Optimal. Leaf size=148 \[ \frac {x \cosh (a+b x)}{4 b^2}-\frac {x \cosh (3 a+3 b x)}{72 b^2}-\frac {x \cosh (5 a+5 b x)}{200 b^2}-\frac {\sinh (a+b x)}{4 b^3}-\frac {x^2 \sinh (a+b x)}{8 b}+\frac {\sinh (3 a+3 b x)}{216 b^3}+\frac {x^2 \sinh (3 a+3 b x)}{48 b}+\frac {\sinh (5 a+5 b x)}{1000 b^3}+\frac {x^2 \sinh (5 a+5 b x)}{80 b} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.13, antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps
used = 11, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {5556, 3377,
2717} \begin {gather*} -\frac {\sinh (a+b x)}{4 b^3}+\frac {\sinh (3 a+3 b x)}{216 b^3}+\frac {\sinh (5 a+5 b x)}{1000 b^3}+\frac {x \cosh (a+b x)}{4 b^2}-\frac {x \cosh (3 a+3 b x)}{72 b^2}-\frac {x \cosh (5 a+5 b x)}{200 b^2}-\frac {x^2 \sinh (a+b x)}{8 b}+\frac {x^2 \sinh (3 a+3 b x)}{48 b}+\frac {x^2 \sinh (5 a+5 b x)}{80 b} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2717
Rule 3377
Rule 5556
Rubi steps
\begin {align*} \int x^2 \cosh ^3(a+b x) \sinh ^2(a+b x) \, dx &=\int \left (-\frac {1}{8} x^2 \cosh (a+b x)+\frac {1}{16} x^2 \cosh (3 a+3 b x)+\frac {1}{16} x^2 \cosh (5 a+5 b x)\right ) \, dx\\ &=\frac {1}{16} \int x^2 \cosh (3 a+3 b x) \, dx+\frac {1}{16} \int x^2 \cosh (5 a+5 b x) \, dx-\frac {1}{8} \int x^2 \cosh (a+b x) \, dx\\ &=-\frac {x^2 \sinh (a+b x)}{8 b}+\frac {x^2 \sinh (3 a+3 b x)}{48 b}+\frac {x^2 \sinh (5 a+5 b x)}{80 b}-\frac {\int x \sinh (5 a+5 b x) \, dx}{40 b}-\frac {\int x \sinh (3 a+3 b x) \, dx}{24 b}+\frac {\int x \sinh (a+b x) \, dx}{4 b}\\ &=\frac {x \cosh (a+b x)}{4 b^2}-\frac {x \cosh (3 a+3 b x)}{72 b^2}-\frac {x \cosh (5 a+5 b x)}{200 b^2}-\frac {x^2 \sinh (a+b x)}{8 b}+\frac {x^2 \sinh (3 a+3 b x)}{48 b}+\frac {x^2 \sinh (5 a+5 b x)}{80 b}+\frac {\int \cosh (5 a+5 b x) \, dx}{200 b^2}+\frac {\int \cosh (3 a+3 b x) \, dx}{72 b^2}-\frac {\int \cosh (a+b x) \, dx}{4 b^2}\\ &=\frac {x \cosh (a+b x)}{4 b^2}-\frac {x \cosh (3 a+3 b x)}{72 b^2}-\frac {x \cosh (5 a+5 b x)}{200 b^2}-\frac {\sinh (a+b x)}{4 b^3}-\frac {x^2 \sinh (a+b x)}{8 b}+\frac {\sinh (3 a+3 b x)}{216 b^3}+\frac {x^2 \sinh (3 a+3 b x)}{48 b}+\frac {\sinh (5 a+5 b x)}{1000 b^3}+\frac {x^2 \sinh (5 a+5 b x)}{80 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.21, size = 105, normalized size = 0.71 \begin {gather*} \frac {-6750 \left (-2 b x \cosh (a+b x)+\left (2+b^2 x^2\right ) \sinh (a+b x)\right )+125 \left (-6 b x \cosh (3 (a+b x))+\left (2+9 b^2 x^2\right ) \sinh (3 (a+b x))\right )+27 \left (-10 b x \cosh (5 (a+b x))+\left (2+25 b^2 x^2\right ) \sinh (5 (a+b x))\right )}{54000 b^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(282\) vs.
