Integrand size = 17, antiderivative size = 120 \[ \int x^m \cosh ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=-\frac {2 b^2 n^2 x^{1+m}}{(1+m) \left ((1+m)^2-4 b^2 n^2\right )}+\frac {(1+m) x^{1+m} \cosh ^2\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-4 b^2 n^2}-\frac {2 b n x^{1+m} \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-4 b^2 n^2} \]
-2*b^2*n^2*x^(1+m)/(1+m)/((1+m)^2-4*b^2*n^2)+(1+m)*x^(1+m)*cosh(a+b*ln(c*x ^n))^2/((1+m)^2-4*b^2*n^2)-2*b*n*x^(1+m)*cosh(a+b*ln(c*x^n))*sinh(a+b*ln(c *x^n))/((1+m)^2-4*b^2*n^2)
Time = 0.20 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.72 \[ \int x^m \cosh ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x^{1+m} \left (1+2 m+m^2-4 b^2 n^2+(1+m)^2 \cosh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )-2 b (1+m) n \sinh \left (2 \left (a+b \log \left (c x^n\right )\right )\right )\right )}{2 (1+m) (1+m-2 b n) (1+m+2 b n)} \]
(x^(1 + m)*(1 + 2*m + m^2 - 4*b^2*n^2 + (1 + m)^2*Cosh[2*(a + b*Log[c*x^n] )] - 2*b*(1 + m)*n*Sinh[2*(a + b*Log[c*x^n])]))/(2*(1 + m)*(1 + m - 2*b*n) *(1 + m + 2*b*n))
Time = 0.27 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.02, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {6056, 15}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int x^m \cosh ^2\left (a+b \log \left (c x^n\right )\right ) \, dx\) |
\(\Big \downarrow \) 6056 |
\(\displaystyle -\frac {2 b^2 n^2 \int x^mdx}{(m+1)^2-4 b^2 n^2}+\frac {(m+1) x^{m+1} \cosh ^2\left (a+b \log \left (c x^n\right )\right )}{-4 b^2 n^2+m^2+2 m+1}-\frac {2 b n x^{m+1} \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{(m+1)^2-4 b^2 n^2}\) |
\(\Big \downarrow \) 15 |
\(\displaystyle \frac {(m+1) x^{m+1} \cosh ^2\left (a+b \log \left (c x^n\right )\right )}{-4 b^2 n^2+m^2+2 m+1}-\frac {2 b n x^{m+1} \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh \left (a+b \log \left (c x^n\right )\right )}{(m+1)^2-4 b^2 n^2}-\frac {2 b^2 n^2 x^{m+1}}{(m+1) \left ((m+1)^2-4 b^2 n^2\right )}\) |
(-2*b^2*n^2*x^(1 + m))/((1 + m)*((1 + m)^2 - 4*b^2*n^2)) + ((1 + m)*x^(1 + m)*Cosh[a + b*Log[c*x^n]]^2)/(1 + 2*m + m^2 - 4*b^2*n^2) - (2*b*n*x^(1 + m)*Cosh[a + b*Log[c*x^n]]*Sinh[a + b*Log[c*x^n]])/((1 + m)^2 - 4*b^2*n^2)
3.3.44.3.1 Defintions of rubi rules used
Int[(a_.)*(x_)^(m_.), x_Symbol] :> Simp[a*(x^(m + 1)/(m + 1)), x] /; FreeQ[ {a, m}, x] && NeQ[m, -1]
Int[Cosh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]^(p_)*((e_.)*(x_))^(m_ .), x_Symbol] :> Simp[(-(m + 1))*(e*x)^(m + 1)*(Cosh[d*(a + b*Log[c*x^n])]^ p/(b^2*d^2*e*n^2*p^2 - e*(m + 1)^2)), x] + (Simp[b*d*n*p*(e*x)^(m + 1)*Sinh [d*(a + b*Log[c*x^n])]*(Cosh[d*(a + b*Log[c*x^n])]^(p - 1)/(b^2*d^2*e*n^2*p ^2 - e*(m + 1)^2)), x] + Simp[b^2*d^2*n^2*p*((p - 1)/(b^2*d^2*n^2*p^2 - (m + 1)^2)) Int[(e*x)^m*Cosh[d*(a + b*Log[c*x^n])]^(p - 2), x], x]) /; FreeQ [{a, b, c, d, e, m, n}, x] && IGtQ[p, 1] && NeQ[b^2*d^2*n^2*p^2 - (m + 1)^2 , 0]
Time = 4.94 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.70
method | result | size |
parallelrisch | \(\frac {\left (2 b^{2} n^{2}+b n \left (1+m \right ) \sinh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )-\frac {\left (1+m \right )^{2} \left (\cosh \left (2 b \ln \left (c \,x^{n}\right )+2 a \right )+1\right )}{2}\right ) x^{1+m}}{\left (4 b^{2} n^{2}-m^{2}-2 m -1\right ) \left (1+m \right )}\) | \(84\) |
