Optimal. Leaf size=204 \[ -\frac {x^5}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{8 b^2 c^6}-\frac {15 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^6}+\frac {5 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^6}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{8 b^2 c^6}+\frac {15 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^6}-\frac {5 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^6} \]
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Rubi [A]
time = 0.29, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps
used = 13, number of rules used = 6, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {5818, 5780,
5556, 3384, 3379, 3382} \begin {gather*} \frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{8 b^2 c^6}-\frac {15 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^6}+\frac {5 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^6}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \sinh ^{-1}(c x)}{b}\right )}{8 b^2 c^6}+\frac {15 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^6}-\frac {5 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 \left (a+b \sinh ^{-1}(c x)\right )}{b}\right )}{16 b^2 c^6}-\frac {x^5}{b c \left (a+b \sinh ^{-1}(c x)\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5780
Rule 5818
Rubi steps
\begin {align*} \int \frac {x^5}{\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^2} \, dx &=-\frac {x^5}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {5 \int \frac {x^4}{a+b \sinh ^{-1}(c x)} \, dx}{b c}\\ &=-\frac {x^5}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {5 \text {Subst}\left (\int \frac {\cosh (x) \sinh ^4(x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{b c^6}\\ &=-\frac {x^5}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {5 \text {Subst}\left (\int \left (\frac {\cosh (x)}{8 (a+b x)}-\frac {3 \cosh (3 x)}{16 (a+b x)}+\frac {\cosh (5 x)}{16 (a+b x)}\right ) \, dx,x,\sinh ^{-1}(c x)\right )}{b c^6}\\ &=-\frac {x^5}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {5 \text {Subst}\left (\int \frac {\cosh (5 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^6}+\frac {5 \text {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^6}-\frac {15 \text {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^6}\\ &=-\frac {x^5}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {\left (5 \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^6}-\frac {\left (15 \cosh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^6}+\frac {\left (5 \cosh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^6}-\frac {\left (5 \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{8 b c^6}+\frac {\left (15 \sinh \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^6}-\frac {\left (5 \sinh \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sinh ^{-1}(c x)\right )}{16 b c^6}\\ &=-\frac {x^5}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {5 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{8 b^2 c^6}-\frac {15 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{16 b^2 c^6}+\frac {5 \cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (\frac {5 a}{b}+5 \sinh ^{-1}(c x)\right )}{16 b^2 c^6}-\frac {5 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )}{8 b^2 c^6}+\frac {15 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \sinh ^{-1}(c x)\right )}{16 b^2 c^6}-\frac {5 \sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (\frac {5 a}{b}+5 \sinh ^{-1}(c x)\right )}{16 b^2 c^6}\\ \end {align*}
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Mathematica [A]
time = 0.25, size = 158, normalized size = 0.77 \begin {gather*} -\frac {x^5}{b c \left (a+b \sinh ^{-1}(c x)\right )}+\frac {5 \left (2 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )-3 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )+\cosh \left (\frac {5 a}{b}\right ) \text {Chi}\left (5 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-2 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\sinh ^{-1}(c x)\right )+3 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )-\sinh \left (\frac {5 a}{b}\right ) \text {Shi}\left (5 \left (\frac {a}{b}+\sinh ^{-1}(c x)\right )\right )\right )}{16 b^2 c^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(632\) vs.
\(2(192)=384\).
time = 7.22, size = 633, normalized size = 3.10
method | result | size |
default | \(-\frac {16 c^{5} x^{5}-16 \sqrt {c^{2} x^{2}+1}\, c^{4} x^{4}+20 c^{3} x^{3}-12 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}+5 c x -\sqrt {c^{2} x^{2}+1}}{32 c^{6} b \left (a +b \arcsinh \left (c x \right )\right )}-\frac {5 \,{\mathrm e}^{\frac {5 a}{b}} \expIntegral \left (1, 5 \arcsinh \left (c x \right )+\frac {5 a}{b}\right )}{32 c^{6} b^{2}}+\frac {-\frac {5 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{8}+\frac {5 c^{3} x^{3}}{8}-\frac {5 \sqrt {c^{2} x^{2}+1}}{32}+\frac {15 c x}{32}}{c^{6} b \left (a +b \arcsinh \left (c x \right )\right )}+\frac {15 \,{\mathrm e}^{\frac {3 a}{b}} \expIntegral \left (1, 3 \arcsinh \left (c x \right )+\frac {3 a}{b}\right )}{32 c^{6} b^{2}}-\frac {5 \left (-\sqrt {c^{2} x^{2}+1}+c x \right )}{16 c^{6} b \left (a +b \arcsinh \left (c x \right )\right )}-\frac {5 \,{\mathrm e}^{\frac {a}{b}} \expIntegral \left (1, \arcsinh \left (c x \right )+\frac {a}{b}\right )}{16 c^{6} b^{2}}-\frac {5 \left (\arcsinh \left (c x \right ) {\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right ) b +{\mathrm e}^{-\frac {a}{b}} \expIntegral \left (1, -\arcsinh \left (c x \right )-\frac {a}{b}\right ) a +b c x +\sqrt {c^{2} x^{2}+1}\, b \right )}{16 c^{6} b^{2} \left (a +b \arcsinh \left (c x \right )\right )}+\frac {\frac {5 b \,c^{3} x^{3}}{8}+\frac {5 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}}{8}+\frac {15 \arcsinh \left (c x \right ) {\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \arcsinh \left (c x \right )-\frac {3 a}{b}\right ) b}{32}+\frac {15 \,{\mathrm e}^{-\frac {3 a}{b}} \expIntegral \left (1, -3 \arcsinh \left (c x \right )-\frac {3 a}{b}\right ) a}{32}+\frac {15 b c x}{32}+\frac {5 \sqrt {c^{2} x^{2}+1}\, b}{32}}{c^{6} b^{2} \left (a +b \arcsinh \left (c x \right )\right )}-\frac {16 b \,c^{5} x^{5}+16 \sqrt {c^{2} x^{2}+1}\, b \,c^{4} x^{4}+20 b \,c^{3} x^{3}+12 \sqrt {c^{2} x^{2}+1}\, b \,c^{2} x^{2}+5 \arcsinh \left (c x \right ) \expIntegral \left (1, -5 \arcsinh \left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {5 a}{b}} b +5 \expIntegral \left (1, -5 \arcsinh \left (c x \right )-\frac {5 a}{b}\right ) {\mathrm e}^{-\frac {5 a}{b}} a +5 b c x +\sqrt {c^{2} x^{2}+1}\, b}{32 c^{6} b^{2} \left (a +b \arcsinh \left (c x \right )\right )}\) | \(633\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {x^{5}}{\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2} \sqrt {c^{2} x^{2} + 1}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 = -0.00 \begin {gather*} \int \frac {x^5}{{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\sqrt {c^2\,x^2+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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