3.3.45 \(\int x^m \cosh ^3(a+b \log (c x^n)) \, dx\) [245]

Optimal. Leaf size=203 \[ -\frac {6 b^2 (1+m) n^2 x^{1+m} \cosh \left (a+b \log \left (c x^n\right )\right )}{\left ((1+m)^2-9 b^2 n^2\right ) \left ((1+m)^2-b^2 n^2\right )}+\frac {(1+m) x^{1+m} \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-9 b^2 n^2}+\frac {6 b^3 n^3 x^{1+m} \sinh \left (a+b \log \left (c x^n\right )\right )}{\left ((1+m)^2-9 b^2 n^2\right ) \left ((1+m)^2-b^2 n^2\right )}-\frac {3 b n x^{1+m} \cosh ^2\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-9 b^2 n^2} \]

[Out]

-6*b^2*(1+m)*n^2*x^(1+m)*cosh(a+b*ln(c*x^n))/((1+m)^2-9*b^2*n^2)/((1+m)^2-b^2*n^2)+(1+m)*x^(1+m)*cosh(a+b*ln(c
*x^n))^3/((1+m)^2-9*b^2*n^2)+6*b^3*n^3*x^(1+m)*sinh(a+b*ln(c*x^n))/((1+m)^2-9*b^2*n^2)/((1+m)^2-b^2*n^2)-3*b*n
*x^(1+m)*cosh(a+b*ln(c*x^n))^2*sinh(a+b*ln(c*x^n))/((1+m)^2-9*b^2*n^2)

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Rubi [A]
time = 0.07, antiderivative size = 203, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {5641, 5639} \begin {gather*} \frac {(m+1) x^{m+1} \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{-9 b^2 n^2+m^2+2 m+1}-\frac {6 b^2 (m+1) n^2 x^{m+1} \cosh \left (a+b \log \left (c x^n\right )\right )}{(-b n+m+1) (b n+m+1) \left ((m+1)^2-9 b^2 n^2\right )}-\frac {3 b n x^{m+1} \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh ^2\left (a+b \log \left (c x^n\right )\right )}{(m+1)^2-9 b^2 n^2}+\frac {6 b^3 n^3 x^{m+1} \sinh \left (a+b \log \left (c x^n\right )\right )}{9 b^4 n^4-10 b^2 (m+1)^2 n^2+(m+1)^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^m*Cosh[a + b*Log[c*x^n]]^3,x]

[Out]

(-6*b^2*(1 + m)*n^2*x^(1 + m)*Cosh[a + b*Log[c*x^n]])/((1 + m - b*n)*(1 + m + b*n)*((1 + m)^2 - 9*b^2*n^2)) +
((1 + m)*x^(1 + m)*Cosh[a + b*Log[c*x^n]]^3)/(1 + 2*m + m^2 - 9*b^2*n^2) + (6*b^3*n^3*x^(1 + m)*Sinh[a + b*Log
[c*x^n]])/((1 + m)^4 - 10*b^2*(1 + m)^2*n^2 + 9*b^4*n^4) - (3*b*n*x^(1 + m)*Cosh[a + b*Log[c*x^n]]^2*Sinh[a +
b*Log[c*x^n]])/((1 + m)^2 - 9*b^2*n^2)

Rule 5639

Int[Cosh[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*(d_.)]*((e_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(m + 1))*(e*x)^(
m + 1)*(Cosh[d*(a + b*Log[c*x^n])]/(b^2*d^2*e*n^2 - e*(m + 1)^2)), x] + Simp[b*d*n*(e*x)^(m + 1)*(Sinh[d*(a +
b*Log[c*x^n])]/(b^2*d^2*e*n^2 - e*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b^2*d^2*n^2 - (m +
1)^2, 0]

Rule 5641

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] + (Dist[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] + 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])
/; 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]

Rubi steps

\begin {align*} \int x^m \cosh ^3\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {(1+m) x^{1+m} \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-9 b^2 n^2}-\frac {3 b n x^{1+m} \cosh ^2\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-9 b^2 n^2}-\frac {\left (6 b^2 n^2\right ) \int x^m \cosh \left (a+b \log \left (c x^n\right )\right ) \, dx}{(1+m)^2-9 b^2 n^2}\\ &=-\frac {6 b^2 (1+m) n^2 x^{1+m} \cosh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^4-10 b^2 (1+m)^2 n^2+9 b^4 n^4}+\frac {(1+m) x^{1+m} \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-9 b^2 n^2}+\frac {6 b^3 n^3 x^{1+m} \sinh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^4-10 b^2 (1+m)^2 n^2+9 b^4 n^4}-\frac {3 b n x^{1+m} \cosh ^2\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-9 b^2 n^2}\\ \end {align*}

