Optimal. Leaf size=66 \[ \frac {5 \cosh ^3(a+b x)}{6 b}+\frac {5 \cosh (a+b x)}{2 b}-\frac {\cosh ^3(a+b x) \coth ^2(a+b x)}{2 b}-\frac {5 \tanh ^{-1}(\cosh (a+b x))}{2 b} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {2592, 288, 302, 206} \[ \frac {5 \cosh ^3(a+b x)}{6 b}+\frac {5 \cosh (a+b x)}{2 b}-\frac {\cosh ^3(a+b x) \coth ^2(a+b x)}{2 b}-\frac {5 \tanh ^{-1}(\cosh (a+b x))}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 206
Rule 288
Rule 302
Rule 2592
Rubi steps
\begin {align*} \int \cosh ^3(a+b x) \coth ^3(a+b x) \, dx &=\frac {\operatorname {Subst}\left (\int \frac {x^6}{\left (1-x^2\right )^2} \, dx,x,\cosh (a+b x)\right )}{b}\\ &=-\frac {\cosh ^3(a+b x) \coth ^2(a+b x)}{2 b}-\frac {5 \operatorname {Subst}\left (\int \frac {x^4}{1-x^2} \, dx,x,\cosh (a+b x)\right )}{2 b}\\ &=-\frac {\cosh ^3(a+b x) \coth ^2(a+b x)}{2 b}-\frac {5 \operatorname {Subst}\left (\int \left (-1-x^2+\frac {1}{1-x^2}\right ) \, dx,x,\cosh (a+b x)\right )}{2 b}\\ &=\frac {5 \cosh (a+b x)}{2 b}+\frac {5 \cosh ^3(a+b x)}{6 b}-\frac {\cosh ^3(a+b x) \coth ^2(a+b x)}{2 b}-\frac {5 \operatorname {Subst}\left (\int \frac {1}{1-x^2} \, dx,x,\cosh (a+b x)\right )}{2 b}\\ &=-\frac {5 \tanh ^{-1}(\cosh (a+b x))}{2 b}+\frac {5 \cosh (a+b x)}{2 b}+\frac {5 \cosh ^3(a+b x)}{6 b}-\frac {\cosh ^3(a+b x) \coth ^2(a+b x)}{2 b}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.04, size = 85, normalized size = 1.29 \[ \frac {9 \cosh (a+b x)}{4 b}+\frac {\cosh (3 (a+b x))}{12 b}-\frac {\text {csch}^2\left (\frac {1}{2} (a+b x)\right )}{8 b}-\frac {\text {sech}^2\left (\frac {1}{2} (a+b x)\right )}{8 b}+\frac {5 \log \left (\tanh \left (\frac {1}{2} (a+b x)\right )\right )}{2 b} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 0.54, size = 1077, normalized size = 16.32 \[ \text {result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [A] time = 0.22, size = 108, normalized size = 1.64 \[ \frac {{\left (27 \, e^{\left (2 \, b x + 2 \, a\right )} + 1\right )} e^{\left (-3 \, b x - 3 \, a\right )} + {\left (e^{\left (3 \, b x + 30 \, a\right )} + 27 \, e^{\left (b x + 28 \, a\right )}\right )} e^{\left (-27 \, a\right )} - \frac {24 \, {\left (e^{\left (3 \, b x + 3 \, a\right )} + e^{\left (b x + a\right )}\right )}}{{\left (e^{\left (2 \, b x + 2 \, a\right )} - 1\right )}^{2}} - 60 \, \log \left (e^{\left (b x + a\right )} + 1\right ) + 60 \, \log \left ({\left | e^{\left (b x + a\right )} - 1 \right |}\right )}{24 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.34, size = 81, normalized size = 1.23 \[ \frac {\frac {\cosh ^{5}\left (b x +a \right )}{3 \sinh \left (b x +a \right )^{2}}+\frac {5 \left (\cosh ^{3}\left (b x +a \right )\right )}{3 \sinh \left (b x +a \right )^{2}}-\frac {5 \cosh \left (b x +a \right )}{\sinh \left (b x +a \right )^{2}}+\frac {5 \,\mathrm {csch}\left (b x +a \right ) \coth \left (b x +a \right )}{2}-5 \arctanh \left ({\mathrm e}^{b x +a}\right )}{b} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [B] time = 0.36, size = 133, normalized size = 2.02 \[ \frac {27 \, e^{\left (-b x - a\right )} + e^{\left (-3 \, b x - 3 \, a\right )}}{24 \, b} - \frac {5 \, \log \left (e^{\left (-b x - a\right )} + 1\right )}{2 \, b} + \frac {5 \, \log \left (e^{\left (-b x - a\right )} - 1\right )}{2 \, b} + \frac {25 \, e^{\left (-2 \, b x - 2 \, a\right )} - 77 \, e^{\left (-4 \, b x - 4 \, a\right )} + 3 \, e^{\left (-6 \, b x - 6 \, a\right )} + 1}{24 \, b {\left (e^{\left (-3 \, b x - 3 \, a\right )} - 2 \, e^{\left (-5 \, b x - 5 \, a\right )} + e^{\left (-7 \, b x - 7 \, a\right )}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [B] time = 1.49, size = 140, normalized size = 2.12 \[ \frac {9\,{\mathrm {e}}^{a+b\,x}}{8\,b}-\frac {5\,\mathrm {atan}\left (\frac {{\mathrm {e}}^{b\,x}\,{\mathrm {e}}^a\,\sqrt {-b^2}}{b}\right )}{\sqrt {-b^2}}+\frac {9\,{\mathrm {e}}^{-a-b\,x}}{8\,b}+\frac {{\mathrm {e}}^{-3\,a-3\,b\,x}}{24\,b}+\frac {{\mathrm {e}}^{3\,a+3\,b\,x}}{24\,b}-\frac {2\,{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{4\,a+4\,b\,x}-2\,{\mathrm {e}}^{2\,a+2\,b\,x}+1\right )}-\frac {{\mathrm {e}}^{a+b\,x}}{b\,\left ({\mathrm {e}}^{2\,a+2\,b\,x}-1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \cosh ^{3}{\left (a + b x \right )} \coth ^{3}{\left (a + b x \right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________