\(2(130)=260\).
time = 1.96, size = 283, normalized size = 1.91
method | result | size |
risch | \(\frac {\left (25 b^{2} x^{2}-10 b x +2\right ) {\mathrm e}^{5 b x +5 a}}{4000 b^{3}}+\frac {\left (9 b^{2} x^{2}-6 b x +2\right ) {\mathrm e}^{3 b x +3 a}}{864 b^{3}}-\frac {\left (b^{2} x^{2}-2 b x +2\right ) {\mathrm e}^{b x +a}}{16 b^{3}}+\frac {\left (b^{2} x^{2}+2 b x +2\right ) {\mathrm e}^{-b x -a}}{16 b^{3}}-\frac {\left (9 b^{2} x^{2}+6 b x +2\right ) {\mathrm e}^{-3 b x -3 a}}{864 b^{3}}-\frac {\left (25 b^{2} x^{2}+10 b x +2\right ) {\mathrm e}^{-5 b x -5 a}}{4000 b^{3}}\) | \(165\) |
default | \(-\frac {\left (b x +a \right )^{2} \sinh \left (b x +a \right )-2 \left (b x +a \right ) \cosh \left (b x +a \right )+2 \sinh \left (b x +a \right )-2 a \left (\left (b x +a \right ) \sinh \left (b x +a \right )-\cosh \left (b x +a \right )\right )+a^{2} \sinh \left (b x +a \right )}{8 b^{3}}+\frac {\left (3 b x +3 a \right )^{2} \sinh \left (3 b x +3 a \right )-2 \left (3 b x +3 a \right ) \cosh \left (3 b x +3 a \right )+2 \sinh \left (3 b x +3 a \right )-6 a \left (\left (3 b x +3 a \right ) \sinh \left (3 b x +3 a \right )-\cosh \left (3 b x +3 a \right )\right )+9 a^{2} \sinh \left (3 b x +3 a \right )}{432 b^{3}}+\frac {\left (5 b x +5 a \right )^{2} \sinh \left (5 b x +5 a \right )-2 \left (5 b x +5 a \right ) \cosh \left (5 b x +5 a \right )+2 \sinh \left (5 b x +5 a \right )-10 a \left (\left (5 b x +5 a \right ) \sinh \left (5 b x +5 a \right )-\cosh \left (5 b x +5 a \right )\right )+25 a^{2} \sinh \left (5 b x +5 a \right )}{2000 b^{3}}\) | \(283\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.27, size = 187, normalized size = 1.26 \begin {gather*} \frac {{\left (25 \, b^{2} x^{2} e^{\left (5 \, a\right )} - 10 \, b x e^{\left (5 \, a\right )} + 2 \, e^{\left (5 \, a\right )}\right )} e^{\left (5 \, b x\right )}}{4000 \, b^{3}} + \frac {{\left (9 \, b^{2} x^{2} e^{\left (3 \, a\right )} - 6 \, b x e^{\left (3 \, a\right )} + 2 \, e^{\left (3 \, a\right )}\right )} e^{\left (3 \, b x\right )}}{864 \, b^{3}} - \frac {{\left (b^{2} x^{2} e^{a} - 2 \, b x e^{a} + 2 \, e^{a}\right )} e^{\left (b x\right )}}{16 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{16 \, b^{3}} - \frac {{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{864 \, b^{3}} - \frac {{\left (25 \, b^{2} x^{2} + 10 \, b x + 2\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{4000 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.37, size = 209, normalized size = 1.41 \begin {gather*} -\frac {270 \, b x \cosh \left (b x + a\right )^{5} + 1350 \, b x \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} - 27 \, {\left (25 \, b^{2} x^{2} + 2\right )} \sinh \left (b x + a\right )^{5} + 750 \, b x \cosh \left (b x + a\right )^{3} - 5 \, {\left (225 \, b^{2} x^{2} + 