Optimal. Leaf size=266 \[ \frac {24 b^4 n^4 x^{1+m}}{(1+m) \left ((1+m)^2-16 b^2 n^2\right ) \left ((1+m)^2-4 b^2 n^2\right )}-\frac {12 b^2 (1+m) n^2 x^{1+m} \cosh ^2\left (a+b \log \left (c x^n\right )\right )}{\left ((1+m)^2-16 b^2 n^2\right ) \left ((1+m)^2-4 b^2 n^2\right )}+\frac {(1+m) x^{1+m} \cosh ^4\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-16 b^2 n^2}+\frac {24 b^3 n^3 x^{1+m} \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{\left ((1+m)^2-16 b^2 n^2\right ) \left ((1+m)^2-4 b^2 n^2\right )}-\frac {4 b n x^{1+m} \cosh ^3\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-16 b^2 n^2} \]
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Rubi [A]
time = 0.10, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {5641, 30}
\begin {gather*} \frac {(m+1) x^{m+1} \cosh ^4\left (a+b \log \left (c x^n\right )\right )}{-16 b^2 n^2+m^2+2 m+1}-\frac {12 b^2 (m+1) n^2 x^{m+1} \cosh ^2\left (a+b \log \left (c x^n\right )\right )}{\left ((m+1)^2-16 b^2 n^2\right ) \left (-4 b^2 n^2+m^2+2 m+1\right )}-\frac {4 b n x^{m+1} \sinh \left (a+b \log \left (c x^n\right )\right ) \cosh ^3\left (a+b \log \left (c x^n\right )\right )}{(m+1)^2-16 b^2 n^2}+\frac {24 b^3 n^3 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 )}{64 b^4 n^4-20 b^2 (m+1)^2 n^2+(m+1)^4}+\frac {24 b^4 n^4 x^{m+1}}{(m+1) \left ((m+1)^2-16 b^2 n^2\right ) \left ((m+1)^2-4 b^2 n^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 30
Rule 5641
Rubi steps
\begin {align*} \int x^m \cosh ^4\left (a+b \log \left (c x^n\right )\right ) \, dx &=\frac {(1+m) x^{1+m} \cosh ^4\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-16 b^2 n^2}-\frac {4 b n x^{1+m} \cosh ^3\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-16 b^2 n^2}-\frac {\left (12 b^2 n^2\right ) \int x^m \cosh ^2\left (a+b \log \left (c x^n\right )\right ) \, dx}{(1+m)^2-16 b^2 n^2}\\ &=-\frac {12 b^2 (1+m) n^2 x^{1+m} \cosh ^2\left (a+b \log \left (c x^n\right )\right )}{(1+m)^4-20 b^2 (1+m)^2 n^2+64 b^4 n^4}+\frac {(1+m) x^{1+m} \cosh ^4\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-16 b^2 n^2}+\frac {24 b^3 n^3 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)^4-20 b^2 (1+m)^2 n^2+64 b^4 n^4}-\frac {4 b n x^{1+m} \cosh ^3\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-16 b^2 n^2}+\frac {\left (24 b^4 n^4\right ) \int x^m \, dx}{(1+m)^4-20 b^2 (1+m)^2 n^2+64 b^4 n^4}\\ &=\frac {24 b^4 n^4 x^{1+m}}{(1+m) \left ((1+m)^4-20 b^2 (1+m)^2 n^2+64 b^4 n^4\right )}-\frac {12 b^2 (1+m) n^2 x^{1+m} \cosh ^2\left (a+b \log \left (c x^n\right )\right )}{(1+m)^4-20 b^2 (1+m)^2 n^2+64 b^4 n^4}+\frac {(1+m) x^{1+m} \cosh ^4\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-16 b^2 n^2}+\frac {24 b^3 n^3 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)^4-20 b^2 (1+m)^2 n^2+64 b^4 n^4}-\frac {4 b n x^{1+m} \cosh ^3\left (a+b \log \left (c x^n\right )\right ) \sinh \left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-16 b^2 n^2}\\ \end {align*}
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Mathematica [A]
time = 2.43, size = 311, normalized size = 1.17 \begin {gather*} \frac {1}{8} x^{1+m} \left (\frac {3}{1+m}+\frac {4 \sinh (2 b n \log (x)) \left (-2 b n \cosh \left (2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+(1+m) \sinh \left (2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )\right )}{(1+m-2 b n) (1+m+2 b n)}+\frac {4 \cosh (2 b n \log (x)) \left ((1+m) \cosh \left (2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )-2 b n \sinh \left (2 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )\right )}{(1+m-2 b n) (1+m+2 b n)}+\frac {\sinh (4 b n \log (x)) \left (-4 b n \cosh \left (4 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )+(1+m) \sinh \left (4 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )\right )}{(1+m-4 b n) (1+m+4 b n)}+\frac {\cosh (4 b n \log (x)) \left ((1+m) \cosh \left (4 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )-4 b n \sinh \left (4 \left (a-b n \log (x)+b \log \left (c x^n\right )\right )\right )\right )}{(1+m-4 b n) (1+m+4 b n)}\right ) \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 2.76, size = 0, normalized size = 0.00 \[\int x^{m} \left (\cosh ^{4}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 161, normalized size = 0.61 \begin {gather*} \frac {c^{4 \, b} x e^{\left (4 \, b \log \left (x^{n}\right ) + m \log \left (x\right ) + 4 \, a\right )}}{16 \, {\left (4 \, b n + m + 1\right )}} + \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 e^{\left (-4 \, b \log \left (x^{n}\right ) + m \log \left (x\right ) - 4 \, a\right )}}{16 \, {\left (4 \, b c^{4 \, b} n - c^{4 \, b} {\left (m + 1\right )}\right )}} + \frac {3 \, x^{m + 1}}{8 \, {\left (m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1123 vs.