(2*b^2*n^2+b*n*(1+m)*sinh(2*b*ln(c*x^n)+2*a)-1/2*(1+m)^2*(cosh(2*b*ln(c*x^ n)+2*a)+1))*x^(1+m)/(4*b^2*n^2-m^2-2*m-1)/(1+m)
Time = 0.26 (sec) , antiderivative size = 250, normalized size of antiderivative = 2.08 \[ \int x^m \cosh ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {{\left (m^{2} + 2 \, m + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} \cosh \left (m \log \left (x\right )\right ) - {\left (4 \, b^{2} n^{2} - m^{2} - 2 \, m - 1\right )} x \cosh \left (m \log \left (x\right )\right ) + {\left ({\left (m^{2} + 2 \, m + 1\right )} x \cosh \left (m \log \left (x\right )\right ) + {\left (m^{2} + 2 \, m + 1\right )} x \sinh \left (m \log \left (x\right )\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - 4 \, {\left ({\left (b m + b\right )} n x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \cosh \left (m \log \left (x\right )\right ) + {\left (b m + b\right )} n x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \sinh \left (m \log \left (x\right )\right )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + {\left ({\left (m^{2} + 2 \, m + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} - {\left (4 \, b^{2} n^{2} - m^{2} - 2 \, m - 1\right )} x\right )} \sinh \left (m \log \left (x\right )\right )}{2 \, {\left (m^{3} - 4 \, {\left (b^{2} m + b^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1\right )}} \]
1/2*((m^2 + 2*m + 1)*x*cosh(b*n*log(x) + b*log(c) + a)^2*cosh(m*log(x)) - (4*b^2*n^2 - m^2 - 2*m - 1)*x*cosh(m*log(x)) + ((m^2 + 2*m + 1)*x*cosh(m*l og(x)) + (m^2 + 2*m + 1)*x*sinh(m*log(x)))*sinh(b*n*log(x) + b*log(c) + a) ^2 - 4*((b*m + b)*n*x*cosh(b*n*log(x) + b*log(c) + a)*cosh(m*log(x)) + (b* m + b)*n*x*cosh(b*n*log(x) + b*log(c) + a)*sinh(m*log(x)))*sinh(b*n*log(x) + b*log(c) + a) + ((m^2 + 2*m + 1)*x*cosh(b*n*log(x) + b*log(c) + a)^2 - (4*b^2*n^2 - m^2 - 2*m - 1)*x)*sinh(m*log(x)))/(m^3 - 4*(b^2*m + b^2)*n^2 + 3*m^2 + 3*m + 1)
\[ \int x^m \cosh ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\begin {cases} \log {\left (x \right )} \cosh ^{2}{\left (a \right )} & \text {for}\: b = 0 \wedge m = -1 \\\int x^{m} \cosh ^{2}{\left (- a + \frac {m \log {\left (c x^{n} \right )}}{2 n} + \frac {\log {\left (c x^{n} \right )}}{2 n} \right )}\, dx & \text {for}\: b = - \frac {m + 1}{2 n} \\\int x^{m} \cosh ^{2}{\left (a + \frac {m \log {\left (c x^{n} \right )}}{2 n} + \frac {\log {\left (c x^{n} \right )}}{2 n} \right )}\, dx & \text {for}\: b = \frac {m + 1}{2 n} \\\int \frac {\cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{x}\, dx & \text {for}\: m = -1 \\- \frac {2 b^{2} n^{2} x x^{m} \sinh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} m n^{2} + 4 b^{2} n^{2} - m^{3} - 3 m^{2} - 3 m - 1} + \frac {2 b^{2} n^{2} x x^{m} \cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} m n^{2} + 4 b^{2} n^{2} - m^{3} - 3 m^{2} - 3 m - 1} + \frac {2 b m n x x^{m} \sinh {\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} m n^{2} + 4 b^{2} n^{2} - m^{3} - 3 m^{2} - 3 m - 1} + \frac {2 b n x x^{m} \sinh {\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} m