54 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} + 50\right )} \sinh \left (b x + a\right )^{3} - 13500 \, b x \cosh \left (b x + a\right ) + 450 \, {\left (6 \, b x \cosh \left (b x + a\right )^{3} + 5 \, b x \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 15 \, {\left (9 \, {\left (25 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{4} - 450 \, b^{2} x^{2} + 25 \, {\left (9 \, b^{2} x^{2} + 2\right )} \cosh \left (b x + a\right )^{2} - 900\right )} \sinh \left (b x + a\right )}{54000 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.65, size = 182, normalized size = 1.23 \begin {gather*} \begin {cases} - \frac {2 x^{2} \sinh ^{5}{\left (a + b x \right )}}{15 b} + \frac {x^{2} \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{3 b} + \frac {4 x \sinh ^{4}{\left (a + b x \right )} \cosh {\left (a + b x \right )}}{15 b^{2}} - \frac {26 x \sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{45 b^{2}} + \frac {52 x \cosh ^{5}{\left (a + b x \right )}}{225 b^{2}} - \frac {856 \sinh ^{5}{\left (a + b x \right )}}{3375 b^{3}} + \frac {338 \sinh ^{3}{\left (a + b x \right )} \cosh ^{2}{\left (a + b x \right )}}{675 b^{3}} - \frac {52 \sinh {\left (a + b x \right )} \cosh ^{4}{\left (a + b x \right )}}{225 b^{3}} & \text {for}\: b \neq 0 \\\frac {x^{3} \sinh ^{2}{\left (a \right )} \cosh ^{3}{\left (a \right )}}{3} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.40, size = 164, normalized size = 1.11 \begin {gather*} \frac {{\left (25 \, b^{2} x^{2} - 10 \, b x + 2\right )} e^{\left (5 \, b x + 5 \, a\right )}}{4000 \, b^{3}} + \frac {{\left (9 \, b^{2} x^{2} - 6 \, b x + 2\right )} e^{\left (3 \, b x + 3 \, a\right )}}{864 \, b^{3}} - \frac {{\left (b^{2} x^{2} - 2 \, b x + 2\right )} e^{\left (b x + a\right )}}{16 \, b^{3}} + \frac {{\left (b^{2} x^{2} + 2 \, b x + 2\right )} e^{\left (-b x - a\right )}}{16 \, b^{3}} - \frac {{\left (9 \, b^{2} x^{2} + 6 \, b x + 2\right )} e^{\left (-3 \, b x - 3 \, a\right )}}{864 \, b^{3}} - \frac {{\left (25 \, b^{2} x^{2} + 10 \, b x + 2\right )} e^{\left (-5 \, b x - 5 \, a\right )}}{4000 \, b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.83, size = 123, normalized size = 0.83 \begin {gather*} \frac {\frac {x^2\,\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{48}+\frac {x^2\,\mathrm {sinh}\left (5\,a+5\,b\,x\right )}{80}-\frac {x^2\,\mathrm {sinh}\left (a+b\,x\right )}{8}}{b}-\frac {\mathrm {sinh}\left (a+b\,x\right )}{4\,b^3}-\frac {\frac {x\,\mathrm {cosh}\left (3\,a+3\,b\,x\right )}{72}-\frac {x\,\mathrm {cosh}\left (a+b\,x\right )}{4}+\frac {x\,\mathrm {cosh}\left (5\,a+5\,b\,x\right )}{200}}{b^2}+\frac {\mathrm {sinh}\left (3\,a+3\,b\,x\right )}{216\,b^3}+\frac {\mathrm {sinh}\left (5\,a+5\,b\,x\right )}{1000\,b^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________