Optimal. Leaf size=124 \[ \frac {\cosh (a+b x)}{8 x}-\frac {\cosh (3 a+3 b x)}{16 x}-\frac {\cosh (5 a+5 b x)}{16 x}-\frac {1}{8} b \text {Chi}(b x) \sinh (a)+\frac {3}{16} b \text {Chi}(3 b x) \sinh (3 a)+\frac {5}{16} b \text {Chi}(5 b x) \sinh (5 a)-\frac {1}{8} b \cosh (a) \text {Shi}(b x)+\frac {3}{16} b \cosh (3 a) \text {Shi}(3 b x)+\frac {5}{16} b \cosh (5 a) \text {Shi}(5 b x) \]
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Rubi [A]
time = 0.21, antiderivative size = 124, normalized size of antiderivative = 1.00, number of steps
used = 14, number of rules used = 5, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {5556, 3378,
3384, 3379, 3382} \begin {gather*} -\frac {1}{8} b \sinh (a) \text {Chi}(b x)+\frac {3}{16} b \sinh (3 a) \text {Chi}(3 b x)+\frac {5}{16} b \sinh (5 a) \text {Chi}(5 b x)-\frac {1}{8} b \cosh (a) \text {Shi}(b x)+\frac {3}{16} b \cosh (3 a) \text {Shi}(3 b x)+\frac {5}{16} b \cosh (5 a) \text {Shi}(5 b x)+\frac {\cosh (a+b x)}{8 x}-\frac {\cosh (3 a+3 b x)}{16 x}-\frac {\cosh (5 a+5 b x)}{16 x} \end {gather*}
Antiderivative was successfully verified.
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Rule 3378
Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rubi steps
\begin {align*} \int \frac {\cosh ^3(a+b x) \sinh ^2(a+b x)}{x^2} \, dx &=\int \left (-\frac {\cosh (a+b x)}{8 x^2}+\frac {\cosh (3 a+3 b x)}{16 x^2}+\frac {\cosh (5 a+5 b x)}{16 x^2}\right ) \, dx\\ &=\frac {1}{16} \int \frac {\cosh (3 a+3 b x)}{x^2} \, dx+\frac {1}{16} \int \frac {\cosh (5 a+5 b x)}{x^2} \, dx-\frac {1}{8} \int \frac {\cosh (a+b x)}{x^2} \, dx\\ &=\frac {\cosh (a+b x)}{8 x}-\frac {\cosh (3 a+3 b x)}{16 x}-\frac {\cosh (5 a+5 b x)}{16 x}-\frac {1}{8} b \int \frac {\sinh (a+b x)}{x} \, dx+\frac {1}{16} (3 b) \int \frac {\sinh (3 a+3 b x)}{x} \, dx+\frac {1}{16} (5 b) \int \frac {\sinh (5 a+5 b x)}{x} \, dx\\ &=\frac {\cosh (a+b x)}{8 x}-\frac {\cosh (3 a+3 b x)}{16 x}-\frac {\cosh (5 a+5 b x)}{16 x}-\frac {1}{8} (b \cosh (a)) \int \frac {\sinh (b x)}{x} \, dx+\frac {1}{16} (3 b \cosh (3 a)) \int \frac {\sinh (3 b x)}{x} \, dx+\frac {1}{16} (5 b \cosh (5 a)) \int \frac {\sinh (5 b x)}{x} \, dx-\frac {1}{8} (b \sinh (a)) \int \frac {\cosh (b x)}{x} \, dx+\frac {1}{16} (3 b \sinh (3 a)) \int \frac {\cosh (3 b x)}{x} \, dx+\frac {1}{16} (5 b \sinh (5 a)) \int \frac {\cosh (5 b x)}{x} \, dx\\ &=\frac {\cosh (a+b x)}{8 x}-\frac {\cosh (3 a+3 b x)}{16 x}-\frac {\cosh (5 a+5 b x)}{16 x}-\frac {1}{8} b \text {Chi}(b x) \sinh (a)+\frac {3}{16} b \text {Chi}(3 b x) \sinh (3 a)+\frac {5}{16} b \text {Chi}(5 b x) \sinh (5 a)-\frac {1}{8} b \cosh (a) \text {Shi}(b x)+\frac {3}{16} b \cosh (3 a) \text {Shi}(3 b x)+\frac {5}{16} b \cosh (5 a) \text {Shi}(5 b x)\\ \end {align*}
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Mathematica [A]
time = 0.23, size = 104, normalized size = 0.84 \begin {gather*} -\frac {-2 \cosh (a+b x)+\cosh (3 (a+b x))+\cosh (5 (a+b x))+2 b x \text {Chi}(b x) \sinh (a)-3 b x \text {Chi}(3 b x) \sinh (3 a)-5 b x \text {Chi}(5 b x) \sinh (5 a)+2 b x \cosh (a) \text {Shi}(b x)-3 b x \cosh (3 a) \text {Shi}(3 b x)-5 b x \cosh (5 a) \text {Shi}(5 b x)}{16 x} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 3.66, size = 158, normalized size = 1.27
method | result | size |
