Optimal. Leaf size=67 \[ \frac {35 c^3 \text {Chi}\left (\sinh ^{-1}(a x)\right )}{64 a}+\frac {21 c^3 \text {Chi}\left (3 \sinh ^{-1}(a x)\right )}{64 a}+\frac {7 c^3 \text {Chi}\left (5 \sinh ^{-1}(a x)\right )}{64 a}+\frac {c^3 \text {Chi}\left (7 \sinh ^{-1}(a x)\right )}{64 a} \]
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Rubi [A]
time = 0.08, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {5791, 3393,
3382} \begin {gather*} \frac {35 c^3 \text {Chi}\left (\sinh ^{-1}(a x)\right )}{64 a}+\frac {21 c^3 \text {Chi}\left (3 \sinh ^{-1}(a x)\right )}{64 a}+\frac {7 c^3 \text {Chi}\left (5 \sinh ^{-1}(a x)\right )}{64 a}+\frac {c^3 \text {Chi}\left (7 \sinh ^{-1}(a x)\right )}{64 a} \end {gather*}
Antiderivative was successfully verified.
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Rule 3382
Rule 3393
Rule 5791
Rubi steps
\begin {align*} \int \frac {\left (c+a^2 c x^2\right )^3}{\sinh ^{-1}(a x)} \, dx &=\frac {c^3 \text {Subst}\left (\int \frac {\cosh ^7(x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{a}\\ &=\frac {c^3 \text {Subst}\left (\int \left (\frac {35 \cosh (x)}{64 x}+\frac {21 \cosh (3 x)}{64 x}+\frac {7 \cosh (5 x)}{64 x}+\frac {\cosh (7 x)}{64 x}\right ) \, dx,x,\sinh ^{-1}(a x)\right )}{a}\\ &=\frac {c^3 \text {Subst}\left (\int \frac {\cosh (7 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a}+\frac {\left (7 c^3\right ) \text {Subst}\left (\int \frac {\cosh (5 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a}+\frac {\left (21 c^3\right ) \text {Subst}\left (\int \frac {\cosh (3 x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a}+\frac {\left (35 c^3\right ) \text {Subst}\left (\int \frac {\cosh (x)}{x} \, dx,x,\sinh ^{-1}(a x)\right )}{64 a}\\ &=\frac {35 c^3 \text {Chi}\left (\sinh ^{-1}(a x)\right )}{64 a}+\frac {21 c^3 \text {Chi}\left (3 \sinh ^{-1}(a x)\right )}{64 a}+\frac {7 c^3 \text {Chi}\left (5 \sinh ^{-1}(a x)\right )}{64 a}+\frac {c^3 \text {Chi}\left (7 \sinh ^{-1}(a x)\right )}{64 a}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 43, normalized size = 0.64 \begin {gather*} \frac {c^3 \left (35 \text {Chi}\left (\sinh ^{-1}(a x)\right )+21 \text {Chi}\left (3 \sinh ^{-1}(a x)\right )+7 \text {Chi}\left (5 \sinh ^{-1}(a x)\right )+\text {Chi}\left (7 \sinh ^{-1}(a x)\right )\right )}{64 a} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.28, size = 42, normalized size = 0.63
method | result | size |
derivativedivides | \(\frac {c^{3} \left (35 \hyperbolicCosineIntegral \left (\arcsinh \left (a x \right )\right )+21 \hyperbolicCosineIntegral \left (3 \arcsinh \left (a x \right )\right )+7 \hyperbolicCosineIntegral \left (5 \arcsinh \left (a x \right )\right )+\hyperbolicCosineIntegral \left (7 \arcsinh \left (a x \right )\right )\right )}{64 a}\) | \(42\) |
default | \(\frac {c^{3} \left (35 \hyperbolicCosineIntegral \left (\arcsinh \left (a x \right )\right )+21 \hyperbolicCosineIntegral \left (3 \arcsinh \left (a x \right )\right )+7 \hyperbolicCosineIntegral \left (5 \arcsinh \left (a x \right )\right )+\hyperbolicCosineIntegral \left (7 \arcsinh \left (a x \right )\right )\right )}{64 a}\) | \(42\) |
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*} c^{3} \left (\int \frac {3 a^{2} x^{2}}{\operatorname {asinh}{\left (a x \right )}}\, dx + \int \frac {3 a^{4} x^{4}}{\operatorname {asinh}{\left (a x \right )}}\, dx + \int \frac {a^{6} x^{6}}{\operatorname {asinh}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {asinh}{\left (a x \right )}}\, dx\right ) \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 = -0.01 \begin {gather*} \int \frac {{\left (c\,a^2\,x^2+c\right )}^3}{\mathrm {asinh}\left (a\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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