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Mathematica [A]
time = 0.93, size = 292, normalized size = 1.44 \begin {gather*} \frac {1}{4} x^{1+m} \left (\frac {3 \sinh (b n \log (x)) \left (-b n \cosh \left (a-b n \log (x)+b \log \left (c x^n\right )\right )+(1+m) \sinh \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )}{(1+m-b n) (1+m+b n)}+\frac {3 \cosh (b n \log (x)) \left ((1+m) \cosh \left (a-b n \log (x)+b \log \left (c x^n\right )\right )-b n \sinh \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )}{(1+m-b n) (1+m+b n)}+\frac {\sinh (3 b n \log (x)) \left (-3 b n \cosh \left (3 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+(1+m) \sinh \left (3 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )\right )}{(1+m-3 b n) (1+m+3 b n)}+\frac {\cosh (3 b n \log (x)) \left ((1+m) \cosh \left (3 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )-3 b n \sinh \left (3 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )\right )}{(1+m-3 b n) (1+m+3 b n)}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^m*Cosh[a + b*Log[c*x^n]]^3,x]

[Out]

(x^(1 + m)*((3*Sinh[b*n*Log[x]]*(-(b*n*Cosh[a - b*n*Log[x] + b*Log[c*x^n]]) + (1 + m)*Sinh[a - b*n*Log[x] + b*
Log[c*x^n]]))/((1 + m - b*n)*(1 + m + b*n)) + (3*Cosh[b*n*Log[x]]*((1 + m)*Cosh[a - b*n*Log[x] + b*Log[c*x^n]]
- b*n*Sinh[a - b*n*Log[x] + b*Log[c*x^n]]))/((1 + m - b*n)*(1 + m + b*n)) + (Sinh[3*b*n*Log[x]]*(-3*b*n*Cosh[
3*(a - b*n*Log[x] + b*Log[c*x^n])] + (1 + m)*Sinh[3*(a - b*n*Log[x] + b*Log[c*x^n])]))/((1 + m - 3*b*n)*(1 + m
+ 3*b*n)) + (Cosh[3*b*n*Log[x]]*((1 + m)*Cosh[3*(a - b*n*Log[x] + b*Log[c*x^n])] - 3*b*n*Sinh[3*(a - b*n*Log[
x] + b*Log[c*x^n])]))/((1 + m - 3*b*n)*(1 + m + 3*b*n))))/4

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Maple [F]
time = 2.52, size = 0, normalized size = 0.00 \[\int x^{m} \left (\cosh ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*cosh(a+b*ln(c*x^n))^3,x)

[Out]

int(x^m*cosh(a+b*ln(c*x^n))^3,x)

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Maxima [A]
time = 0.30, size = 138, normalized size = 0.68 \begin {gather*} \frac {c^{3 \, b} x e^{\left (3 \, b \log \left (x^{n}\right ) + m \log \left (x\right ) + 3 \, a\right )}}{8 \, {\left (3 \, b n + m + 1\right )}} + \frac {3 \, c^{b} x e^{\left (b \log \left (x^{n}\right ) + m \log \left (x\right ) + a\right )}}{8 \, {\left (b n + m + 1\right )}} - \frac {3 \, x e^{\left (-b \log \left (x^{n}\right ) + m \log \left (x\right ) - a\right )}}{8 \, {\left (b c^{b} n - c^{b} {\left (m + 1\right )}\right )}} - \frac {x e^{\left (-3 \, b \log \left (x^{n}\right ) + m \log \left (x\right ) - 3 \, a\right )}}{8 \, {\left (3 \, b c^{3 \, b} n - c^{3 \, b} {\left (m + 1\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*cosh(a+b*log(c*x^n))^3,x, algorithm="maxima")

[Out]

1/8*c^(3*b)*x*e^(3*b*log(x^n) + m*log(x) + 3*a)/(3*b*n + m + 1) + 3/8*c^b*x*e^(b*log(x^n) + m*log(x) + a)/(b*n
+ m + 1) - 3/8*x*e^(-b*log(x^n) + m*log(x) - a)/(b*c^b*n - c^b*(m + 1)) - 1/8*x*e^(-3*b*log(x^n) + m*log(x) -
3*a)/(3*b*c^(3*b)*n - c^(3*b)*(m + 1))