risch | \(-\frac {{\mathrm e}^{-5 b x -5 a}}{32 x}+\frac {5 b \,{\mathrm e}^{-5 a} \expIntegral \left (1, 5 b x \right )}{32}-\frac {{\mathrm e}^{-3 b x -3 a}}{32 x}+\frac {3 b \,{\mathrm e}^{-3 a} \expIntegral \left (1, 3 b x \right )}{32}+\frac {{\mathrm e}^{-b x -a}}{16 x}-\frac {b \,{\mathrm e}^{-a} \expIntegral \left (1, b x \right )}{16}+\frac {{\mathrm e}^{b x +a}}{16 x}+\frac {b \,{\mathrm e}^{a} \expIntegral \left (1, -b x \right )}{16}-\frac {{\mathrm e}^{3 b x +3 a}}{32 x}-\frac {3 b \,{\mathrm e}^{3 a} \expIntegral \left (1, -3 b x \right )}{32}-\frac {{\mathrm e}^{5 b x +5 a}}{32 x}-\frac {5 b \,{\mathrm e}^{5 a} \expIntegral \left (1, -5 b x \right )}{32}\) | \(158\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.34, size = 76, normalized size = 0.61 \begin {gather*} -\frac {5}{32} \, b e^{\left (-5 \, a\right )} \Gamma \left (-1, 5 \, b x\right ) - \frac {3}{32} \, b e^{\left (-3 \, a\right )} \Gamma \left (-1, 3 \, b x\right ) + \frac {1}{16} \, b e^{\left (-a\right )} \Gamma \left (-1, b x\right ) - \frac {1}{16} \, b e^{a} \Gamma \left (-1, -b x\right ) + \frac {3}{32} \, b e^{\left (3 \, a\right )} \Gamma \left (-1, -3 \, b x\right ) + \frac {5}{32} \, b e^{\left (5 \, a\right )} \Gamma \left (-1, -5 \, b x\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. 214 vs.
\(2 (106) = 212\).
time = 0.36, size = 214, normalized size = 1.73 \begin {gather*} -\frac {2 \, \cosh \left (b x + a\right )^{5} + 10 \, \cosh \left (b x + a\right ) \sinh \left (b x + a\right )^{4} + 2 \, \cosh \left (b x + a\right )^{3} + 2 \, {\left (10 \, \cosh \left (b x + a\right )^{3} + 3 \, \cosh \left (b x + a\right )\right )} \sinh \left (b x + a\right )^{2} - 5 \, {\left (b x {\rm Ei}\left (5 \, b x\right ) - b x {\rm Ei}\left (-5 \, b x\right )\right )} \cosh \left (5 \, a\right ) - 3 \, {\left (b x {\rm Ei}\left (3 \, b x\right ) - b x {\rm Ei}\left (-3 \, b x\right )\right )} \cosh \left (3 \, a\right ) + 2 \, {\left (b x {\rm Ei}\left (b x\right ) - b x {\rm Ei}\left (-b x\right )\right )} \cosh \left (a\right ) - 5 \, {\left (b x {\rm Ei}\left (5 \, b x\right ) + b x {\rm Ei}\left (-5 \, b x\right )\right )} \sinh \left (5 \, a\right ) - 3 \, {\left (b x {\rm Ei}\left (3 \, b x\right ) + b x {\rm Ei}\left (-3 \, b x\right )\right )} \sinh \left (3 \, a\right ) + 2 \, {\left (b x {\rm Ei}\left (b x\right ) + b x {\rm Ei}\left (-b x\right )\right )} \sinh \left (a\right ) - 4 \, \cosh \left (b x + a\right )}{32 \, x} \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*} \int \frac {\sinh ^{2}{\left (a + b x \right )} \cosh ^{3}{\left (a + b x \right )}}{x^{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.41, size = 144, normalized size = 1.16 \begin {gather*} \frac {5 \, b x {\rm Ei}\left (5 \, b x\right ) e^{\left (5 \, a\right )} + 3 \, b x {\rm Ei}\left (3 \, b x\right ) e^{\left (3 \, a\right )} + 2 \, b x {\rm Ei}\left (-b x\right ) e^{\left (-a\right )} - 3 \, b x {\rm Ei}\left (-3 \, b x\right ) e^{\left (-3 \, a\right )} - 5 \, b x {\rm Ei}\left (-5 \, b x\right ) e^{\left (-5 \, a\right )} - 2 \, b x {\rm Ei}\left (b x\right ) e^{a} - e^{\left (5 \, b x + 5 \, a\right )} - e^{\left (3 \, b x + 3 \, a\right )} + 2 \, e^{\left (b x + a\right )} + 2 \, e^{\left (-b x - a\right )} - e^{\left (-3 \, b x - 3 \, a\right )} - e^{\left (-5 \, b x - 5 \, a\right )}}{32 \, x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\mathrm {cosh}\left (a+b\,x\right )}^3\,{\mathrm {sinh}\left (a+b\,x\right )}^2}{x^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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