n^{2} + 4 b^{2} n^{2} - m^{3} - 3 m^{2} - 3 m - 1} - \frac {m^{2} x x^{m} \cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} m n^{2} + 4 b^{2} n^{2} - m^{3} - 3 m^{2} - 3 m - 1} - \frac {2 m x x^{m} \cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} m n^{2} + 4 b^{2} n^{2} - m^{3} - 3 m^{2} - 3 m - 1} - \frac {x x^{m} \cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{4 b^{2} m n^{2} + 4 b^{2} n^{2} - m^{3} - 3 m^{2} - 3 m - 1} & \text {otherwise} \end {cases} \]
Piecewise((log(x)*cosh(a)**2, Eq(b, 0) & Eq(m, -1)), (Integral(x**m*cosh(- a + m*log(c*x**n)/(2*n) + log(c*x**n)/(2*n))**2, x), Eq(b, -(m + 1)/(2*n)) ), (Integral(x**m*cosh(a + m*log(c*x**n)/(2*n) + log(c*x**n)/(2*n))**2, x) , Eq(b, (m + 1)/(2*n))), (Integral(cosh(a + b*log(c*x**n))**2/x, x), Eq(m, -1)), (-2*b**2*n**2*x*x**m*sinh(a + b*log(c*x**n))**2/(4*b**2*m*n**2 + 4* b**2*n**2 - m**3 - 3*m**2 - 3*m - 1) + 2*b**2*n**2*x*x**m*cosh(a + b*log(c *x**n))**2/(4*b**2*m*n**2 + 4*b**2*n**2 - m**3 - 3*m**2 - 3*m - 1) + 2*b*m *n*x*x**m*sinh(a + b*log(c*x**n))*cosh(a + b*log(c*x**n))/(4*b**2*m*n**2 + 4*b**2*n**2 - m**3 - 3*m**2 - 3*m - 1) + 2*b*n*x*x**m*sinh(a + b*log(c*x* *n))*cosh(a + b*log(c*x**n))/(4*b**2*m*n**2 + 4*b**2*n**2 - m**3 - 3*m**2 - 3*m - 1) - m**2*x*x**m*cosh(a + b*log(c*x**n))**2/(4*b**2*m*n**2 + 4*b** 2*n**2 - m**3 - 3*m**2 - 3*m - 1) - 2*m*x*x**m*cosh(a + b*log(c*x**n))**2/ (4*b**2*m*n**2 + 4*b**2*n**2 - m**3 - 3*m**2 - 3*m - 1) - x*x**m*cosh(a + b*log(c*x**n))**2/(4*b**2*m*n**2 + 4*b**2*n**2 - m**3 - 3*m**2 - 3*m - 1), True))
Time = 0.21 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.72 \[ \int x^m \cosh ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {c^{2 \, b} x e^{\left (2 \, b \log \left (x^{n}\right ) + m \log \left (x\right ) + 2 \, a\right )}}{4 \, {\left (2 \, b n + m + 1\right )}} - \frac {x e^{\left (-2 \, b \log \left (x^{n}\right ) + m \log \left (x\right ) - 2 \, a\right )}}{4 \, {\left (2 \, b c^{2 \, b} n - c^{2 \, b} {\left (m + 1\right )}\right )}} + \frac {x^{m + 1}}{2 \, {\left (m + 1\right )}} \]
1/4*c^(2*b)*x*e^(2*b*log(x^n) + m*log(x) + 2*a)/(2*b*n + m + 1) - 1/4*x*e^ (-2*b*log(x^n) + m*log(x) - 2*a)/(2*b*c^(2*b)*n - c^(2*b)*(m + 1)) + 1/2*x ^(m + 1)/(m + 1)
Leaf count of result is larger than twice the leaf count of optimal. 759 vs. \(2 (127) = 254\).
Time = 0.30 (sec) , antiderivative size = 759, normalized size of antiderivative = 6.32 \[ \int x^m \cosh ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {b c^{2 \, b} m n x x^{2 \, b n} x^{m} e^{\left (2 \, a\right )}}{2 \, {\left (4 \, b^{2} m n^{2} + 4 \, b^{2} n^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1\right )}} - \frac {c^{2 \, b} m^{2} x x^{2 \, b n} x^{m} e^{\left (2 \, a\right )}}{4 \, {\left (4 \, b^{2} m n^{2} + 4 \, b^{2} n^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1\right )}} + \frac {b c^{2 \, b} n x x^{2 \, b n} x^{m} e^{\left (2 \, a\right )}}{2 \, {\left (4 \, b^{2} m n^{2} + 4 \, b^{2} n^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1\right )}} + \frac {2 \, b^{2} n^{2} x x^{m}}{4 \, b^{2} m n^{2} + 4 \, b^{2} n^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1} - \frac {c^{2 \, b} m x x^{2 \, b n} x^{m} e^{\left (2 \, a\right )}}{2 \, {\left (4 \, b^{2} m n^{2} + 4 \, b^{2} n^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1\right )}} - \frac {c^{2 \, b} x x^{2 \, b n} x^{m} e^{\left (2 \, a\right )}}{4 \, {\left (4 \, b^{2} m n^{2} + 4 \, b^{2} n^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1\right )}} - \frac {m^{2} x x^{m}}{2 \, {\left (4 \, b^{2} m n^{2} + 4 \, b^{2} n^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1\right )}} - \frac {b m n x x^{m} e^{\left (-2 \, a\right )}}{2 \, {\left (4 \, b^{2} m n^{2} + 4 \, b^{2} n^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1\right )} c^{2 \, b} x^{2 \, b n}} - \frac {m x x^{m}}{4 \, b^{2} m n^{2} + 4 \, b^{2} n^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1} - \frac {m^{2} x x^{m} e^{\left (-2 \, a\right )}}{4 \, {\left (4 \, b^{2} m n^{2} + 4 \, b^{2} n^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1\right )} c^{2 \, b} x^{2 \, b n}} - \frac {b n x x^{m} e^{\left (-2 \, a\right )}}{2 \, {\left (4 \, b^{2} m n^{2} + 4 \, b^{2} n^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1\right )} c^{2 \, b} x^{2 \, b n}} - \frac {x x^{m}}{2 \, {\left (4 \, b^{2} m n^{2} + 4 \, b^{2} n^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1\right )}} - \frac {m x x^{m} e^{\left (-2 \, a\right )}}{2 \, {\left (4 \, b^{2} m n^{2} + 4 \, b^{2} n^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1\right )} c^{2 \, b} x^{2 \, b n}} - \frac {x x^{m} e^{\left (-2 \, a\right )}}{4 \, {\left (4 \, b^{2} m n^{2} + 4 \, b^{2} n^{2} - m^{3} - 3 \, m^{2} - 3 \, m - 1\right )} c^{2 \, b} x^{2 \, b n}} \]
1/2*b*c^(2*b)*m*n*x*x^(2*b*n)*x^m*e^(2*a)/(4*b^2*m*n^2 + 4*b^2*n^2 - m^3 - 3*m^2 - 3*m - 1) - 1/4*c^(2*b)*m^2*x*x^(2*b*n)*x^m*e^(2*a)/(4*b^2*m*n^2 + 4*b^2*n^2 - m^3 - 3*m^2 - 3*m - 1) + 1/2*b*c^(2*b)*n*x*x^(2*b*n)*x^m*e^(2 *a)/(4*b^2*m*n^2 + 4*b^2*n^2 - m^3 - 3*m^2 - 3*m - 1) + 2*b^2*n^2*x*x^m/(4 *b^2*m*n^2 + 4*b^2*n^2 - m^3 - 3*m^2 - 3*m - 1) - 1/2*c^(2*b)*m*x*x^(2*b*n )*x^m*e^(2*a)/(4*b^2*m*n^2 + 4*b^2*n^2 - m^3 - 3*m^2 - 3*m - 1) - 1/4*c^(2 *b)*x*x^(2*b*n)*x^m*e^(2*a)/(4*b^2*m*n^2 + 4*b^2*n^2 - m^3 - 3*m^2 - 3*m - 1) - 1/2*m^2*x*x^m/(4*b^2*m*n^2 + 4*b^2*n^2 - m^3 - 3*m^2 - 3*m - 1) - 1/ 2*b*m*n*x*x^m*e^(-2*a)/((4*b^2*m*n^2 + 4*b^2*n^2 - m^3 - 3*m^2 - 3*m - 1)* c^(2*b)*x^(2*b*n)) - m*x*x^m/(4*b^2*m*n^2 + 4*b^2*n^2 - m^3 - 3*m^2 - 3*m - 1) - 1/4*m^2*x*x^m*e^(-2*a)/((4*b^2*m*n^2 + 4*b^2*n^2 - m^3 - 3*m^2 - 3* m - 1)*c^(2*b)*x^(2*b*n)) - 1/2*b*n*x*x^m*e^(-2*a)/((4*b^2*m*n^2 + 4*b^2*n ^2 - m^3 - 3*m^2 - 3*m - 1)*c^(2*b)*x^(2*b*n)) - 1/2*x*x^m/(4*b^2*m*n^2 + 4*b^2*n^2 - m^3 - 3*m^2 - 3*m - 1) - 1/2*m*x*x^m*e^(-2*a)/((4*b^2*m*n^2 + 4*b^2*n^2 - m^3 - 3*m^2 - 3*m - 1)*c^(2*b)*x^(2*b*n)) - 1/4*x*x^m*e^(-2*a) /((4*b^2*m*n^2 + 4*b^2*n^2 - m^3 - 3*m^2 - 3*m - 1)*c^(2*b)*x^(2*b*n))
Time = 1.84 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.61 \[ \int x^m \cosh ^2\left (a+b \log \left (c x^n\right )\right ) \, dx=\frac {x\,x^m}{2\,m+2}+\frac {x\,x^m\,{\mathrm {e}}^{-2\,a}}{{\left (c\,x^n\right )}^{2\,b}\,\left (4\,m-8\,b\,n+4\right )}+\frac {x\,x^m\,{\mathrm {e}}^{2\,a}\,{\left (c\,x^n\right )}^{2\,b}}{4\,m+8\,b\,n+4} \]