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 584 vs. \(2 (214) = 428\).
time = 0.40, size = 584, normalized size = 2.88 \begin {gather*} \frac {{\left (m^{3} - {\left (b^{2} m + b^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} \cosh \left (m \log \left (x\right )\right ) + 3 \, {\left (m^{3} - 9 \, {\left (b^{2} m + b^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) \cosh \left (m \log \left (x\right )\right ) + 3 \, {\left ({\left (b^{3} n^{3} - {\left (b m^{2} + 2 \, b m + b\right )} n\right )} x \cosh \left (m \log \left (x\right )\right ) + {\left (b^{3} n^{3} - {\left (b m^{2} + 2 \, b m + b\right )} n\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 )^{3} + 3 \, {\left ({\left (m^{3} - {\left (b^{2} m + b^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1\right )} 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 (m^{3} - {\left (b^{2} m + b^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1\right )} 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 )^{2} + 3 \, {\left (3 \, {\left (b^{3} n^{3} - {\left (b m^{2} + 2 \, b m + b\right )} n\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 (9 \, b^{3} n^{3} - {\left (b m^{2} + 2 \, b m + b\right )} n\right )} x \cosh \left (m \log \left (x\right )\right ) + {\left (3 \, {\left (b^{3} n^{3} - {\left (b m^{2} + 2 \, b m + b\right )} 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} - {\left (b m^{2} + 2 \, b m + b\right )} n\right )} x\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^{3} - {\left (b^{2} m + b^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + 3 \, {\left (m^{3} - 9 \, {\left (b^{2} m + b^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )\right )} \sinh \left (m \log \left (x\right )\right )}{4 \, {\left (9 \, b^{4} n^{4} + m^{4} + 4 \, m^{3} - 10 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, m + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*cosh(a+b*log(c*x^n))^3,x, algorithm="fricas")

[Out]

