Optimal. Leaf size=149 \[ \frac{6 b^3 n^3 x \sinh \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4-10 b^2 n^2+1}+\frac{x \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2}-\frac{6 b^2 n^2 x \cosh \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4-10 b^2 n^2+1}-\frac{3 b n x \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh ^2\left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.0388181, antiderivative size = 149, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {5520, 5518} \[ \frac{6 b^3 n^3 x \sinh \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4-10 b^2 n^2+1}+\frac{x \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2}-\frac{6 b^2 n^2 x \cosh \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4-10 b^2 n^2+1}-\frac{3 b n x \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh ^2\left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 5520
Rule 5518
Rubi steps
\begin{align*} \int \cosh ^3\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac{x \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2}-\frac{3 b n x \cosh ^2\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2}-\frac{\left (6 b^2 n^2\right ) \int \cosh \left (a+b \log \left (c x^n\right )\right ) \, dx}{1-9 b^2 n^2}\\ &=-\frac{6 b^2 n^2 x \cosh \left (a+b \log \left (c x^n\right )\right )}{1-10 b^2 n^2+9 b^4 n^4}+\frac{x \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2}+\frac{6 b^3 n^3 x \sinh \left (a+b \log \left (c x^n\right )\right )}{1-10 b^2 n^2+9 b^4 n^4}-\frac{3 b n x \cosh ^2\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{1-9 b^2 n^2}\\ \end{align*}
Mathematica [A] time = 0.520587, size = 117, normalized size = 0.79 \[ \frac{x \left (\left (3-27 b^2 n^2\right ) \cosh \left (a+b \log \left (c x^n\right )\right )+\left (1-b^2 n^2\right ) \cosh \left (3 \left (a+b \log \left (c x^n\right )\right )\right )+6 b n \sinh \left (a+b \log \left (c x^n\right )\right ) \left (\left (b^2 n^2-1\right ) \cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )+5 b^2 n^2-1\right )\right )}{36 b^4 n^4-40 b^2 n^2+4} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F] time = 0.115, size = 0, normalized size = 0. \begin{align*} \int \left ( \cosh \left ( a+b\ln \left ( c{x}^{n} \right ) \right ) \right ) ^{3}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A] time = 1.19777, size = 155, normalized size = 1.04 \begin{align*} \frac{c^{3 \, b} x e^{\left (3 \, b \log \left (x^{n}\right ) + 3 \, a\right )}}{8 \,{\left (3 \, b n + 1\right )}} + \frac{3 \, c^{b} x e^{\left (b \log \left (x^{n}\right ) + a\right )}}{8 \,{\left (b n + 1\right )}} - \frac{3 \, x e^{\left (-b \log \left (x^{n}\right ) - a\right )}}{8 \,{\left (b c^{b} n - c^{b}\right )}} - \frac{x e^{\left (-3 \, a\right )}}{8 \,{\left (3 \, b c^{3 \, b} n - c^{3 \, b}\right )}{\left (x^{n}\right )}^{3 \, b}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.87502, size = 536, normalized size = 3.6 \begin{align*} -\frac{{\left (b^{2} n^{2} - 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \,{\left (b^{2} n^{2} - 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 3 \,{\left (b^{3} n^{3} - b n\right )} x \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \,{\left (9 \, b^{2} n^{2} - 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) - 3 \,{\left (3 \,{\left (b^{3} n^{3} - b n\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} +{\left (9 \, b^{3} n^{3} - b n\right )} x\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )}{4 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: TypeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.36022, size = 898, normalized size = 6.03 \begin{align*} \frac{3 \, b^{3} c^{3 \, b} n^{3} x x^{3 \, b n} e^{\left (3 \, a\right )}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} + \frac{27 \, b^{3} c^{b} n^{3} x x^{b n} e^{a}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} - \frac{b^{2} c^{3 \, b} n^{2} x x^{3 \, b n} e^{\left (3 \, a\right )}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} - \frac{27 \, b^{2} c^{b} n^{2} x x^{b n} e^{a}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} - \frac{3 \, b c^{3 \, b} n x x^{3 \, b n} e^{\left (3 \, a\right )}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} - \frac{27 \, b^{3} n^{3} x e^{\left (-a\right )}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{b} x^{b n}} - \frac{3 \, b^{3} n^{3} x e^{\left (-3 \, a\right )}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{3 \, b} x^{3 \, b n}} - \frac{3 \, b c^{b} n x x^{b n} e^{a}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} + \frac{c^{3 \, b} x x^{3 \, b n} e^{\left (3 \, a\right )}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} - \frac{27 \, b^{2} n^{2} x e^{\left (-a\right )}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{b} x^{b n}} - \frac{b^{2} n^{2} x e^{\left (-3 \, a\right )}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{3 \, b} x^{3 \, b n}} + \frac{3 \, c^{b} x x^{b n} e^{a}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )}} + \frac{3 \, b n x e^{\left (-a\right )}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{b} x^{b n}} + \frac{3 \, b n x e^{\left (-3 \, a\right )}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{3 \, b} x^{3 \, b n}} + \frac{3 \, x e^{\left (-a\right )}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{b} x^{b n}} + \frac{x e^{\left (-3 \, a\right )}}{8 \,{\left (9 \, b^{4} n^{4} - 10 \, b^{2} n^{2} + 1\right )} c^{3 \, b} x^{3 \, b n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]