Optimal. Leaf size=180 \[ \frac {73 b d^2 x \sqrt {1+c^2 x^2}}{3072 c^3}-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2}}{4608 c}-\frac {43 b c d^2 x^5 \sqrt {1+c^2 x^2}}{1152}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {1+c^2 x^2}-\frac {73 b d^2 \sinh ^{-1}(c x)}{3072 c^4}+\frac {1}{4} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^2 d^2 x^6 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sinh ^{-1}(c x)\right ) \]
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Rubi [A]
time = 0.11, antiderivative size = 180, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 8, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {272, 45, 5803,
12, 1281, 470, 327, 221} \begin {gather*} \frac {1}{8} c^4 d^2 x^8 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^2 d^2 x^6 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{4} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )-\frac {73 b d^2 \sinh ^{-1}(c x)}{3072 c^4}-\frac {43 b c d^2 x^5 \sqrt {c^2 x^2+1}}{1152}-\frac {73 b d^2 x^3 \sqrt {c^2 x^2+1}}{4608 c}+\frac {73 b d^2 x \sqrt {c^2 x^2+1}}{3072 c^3}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {c^2 x^2+1} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 45
Rule 221
Rule 272
Rule 327
Rule 470
Rule 1281
Rule 5803
Rubi steps
\begin {align*} \int x^3 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {1}{4} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^2 d^2 x^6 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {d^2 x^4 \left (6+8 c^2 x^2+3 c^4 x^4\right )}{24 \sqrt {1+c^2 x^2}} \, dx\\ &=\frac {1}{4} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^2 d^2 x^6 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{24} \left (b c d^2\right ) \int \frac {x^4 \left (6+8 c^2 x^2+3 c^4 x^4\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=-\frac {1}{64} b c^3 d^2 x^7 \sqrt {1+c^2 x^2}+\frac {1}{4} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^2 d^2 x^6 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (b d^2\right ) \int \frac {x^4 \left (48 c^2+43 c^4 x^2\right )}{\sqrt {1+c^2 x^2}} \, dx}{192 c}\\ &=-\frac {43 b c d^2 x^5 \sqrt {1+c^2 x^2}}{1152}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {1+c^2 x^2}+\frac {1}{4} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^2 d^2 x^6 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (73 b c d^2\right ) \int \frac {x^4}{\sqrt {1+c^2 x^2}} \, dx}{1152}\\ &=-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2}}{4608 c}-\frac {43 b c d^2 x^5 \sqrt {1+c^2 x^2}}{1152}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {1+c^2 x^2}+\frac {1}{4} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^2 d^2 x^6 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sinh ^{-1}(c x)\right )+\frac {\left (73 b d^2\right ) \int \frac {x^2}{\sqrt {1+c^2 x^2}} \, dx}{1536 c}\\ &=\frac {73 b d^2 x \sqrt {1+c^2 x^2}}{3072 c^3}-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2}}{4608 c}-\frac {43 b c d^2 x^5 \sqrt {1+c^2 x^2}}{1152}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {1+c^2 x^2}+\frac {1}{4} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^2 d^2 x^6 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (73 b d^2\right ) \int \frac {1}{\sqrt {1+c^2 x^2}} \, dx}{3072 c^3}\\ &=\frac {73 b d^2 x \sqrt {1+c^2 x^2}}{3072 c^3}-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2}}{4608 c}-\frac {43 b c d^2 x^5 \sqrt {1+c^2 x^2}}{1152}-\frac {1}{64} b c^3 d^2 x^7 \sqrt {1+c^2 x^2}-\frac {73 b d^2 \sinh ^{-1}(c x)}{3072 c^4}+\frac {1}{4} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^2 d^2 x^6 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} c^4 d^2 x^8 \left (a+b \sinh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 115, normalized size = 0.64 \begin {gather*} \frac {d^2 \left (384 a c^4 x^4 \left (6+8 c^2 x^2+3 c^4 x^4\right )-b c x \sqrt {1+c^2 x^2} \left (-219+146 c^2 x^2+344 c^4 x^4+144 c^6 x^6\right )+3 b \left (-73+768 c^4 x^4+1024 c^6 x^6+384 c^8 x^8\right ) \sinh ^{-1}(c x)\right )}{9216 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.75, size = 146, normalized size = 0.81