1/4*((m^3 - (b^2*m + b^2)*n^2 + 3*m^2 + 3*m + 1)*x*cosh(b*n*log(x) + b*log(c) + a)^3*cosh(m*log(x)) + 3*(m^3 -
9*(b^2*m + b^2)*n^2 + 3*m^2 + 3*m + 1)*x*cosh(b*n*log(x) + b*log(c) + a)*cosh(m*log(x)) + 3*((b^3*n^3 - (b*m^
2 + 2*b*m + b)*n)*x*cosh(m*log(x)) + (b^3*n^3 - (b*m^2 + 2*b*m + b)*n)*x*sinh(m*log(x)))*sinh(b*n*log(x) + b*l
og(c) + a)^3 + 3*((m^3 - (b^2*m + b^2)*n^2 + 3*m^2 + 3*m + 1)*x*cosh(b*n*log(x) + b*log(c) + a)*cosh(m*log(x))
+ (m^3 - (b^2*m + b^2)*n^2 + 3*m^2 + 3*m + 1)*x*cosh(b*n*log(x) + b*log(c) + a)*sinh(m*log(x)))*sinh(b*n*log(
x) + b*log(c) + a)^2 + 3*(3*(b^3*n^3 - (b*m^2 + 2*b*m + b)*n)*x*cosh(b*n*log(x) + b*log(c) + a)^2*cosh(m*log(x
)) + (9*b^3*n^3 - (b*m^2 + 2*b*m + b)*n)*x*cosh(m*log(x)) + (3*(b^3*n^3 - (b*m^2 + 2*b*m + b)*n)*x*cosh(b*n*lo
g(x) + b*log(c) + a)^2 + (9*b^3*n^3 - (b*m^2 + 2*b*m + b)*n)*x)*sinh(m*log(x)))*sinh(b*n*log(x) + b*log(c) + a
) + ((m^3 - (b^2*m + b^2)*n^2 + 3*m^2 + 3*m + 1)*x*cosh(b*n*log(x) + b*log(c) + a)^3 + 3*(m^3 - 9*(b^2*m + b^2
)*n^2 + 3*m^2 + 3*m + 1)*x*cosh(b*n*log(x) + b*log(c) + a))*sinh(m*log(x)))/(9*b^4*n^4 + m^4 + 4*m^3 - 10*(b^2
*m^2 + 2*b^2*m + b^2)*n^2 + 6*m^2 + 4*m + 1)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \log {\left (x \right )} \cosh ^{3}{\left (a \right )} & \text {for}\: b = 0 \wedge m = -1 \\\int x^{m} \cosh ^{3}{\left (- a + \frac {m \log {\left (c x^{n} \right )}}{3 n} + \frac {\log {\left (c x^{n} \right )}}{3 n} \right )}\, dx & \text {for}\: b = \frac {- m - 1}{3 n} \\\int x^{m} \cosh ^{3}{\left (- a + \frac {m \log {\left (c x^{n} \right )}}{n} + \frac {\log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {- m - 1}{n} \\\int x^{m} \cosh ^{3}{\left (a + \frac {m \log {\left (c x^{n} \right )}}{3 n} + \frac {\log {\left (c x^{n} \right )}}{3 n} \right )}\, dx & \text {for}\: b = \frac {m + 1}{3 n} \\\int x^{m} \cosh ^{3}{\left (a + \frac {m \log {\left (c x^{n} \right )}}{n} + \frac {\log {\left (c x^{n} \right )}}{n} \right )}\, dx & \text {for}\: b = \frac {m + 1}{n} \\- \frac {6 b^{3} n^{3} x x^{m} \sinh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} m^{2} n^{2} - 20 b^{2} m n^{2} - 10 b^{2} n^{2} + m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1} + \frac {9 b^{3} n^{3} x x^{m} \sinh {\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} m^{2} n^{2} - 20 b^{2} m n^{2} - 10 b^{2} n^{2} + m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1} + \frac {6 b^{2} m n^{2} x x^{m} \sinh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} m^{2} n^{2} - 20 b^{2} m n^{2} - 10 b^{2} n^{2} + m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1} - \frac {7 b^{2} m n^{2} x x^{m} \cosh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} m^{2} n^{2} - 20 b^{2} m n^{2} - 10 b^{2} n^{2} + m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1} + \frac {6 b^{2} n^{2} x x^{m} \sinh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh {\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} m^{2} n^{2} - 20 b^{2} m n^{2} - 10 b^{2} n^{2} + m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1} - \frac {7 b^{2} n^{2} x x^{m} \cosh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} m^{2} n^{2} - 20 b^{2} m n^{2} - 10 b^{2} n^{2} + m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1} - \frac {3 b m^{2} n x x^{m} \sinh {\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} m^{2} n^{2} - 20 b^{2} m n^{2} - 10 b^{2} n^{2} + m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1} - \frac {6 b m n x x^{m} \sinh {\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} m^{2} n^{2} - 20 b^{2} m n^{2} - 10 b^{2} n^{2} + m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1} - \frac {3 b n x x^{m} \sinh {\left (a + b \log {\left (c x^{n} \right )} \right )} \cosh ^{2}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} m^{2} n^{2} - 20 b^{2} m n^{2} - 10 b^{2} n^{2} + m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1} + \frac {m^{3} x x^{m} \cosh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} m^{2} n^{2} - 20 b^{2} m n^{2} - 10 b^{2} n^{2} + m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1} + \frac {3 m^{2} x x^{m} \cosh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} m^{2} n^{2} - 20 b^{2} m n^{2} - 10 b^{2} n^{2} + m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1} + \frac {3 m x x^{m} \cosh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} m^{2} n^{2} - 20 b^{2} m n^{2} - 10 b^{2} n^{2} + m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1} + \frac {x x^{m} \cosh ^{3}{\left (a + b \log {\left (c x^{n} \right )} \right )}}{9 b^{4} n^{4} - 10 b^{2} m^{2} n^{2} - 20 b^{2} m n^{2} - 10 b^{2} n^{2} + m^{4} + 4 m^{3} + 6 m^{2} + 4 m + 1} & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*cosh(a+b*ln(c*x**n))**3,x)

[Out]

