Optimal. Leaf size=397 \[ -\frac {3 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{128 b c^4 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^4 \sqrt {-1+c x}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{256 b c^4 \sqrt {-1+c x}}+\frac {\sqrt {1-c x} \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{256 b c^4 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{128 b c^4 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^4 \sqrt {-1+c x}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{256 b c^4 \sqrt {-1+c x}}-\frac {\sqrt {1-c x} \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{256 b c^4 \sqrt {-1+c x}} \]
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Rubi [A]
time = 0.35, antiderivative size = 397, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {5952, 5556,
3384, 3379, 3382} \begin {gather*} -\frac {3 \sqrt {1-c x} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{128 b c^4 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^4 \sqrt {c x-1}}-\frac {3 \sqrt {1-c x} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{256 b c^4 \sqrt {c x-1}}+\frac {\sqrt {1-c x} \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{256 b c^4 \sqrt {c x-1}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a+b \cosh ^{-1}(c x)}{b}\right )}{128 b c^4 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{32 b c^4 \sqrt {c x-1}}+\frac {3 \sqrt {1-c x} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{256 b c^4 \sqrt {c x-1}}-\frac {\sqrt {1-c x} \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 \left (a+b \cosh ^{-1}(c x)\right )}{b}\right )}{256 b c^4 \sqrt {c x-1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3379
Rule 3382
Rule 3384
Rule 5556
Rule 5952
Rubi steps
\begin {align*} \int \frac {x^3 \left (1-c^2 x^2\right )^{5/2}}{a+b \cosh ^{-1}(c x)} \, dx &=\frac {\sqrt {1-c^2 x^2} \int \frac {x^3 (-1+c x)^{5/2} (1+c x)^{5/2}}{a+b \cosh ^{-1}(c x)} \, dx}{\sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\cosh ^3(x) \sinh ^6(x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \left (-\frac {3 \cosh (x)}{128 (a+b x)}+\frac {\cosh (3 x)}{32 (a+b x)}-\frac {3 \cosh (7 x)}{256 (a+b x)}+\frac {\cosh (9 x)}{256 (a+b x)}\right ) \, dx,x,\cosh ^{-1}(c x)\right )}{c^4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\cosh (9 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{256 c^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cosh (7 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{256 c^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 \sqrt {1-c^2 x^2}\right ) \text {Subst}\left (\int \frac {\cosh (x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \text {Subst}\left (\int \frac {\cosh (3 x)}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{32 c^4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {\left (3 \sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \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,\cosh ^{-1}(c x)\right )}{32 c^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (3 \sqrt {1-c^2 x^2} \cosh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{256 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (\sqrt {1-c^2 x^2} \cosh \left (\frac {9 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cosh \left (\frac {9 a}{b}+9 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{256 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 \sqrt {1-c^2 x^2} \sinh \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{128 c^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \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,\cosh ^{-1}(c x)\right )}{32 c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\left (3 \sqrt {1-c^2 x^2} \sinh \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{256 c^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\left (\sqrt {1-c^2 x^2} \sinh \left (\frac {9 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sinh \left (\frac {9 a}{b}+9 x\right )}{a+b x} \, dx,x,\cosh ^{-1}(c x)\right )}{256 c^4 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=-\frac {3 \sqrt {1-c^2 x^2} \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{128 b c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{32 b c^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {3 \sqrt {1-c^2 x^2} \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (\frac {7 a}{b}+7 \cosh ^{-1}(c x)\right )}{256 b c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {\sqrt {1-c^2 x^2} \cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (\frac {9 a}{b}+9 \cosh ^{-1}(c x)\right )}{256 b c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 \sqrt {1-c^2 x^2} \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )}{128 b c^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (\frac {3 a}{b}+3 \cosh ^{-1}(c x)\right )}{32 b c^4 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 \sqrt {1-c^2 x^2} \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (\frac {7 a}{b}+7 \cosh ^{-1}(c x)\right )}{256 b c^4 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {\sqrt {1-c^2 x^2} \sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (\frac {9 a}{b}+9 \cosh ^{-1}(c x)\right )}{256 b c^4 \sqrt {-1+c x} \sqrt {1+c x}}\\ \end {align*}
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Mathematica [A]
time = 0.95, size = 216, normalized size = 0.54 \begin {gather*} \frac {\sqrt {1-c^2 x^2} \left (-6 \cosh \left (\frac {a}{b}\right ) \text {Chi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )+8 \cosh \left (\frac {3 a}{b}\right ) \text {Chi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-3 \cosh \left (\frac {7 a}{b}\right ) \text {Chi}\left (7 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+\cosh \left (\frac {9 a}{b}\right ) \text {Chi}\left (9 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+6 \sinh \left (\frac {a}{b}\right ) \text {Shi}\left (\frac {a}{b}+\cosh ^{-1}(c x)\right )-8 \sinh \left (\frac {3 a}{b}\right ) \text {Shi}\left (3 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )+3 \sinh \left (\frac {7 a}{b}\right ) \text {Shi}\left (7 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )-\sinh \left (\frac {9 a}{b}\right ) \text {Shi}\left (9 \left (\frac {a}{b}+\cosh ^{-1}(c x)\right )\right )\right )}{256 c^4 \sqrt {\frac {-1+c x}{1+c x}} (b+b c x)} \end {gather*}
Warning: Unable to verify antiderivative.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(730\) vs.
\(2(349)=698\).
time = 4.29, size = 731, normalized size = 1.84
method | result | size |
default | \(\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \expIntegral \left (1, 9 \,\mathrm {arccosh}\left (c x \right )+\frac {9 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+9 a}{b}}}{512 \left (c x +1\right ) c^{4} \left (c x -1\right ) b}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \expIntegral \left (1, -9 \,\mathrm {arccosh}\left (c x \right )-\frac {9 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\mathrm {arccosh}\left (c x \right )+9 a}{b}}}{512 \left (c x +1\right ) c^{4} \left (c x -1\right ) b}-\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \expIntegral \left (1, 7 \,\mathrm {arccosh}\left (c x \right )+\frac {7 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+7 a}{b}}}{512 \left (c x +1\right ) c^{4} \left (c x -1\right ) b}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \expIntegral \left (1, 3 \,\mathrm {arccosh}\left (c x \right )+\frac {3 a}{b}\right ) {\mathrm e}^{\frac {b \,\mathrm {arccosh}\left (c x \right )+3 a}{b}}}{64 \left (c x +1\right ) c^{4} \left (c x -1\right ) b}-\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \expIntegral \left (1, \mathrm {arccosh}\left (c x \right )+\frac {a}{b}\right ) {\mathrm e}^{\frac {a +b \,\mathrm {arccosh}\left (c x \right )}{b}}}{256 \left (c x +1\right ) c^{4} \left (c x -1\right ) b}-\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \expIntegral \left (1, -\mathrm {arccosh}\left (c x \right )-\frac {a}{b}\right ) {\mathrm e}^{-\frac {-b \,\mathrm {arccosh}\left (c x \right )+a}{b}}}{256 \left (c x +1\right ) c^{4} \left (c x -1\right ) b}+\frac {\sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \expIntegral \left (1, -3 \,\mathrm {arccosh}\left (c x \right )-\frac {3 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\mathrm {arccosh}\left (c x \right )+3 a}{b}}}{64 \left (c x +1\right ) c^{4} \left (c x -1\right ) b}-\frac {3 \sqrt {-c^{2} x^{2}+1}\, \left (-\sqrt {c x +1}\, \sqrt {c x -1}\, x c +c^{2} x^{2}-1\right ) \expIntegral \left (1, -7 \,\mathrm {arccosh}\left (c x \right )-\frac {7 a}{b}\right ) {\mathrm e}^{-\frac {-b \,\mathrm {arccosh}\left (c x \right )+7 a}{b}}}{512 \left (c x +1\right ) c^{4} \left (c x -1\right ) b}\) | \(731\) |
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(-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 [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^3\,{\left (1-c^2\,x^2\right )}^{5/2}}{a+b\,\mathrm {acosh}\left (c\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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