method | result | size |
derivativedivides | \(\frac {d^{2} a \left (\frac {\left (c^{2} x^{2}+1\right )^{4}}{8}-\frac {\left (c^{2} x^{2}+1\right )^{3}}{6}\right )+d^{2} b \left (\frac {\arcsinh \left (c x \right ) c^{8} x^{8}}{8}+\frac {\arcsinh \left (c x \right ) c^{6} x^{6}}{3}+\frac {\arcsinh \left (c x \right ) c^{4} x^{4}}{4}-\frac {73 \arcsinh \left (c x \right )}{3072}-\frac {c x \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}+\frac {11 c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1152}+\frac {55 c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{4608}+\frac {55 \sqrt {c^{2} x^{2}+1}\, c x}{3072}\right )}{c^{4}}\) | \(146\) |
default | \(\frac {d^{2} a \left (\frac {\left (c^{2} x^{2}+1\right )^{4}}{8}-\frac {\left (c^{2} x^{2}+1\right )^{3}}{6}\right )+d^{2} b \left (\frac {\arcsinh \left (c x \right ) c^{8} x^{8}}{8}+\frac {\arcsinh \left (c x \right ) c^{6} x^{6}}{3}+\frac {\arcsinh \left (c x \right ) c^{4} x^{4}}{4}-\frac {73 \arcsinh \left (c x \right )}{3072}-\frac {c x \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{64}+\frac {11 c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{1152}+\frac {55 c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{4608}+\frac {55 \sqrt {c^{2} x^{2}+1}\, c x}{3072}\right )}{c^{4}}\) | \(146\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 292, normalized size = 1.62 \begin {gather*} \frac {1}{8} \, a c^{4} d^{2} x^{8} + \frac {1}{3} \, a c^{2} d^{2} x^{6} + \frac {1}{3072} \, {\left (384 \, x^{8} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{7}}{c^{2}} - \frac {56 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{4}} + \frac {70 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{6}} - \frac {105 \, \sqrt {c^{2} x^{2} + 1} x}{c^{8}} + \frac {105 \, \operatorname {arsinh}\left (c x\right )}{c^{9}}\right )} c\right )} b c^{4} d^{2} + \frac {1}{4} \, a d^{2} x^{4} + \frac {1}{144} \, {\left (48 \, x^{6} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{5}}{c^{2}} - \frac {10 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{4}} + \frac {15 \, \sqrt {c^{2} x^{2} + 1} x}{c^{6}} - \frac {15 \, \operatorname {arsinh}\left (c x\right )}{c^{7}}\right )} c\right )} b c^{2} d^{2} + \frac {1}{32} \, {\left (8 \, x^{4} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {2 \, \sqrt {c^{2} x^{2} + 1} x^{3}}{c^{2}} - \frac {3 \, \sqrt {c^{2} x^{2} + 1} x}{c^{4}} + \frac {3 \, \operatorname {arsinh}\left (c x\right )}{c^{5}}\right )} c\right )} b d^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.37, size = 161, normalized size = 0.89 \begin {gather*} \frac {1152 \, a c^{8} d^{2} x^{8} + 3072 \, a c^{6} d^{2} x^{6} + 2304 \, a c^{4} d^{2} x^{4} + 3 \, {\left (384 \, b c^{8} d^{2} x^{8} + 1024 \, b c^{6} d^{2} x^{6} + 768 \, b c^{4} d^{2} x^{4} - 73 \, b d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (144 \, b c^{7} d^{2} x^{7} + 344 \, b c^{5} d^{2} x^{5} + 146 \, b c^{3} d^{2} x^{3} - 219 \, b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}{9216 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.07, size = 218, normalized size = 1.21 \begin {gather*} \begin {cases} \frac {a c^{4} d^{2} x^{8}}{8} + \frac {a c^{2} d^{2} x^{6}}{3} + \frac {a d^{2} x^{4}}{4} + \frac {b c^{4} d^{2} x^{8} \operatorname {asinh}{\left (c x \right )}}{8} - \frac {b c^{3} d^{2} x^{7} \sqrt {c^{2} x^{2} + 1}}{64} + \frac {b c^{2} d^{2} x^{6} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {43 b c d^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{1152} + \frac {b d^{2} x^{4} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {73 b d^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{4608 c} + \frac {73 b d^{2} x \sqrt {c^{2} x^{2} + 1}}{3072 c^{3}} - \frac {73 b d^{2} \operatorname {asinh}{\left (c x \right )}}{3072 c^{4}} & \text {for}\: c \neq 0 \\\frac {a d^{2} x^{4}}{4} & \text {otherwise} \end {cases} \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 = -0.01 \begin {gather*} \int x^3\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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