Piecewise((log(x)*cosh(a)**3, Eq(b, 0) & Eq(m, -1)), (Integral(x**m*cosh(-a + m*log(c*x**n)/(3*n) + log(c*x**n
)/(3*n))**3, x), Eq(b, (-m - 1)/(3*n))), (Integral(x**m*cosh(-a + m*log(c*x**n)/n + log(c*x**n)/n)**3, x), Eq(
b, (-m - 1)/n)), (Integral(x**m*cosh(a + m*log(c*x**n)/(3*n) + log(c*x**n)/(3*n))**3, x), Eq(b, (m + 1)/(3*n))
), (Integral(x**m*cosh(a + m*log(c*x**n)/n + log(c*x**n)/n)**3, x), Eq(b, (m + 1)/n)), (-6*b**3*n**3*x*x**m*si
nh(a + b*log(c*x**n))**3/(9*b**4*n**4 - 10*b**2*m**2*n**2 - 20*b**2*m*n**2 - 10*b**2*n**2 + m**4 + 4*m**3 + 6*
m**2 + 4*m + 1) + 9*b**3*n**3*x*x**m*sinh(a + b*log(c*x**n))*cosh(a + b*log(c*x**n))**2/(9*b**4*n**4 - 10*b**2
*m**2*n**2 - 20*b**2*m*n**2 - 10*b**2*n**2 + m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 6*b**2*m*n**2*x*x**m*sinh(a +
b*log(c*x**n))**2*cosh(a + b*log(c*x**n))/(9*b**4*n**4 - 10*b**2*m**2*n**2 - 20*b**2*m*n**2 - 10*b**2*n**2 +
m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 7*b**2*m*n**2*x*x**m*cosh(a + b*log(c*x**n))**3/(9*b**4*n**4 - 10*b**2*m**
2*n**2 - 20*b**2*m*n**2 - 10*b**2*n**2 + m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 6*b**2*n**2*x*x**m*sinh(a + b*log
(c*x**n))**2*cosh(a + b*log(c*x**n))/(9*b**4*n**4 - 10*b**2*m**2*n**2 - 20*b**2*m*n**2 - 10*b**2*n**2 + m**4 +
4*m**3 + 6*m**2 + 4*m + 1) - 7*b**2*n**2*x*x**m*cosh(a + b*log(c*x**n))**3/(9*b**4*n**4 - 10*b**2*m**2*n**2 -
20*b**2*m*n**2 - 10*b**2*n**2 + m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 3*b*m**2*n*x*x**m*sinh(a + b*log(c*x**n))
*cosh(a + b*log(c*x**n))**2/(9*b**4*n**4 - 10*b**2*m**2*n**2 - 20*b**2*m*n**2 - 10*b**2*n**2 + m**4 + 4*m**3 +
6*m**2 + 4*m + 1) - 6*b*m*n*x*x**m*sinh(a + b*log(c*x**n))*cosh(a + b*log(c*x**n))**2/(9*b**4*n**4 - 10*b**2*
m**2*n**2 - 20*b**2*m*n**2 - 10*b**2*n**2 + m**4 + 4*m**3 + 6*m**2 + 4*m + 1) - 3*b*n*x*x**m*sinh(a + b*log(c*
x**n))*cosh(a + b*log(c*x**n))**2/(9*b**4*n**4 - 10*b**2*m**2*n**2 - 20*b**2*m*n**2 - 10*b**2*n**2 + m**4 + 4*
m**3 + 6*m**2 + 4*m + 1) + m**3*x*x**m*cosh(a + b*log(c*x**n))**3/(9*b**4*n**4 - 10*b**2*m**2*n**2 - 20*b**2*m
*n**2 - 10*b**2*n**2 + m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*m**2*x*x**m*cosh(a + b*log(c*x**n))**3/(9*b**4*n*
*4 - 10*b**2*m**2*n**2 - 20*b**2*m*n**2 - 10*b**2*n**2 + m**4 + 4*m**3 + 6*m**2 + 4*m + 1) + 3*m*x*x**m*cosh(a
+ b*log(c*x**n))**3/(9*b**4*n**4 - 10*b**2*m**2*n**2 - 20*b**2*m*n**2 - 10*b**2*n**2 + m**4 + 4*m**3 + 6*m**2
+ 4*m + 1) + x*x**m*cosh(a + b*log(c*x**n))**3/(9*b**4*n**4 - 10*b**2*m**2*n**2 - 20*b**2*m*n**2 - 10*b**2*n*
*2 + m**4 + 4*m**3 + 6*m**2 + 4*m + 1), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 3225 vs. \(2 (214) = 428\).
time = 0.53, size = 3225, normalized size = 15.89 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*cosh(a+b*log(c*x^n))^3,x, algorithm="giac")

[Out]

