (\frac {i}{\left (1+{\mathrm e}^{b x +a}\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right ) \mathrm {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{\left (1+{\mathrm e}^{b x +a}\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right ) \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \pi -i \mathrm {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{\left (1+{\mathrm e}^{b x +a}\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right )^{3} \pi +i \mathrm {csgn}\left (\frac {i {\mathrm e}^{b x +a}}{\left (1+{\mathrm e}^{b x +a}\right ) \left ({\mathrm e}^{b x +a}-1\right )}\right )^{2} \mathrm {csgn}\left (i {\mathrm e}^{b x +a}\right ) \pi +2 \ln \left ({\mathrm e}^{b x +a}\right )-2 \ln \left ({\mathrm e}^{b x +a}-1\right )+2 \ln \relax (2)-2 \ln \left (1+{\mathrm e}^{b x +a}\right )\right )}{2}}}{b n \left (n +2\right ) \left ({\mathrm e}^{2 b x +2 a}-1\right )^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.50, size = 414, normalized size = 11.19 \[ -\frac {2^{n} n e^{\left (-{\left (b x + a\right )} n - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right )\right )}}{{\left (n^{2} - 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} - \frac {{\left (2^{n + 1} n - 2^{n + 2}\right )} e^{\left (-{\left (b x + a\right )} n - 2 \, b x - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right ) - 2 \, a\right )}}{{\left (n^{2} - 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} - \frac {{\left (2^{n} n + 2^{n + 1}\right )} e^{\left (-{\left (b x + a\right )} n - 4 \, b x - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right ) - 4 \, a\right )}}{{\left (n^{2} - 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} - \frac {2^{n + 1} e^{\left (-{\left (b x + a\right )} n - n \log \left (e^{\left (-b x - a\right )} + 1\right ) - n \log \left (-e^{\left (-b x - a\right )} + 1\right )\right )}}{{\left (n^{2} - 2 \, {\left (n^{2} + 2 \, n\right )} e^{\left (-2 \, b x - 2 \, a\right )} + {\left (n^{2} + 2 \, n\right )} e^{\left (-4 \, b x - 4 \, a\right )} + 2 \, n\right )} b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.55, size = 100, normalized size = 2.70 \[ -\frac {{\left (\frac {1}{\frac {{\mathrm {e}}^{a+b\,x}}{2}-\frac {{\mathrm {e}}^{-a-b\,x}}{2}}\right )}^n\,\left (\frac {1}{b\,n}+\frac {{\mathrm {e}}^{4\,a+4\,b\,x}}{b\,n}+\frac {{\mathrm {e}}^{2\,a+2\,b\,x}\,\left (2\,n-4\right )}{b\,n\,\left (n+2\right )}\right )}{{\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \coth ^{3}{\left (a + b x \right )} \operatorname {csch}^{n}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________