\(2 (283) = 566\).
time = 0.47, size = 1123, normalized size = 4.22 \begin {gather*} \frac {{\left (m^{4} + 4 \, m^{3} - 4 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, m + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} \cosh \left (m \log \left (x\right )\right ) + 4 \, {\left (m^{4} + 4 \, m^{3} - 16 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, 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 ({\left (m^{4} + 4 \, m^{3} - 4 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, m + 1\right )} x \cosh \left (m \log \left (x\right )\right ) + {\left (m^{4} + 4 \, m^{3} - 4 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, 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 )^{4} + 16 \, {\left ({\left (4 \, {\left (b^{3} m + b^{3}\right )} n^{3} - {\left (b m^{3} + 3 \, b m^{2} + 3 \, b m + b\right )} n\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 (4 \, {\left (b^{3} m + b^{3}\right )} n^{3} - {\left (b m^{3} + 3 \, b m^{2} + 3 \, b m + b\right )} n\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 )^{3} + 3 \, {\left (64 \, b^{4} n^{4} + m^{4} + 4 \, m^{3} - 20 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, m + 1\right )} x \cosh \left (m \log \left (x\right )\right ) + 2 \, {\left (3 \, {\left (m^{4} + 4 \, m^{3} - 4 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, 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 ) + 2 \, {\left (m^{4} + 4 \, m^{3} - 16 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, m + 1\right )} x \cosh \left (m \log \left (x\right )\right ) + {\left (3 \, {\left (m^{4} + 4 \, m^{3} - 4 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, m + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 2 \, {\left (m^{4} + 4 \, m^{3} - 16 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, m + 1\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 )^{2} + 16 \, {\left ({\left (4 \, {\left (b^{3} m + b^{3}\right )} n^{3} - {\left (b m^{3} + 3 \, b m^{2} + 3 \, b m + b\right )} n\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 ) + {\left (16 \, {\left (b^{3} m + b^{3}\right )} n^{3} - {\left (b m^{3} + 3 \, b m^{2} + 3 \, b m + b\right )} n\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 ({\left (4 \, {\left (b^{3} m + b^{3}\right )} n^{3} - {\left (b m^{3} + 3 \, b m^{2} + 3 \, b m + b\right )} n\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{3} + {\left (16 \, {\left (b^{3} m + b^{3}\right )} n^{3} - {\left (b m^{3} + 3 \, b m^{2} + 3 \, b m + b\right )} n\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 )\right )} \sinh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right ) + {\left ({\left (m^{4} + 4 \, m^{3} - 4 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, m + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{4} + 4 \, {\left (m^{4} + 4 \, m^{3} - 16 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, m + 1\right )} x \cosh \left (b n \log \left (x\right ) + b \log \left (c\right ) + a\right )^{2} + 3 \, {\left (64 \, b^{4} n^{4} + m^{4} + 4 \, m^{3} - 20 \, {\left (b^{2} m^{2} + 2 \, b^{2} m + b^{2}\right )} n^{2} + 6 \, m^{2} + 4 \, m + 1\right )} x\right )} \sinh \left (m \log \left (x\right )\right )}{8 \, {\left (m^{5} + 64 \, {\left (b^{4} m + b^{4}\right )} n^{4} + 5 \, m^{4} + 10 \, m^{3} - 20 \, {\left (b^{2} m^{3} + 3 \, b^{2} m^{2} + 3 \, b^{2} m + b^{2}\right )} n^{2} + 10 \, m^{2} + 5 \, m + 1\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 6880 vs.
\(2 (283) = 566\).
time = 0.55, size = 6880, normalized size = 25.86 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.19, size = 134, normalized size = 0.50 \begin {gather*} \frac {3\,x\,x^m}{8\,m+8}+\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}+\frac {x\,x^m\,{\mathrm {e}}^{-4\,a}}{{\left (c\,x^n\right )}^{4\,b}\,\left (16\,m-64\,b\,n+16\right )}+\frac {x\,x^m\,{\mathrm {e}}^{4\,a}\,{\left (c\,x^n\right )}^{4\,b}}{16\,m+64\,b\,n+16} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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