3/8*b^3*c^(3*b)*n^3*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*
m^3 + 6*m^2 + 4*m + 1) + 27/8*b^3*c^b*n^3*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 -
10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 1/8*b^2*c^(3*b)*m*n^2*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2
*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 27/8*b^2*c^b*m*n^2*x*x^(b*n)*x^m*e^a/(9*b^
4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 3/8*b*c^(3*b)*m^2*n*x*x^
(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) -
1/8*b^2*c^(3*b)*n^2*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4
*m^3 + 6*m^2 + 4*m + 1) - 3/8*b*c^b*m^2*n*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 -
10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 27/8*b^2*c^b*n^2*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b
^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) + 1/8*c^(3*b)*m^3*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 -
10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 3/4*b*c^(3*b)*m*n*x*x^(3*b*n)*x^
m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) + 3/8*c^b*m
^3*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)
- 3/4*b*c^b*m*n*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^
2 + 4*m + 1) + 3/8*c^(3*b)*m^2*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b
^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 3/8*b*c^(3*b)*n*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b
^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 27/8*b^3*n^3*x*x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^
2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^b*x^(b*n)) - 3/8*b^3*n^3*x*x^m*e^(-3*a)/((9*b
^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) + 9/8*
c^b*m^2*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m
+ 1) - 3/8*b*c^b*n*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6
*m^2 + 4*m + 1) + 3/8*c^(3*b)*m*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*
b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) - 27/8*b^2*m*n^2*x*x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 +
m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^b*x^(b*n)) - 1/8*b^2*m*n^2*x*x^m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*
m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) + 9/8*c^b*m*x*x^(b*n)*
x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) + 1/8*c^(3*b)
*x*x^(3*b*n)*x^m*e^(3*a)/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m +
1) + 3/8*b*m^2*n*x*x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2
+ 4*m + 1)*c^b*x^(b*n)) - 27/8*b^2*n^2*x*x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2
*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^b*x^(b*n)) + 3/8*b*m^2*n*x*x^m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^
2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) - 1/8*b^2*n^2*x*x^m*e^(-3*a)/((9*b^4*
n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) + 3/8*c^b
*x*x^(b*n)*x^m*e^a/(9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1) +
3/8*m^3*x*x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)
*c^b*x^(b*n)) + 3/4*b*m*n*x*x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3
+ 6*m^2 + 4*m + 1)*c^b*x^(b*n)) + 1/8*m^3*x*x^m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 1
0*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) + 3/4*b*m*n*x*x^m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*m^2*n^
2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) + 9/8*m^2*x*x^m*e^(-a)/((9*b
^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^b*x^(b*n)) + 3/8*b*n*x*
x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^b*x^(b*
n)) + 3/8*m^2*x*x^m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 +
4*m + 1)*c^(3*b)*x^(3*b*n)) + 3/8*b*n*x*x^m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^
2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) + 9/8*m*x*x^m*e^(-a)/((9*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2
*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^b*x^(b*n)) + 3/8*m*x*x^m*e^(-3*a)/((9*b^4*n^4 - 10*b^2*
m^2*n^2 - 20*b^2*m*n^2 + m^4 - 10*b^2*n^2 + 4*m^3 + 6*m^2 + 4*m + 1)*c^(3*b)*x^(3*b*n)) + 3/8*x*x^m*e^(-a)/((9
*b^4*n^4 - 10*b^2*m^2*n^2 - 20*b^2*m*n^2 + m^4 ...

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Mupad [B]
time = 1.18, size = 117, normalized size = 0.58 \begin {gather*} \frac {3\,x\,x^m\,{\mathrm {e}}^{-a}}{{\left (c\,x^n\right )}^b\,\left (8\,m-8\,b\,n+8\right )}+\frac {x\,x^m\,{\mathrm {e}}^{-3\,a}}{{\left (c\,x^n\right )}^{3\,b}\,\left (8\,m-24\,b\,n+8\right )}+\frac {x\,x^m\,{\mathrm {e}}^{3\,a}\,{\left (c\,x^n\right )}^{3\,b}}{8\,m+24\,b\,n+8}+\frac {3\,x\,x^m\,{\mathrm {e}}^a\,{\left (c\,x^n\right )}^b}{8\,m+8\,b\,n+8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^m*cosh(a + b*log(c*x^n))^3,x)

[Out]

(3*x*x^m*exp(-a))/((c*x^n)^b*(8*m - 8*b*n + 8)) + (x*x^m*exp(-3*a))/((c*x^n)^(3*b)*(8*m - 24*b*n + 8)) + (x*x^
m*exp(3*a)*(c*x^n)^(3*b))/(8*m + 24*b*n + 8) + (3*x*x^m*exp(a)*(c*x^n)^b)/(8*m + 8*b*n + 8)

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