Optimal. Leaf size=203 \[ -\frac {6 b^3 n^3 x^{1+m} \cosh \left (a+b \log \left (c x^n\right )\right )}{\left ((1+m)^2-9 b^2 n^2\right ) \left ((1+m)^2-b^2 n^2\right )}+\frac {6 b^2 (1+m) n^2 x^{1+m} \sinh \left (a+b \log \left (c x^n\right )\right )}{\left ((1+m)^2-9 b^2 n^2\right ) \left ((1+m)^2-b^2 n^2\right )}-\frac {3 b n x^{1+m} \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-9 b^2 n^2}+\frac {(1+m) x^{1+m} \sinh ^3\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-9 b^2 n^2} \]
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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 = {5640, 5638}
\begin {gather*} \frac {(m+1) x^{m+1} \sinh ^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} \sinh \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 ^2\left (a+b \log \left (c x^n\right )\right ) \cosh \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} \cosh \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.
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Rule 5638
Rule 5640
Rubi steps
\begin {align*} \int x^m \sinh ^3\left (a+b \log \left (c x^n\right )\right ) \, dx &=-\frac {3 b n x^{1+m} \cosh \left (a+b \log \left (c x^n\right )\right ) \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-9 b^2 n^2}+\frac {(1+m) x^{1+m} \sinh ^3\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 \sinh \left (a+b \log \left (c x^n\right )\right ) \, dx}{(1+m)^2-9 b^2 n^2}\\ &=-\frac {6 b^3 n^3 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 {6 b^2 (1+m) n^2 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 \left (a+b \log \left (c x^n\right )\right ) \sinh ^2\left (a+b \log \left (c x^n\right )\right )}{(1+m)^2-9 b^2 n^2}+\frac {(1+m) x^{1+m} \sinh ^3\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 = 0.97, size = 292, normalized size = 1.44 \begin {gather*} \frac {1}{4} x^{1+m} \left (-\frac {3 \cosh (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 \sinh (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 {\cosh (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 {\sinh (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.
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Maple [F]
time = 2.15, size = 0, normalized size = 0.00 \[\int x^{m} \left (\sinh ^{3}\left (a +b \ln \left (c \,x^{n}\right )\right )\right )\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, 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.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 585 vs.
\(2 (214) = 428\).
time = 0.39, size = 585, normalized size = 2.88 \begin {gather*} \frac {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 )^{3} \cosh \left (m \log \left (x\right )\right ) - 3 \, {\left (9 \, 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 ) \cosh \left (m \log \left (x\right )\right ) + {\left ({\left (m^{3} - {\left (b^{2} m + b^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1\right )} x \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 \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} + 9 \, {\left ({\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 ) \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 \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 ({\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 )^{2} \cosh \left (m \log \left (x\right )\right ) - {\left (m^{3} - 9 \, {\left (b^{2} m + b^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, m + 1\right )} x \cosh \left (m \log \left (x\right )\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 )^{2} - {\left (m^{3} - 9 \, {\left (b^{2} m + b^{2}\right )} n^{2} + 3 \, m^{2} + 3 \, 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 ) + 3 \, {\left ({\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 )^{3} - {\left (9 \, 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 )\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.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \begin {cases} \log {\left (x \right )} \sinh ^{3}{\left (a \right )} & \text {for}\: b = 0 \wedge m = -1 \\- \int x^{m} \sinh ^{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} \sinh ^{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} \sinh ^{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} \sinh ^{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 {9 b^{3} n^{3} 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 {6 b^{3} n^{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 {7 b^{2} m n^{2} 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 {6 b^{2} m n^{2} 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 {7 b^{2} n^{2} 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 {6 b^{2} n^{2} 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 m^{2} n 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 {6 b m n 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 {3 b n 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 {m^{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 {3 m^{2} 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 {3 m 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 {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} & \text {otherwise} \end {cases} \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. 3225 vs.
\(2 (214) = 428\).
time = 0.50, 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.
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Mupad [B]
time = 0.85, size = 118, 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.
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