Optimal. Leaf size=296 \[ -\frac {73 b^2 d^2 x^2}{3072 c^2}+\frac {73 b^2 d^2 x^4}{9216}+\frac {43 b^2 c^2 d^2 x^6}{3456}+\frac {1}{256} b^2 c^4 d^2 x^8+\frac {73 b d^2 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{1536 c^3}-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2304 c}-\frac {25}{576} b c d^2 x^5 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{32} b c d^2 x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {73 d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3072 c^4}+\frac {1}{24} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{12} d^2 x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \]
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Rubi [A]
time = 0.70, antiderivative size = 296, normalized size of antiderivative = 1.00, number of steps
used = 25, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {5808, 5776,
5812, 5783, 30, 5806, 14} \begin {gather*} -\frac {73 d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3072 c^4}-\frac {1}{32} b c d^2 x^5 \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {25}{576} b c d^2 x^5 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{8} d^2 x^4 \left (c^2 x^2+1\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{12} d^2 x^4 \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {73 b d^2 x^3 \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{2304 c}+\frac {73 b d^2 x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{1536 c^3}+\frac {1}{24} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{256} b^2 c^4 d^2 x^8+\frac {43 b^2 c^2 d^2 x^6}{3456}-\frac {73 b^2 d^2 x^2}{3072 c^2}+\frac {73 b^2 d^2 x^4}{9216} \end {gather*}
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 5776
Rule 5783
Rule 5806
Rule 5808
Rule 5812
Rubi steps
\begin {align*} \int x^3 \left (d+c^2 d x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{2} d \int x^3 \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {1}{4} \left (b c d^2\right ) \int x^4 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac {1}{32} b c d^2 x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{12} d^2 x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{6} d^2 \int x^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {1}{32} \left (3 b c d^2\right ) \int x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx-\frac {1}{6} \left (b c d^2\right ) \int x^4 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx+\frac {1}{32} \left (b^2 c^2 d^2\right ) \int x^5 \left (1+c^2 x^2\right ) \, dx\\ &=-\frac {25}{576} b c d^2 x^5 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{32} b c d^2 x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{24} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{12} d^2 x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {1}{64} \left (b c d^2\right ) \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{36} \left (b c d^2\right ) \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx-\frac {1}{12} \left (b c d^2\right ) \int \frac {x^4 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx+\frac {1}{64} \left (b^2 c^2 d^2\right ) \int x^5 \, dx+\frac {1}{36} \left (b^2 c^2 d^2\right ) \int x^5 \, dx+\frac {1}{32} \left (b^2 c^2 d^2\right ) \int \left (x^5+c^2 x^7\right ) \, dx\\ &=\frac {43 b^2 c^2 d^2 x^6}{3456}+\frac {1}{256} b^2 c^4 d^2 x^8-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2304 c}-\frac {25}{576} b c d^2 x^5 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{32} b c d^2 x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{24} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{12} d^2 x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{256} \left (b^2 d^2\right ) \int x^3 \, dx+\frac {1}{144} \left (b^2 d^2\right ) \int x^3 \, dx+\frac {1}{48} \left (b^2 d^2\right ) \int x^3 \, dx+\frac {\left (3 b d^2\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{256 c}+\frac {\left (b d^2\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{48 c}+\frac {\left (b d^2\right ) \int \frac {x^2 \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx}{16 c}\\ &=\frac {73 b^2 d^2 x^4}{9216}+\frac {43 b^2 c^2 d^2 x^6}{3456}+\frac {1}{256} b^2 c^4 d^2 x^8+\frac {73 b d^2 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{1536 c^3}-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2304 c}-\frac {25}{576} b c d^2 x^5 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{32} b c d^2 x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{24} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{12} d^2 x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {\left (3 b d^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{512 c^3}-\frac {\left (b d^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{96 c^3}-\frac {\left (b d^2\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{32 c^3}-\frac {\left (3 b^2 d^2\right ) \int x \, dx}{512 c^2}-\frac {\left (b^2 d^2\right ) \int x \, dx}{96 c^2}-\frac {\left (b^2 d^2\right ) \int x \, dx}{32 c^2}\\ &=-\frac {73 b^2 d^2 x^2}{3072 c^2}+\frac {73 b^2 d^2 x^4}{9216}+\frac {43 b^2 c^2 d^2 x^6}{3456}+\frac {1}{256} b^2 c^4 d^2 x^8+\frac {73 b d^2 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{1536 c^3}-\frac {73 b d^2 x^3 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{2304 c}-\frac {25}{576} b c d^2 x^5 \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {1}{32} b c d^2 x^5 \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )-\frac {73 d^2 \left (a+b \sinh ^{-1}(c x)\right )^2}{3072 c^4}+\frac {1}{24} d^2 x^4 \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{12} d^2 x^4 \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{8} d^2 x^4 \left (1+c^2 x^2\right )^2 \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A]
time = 0.22, size = 237, normalized size = 0.80 \begin {gather*} \frac {d^2 \left (c x \left (1152 a^2 c^3 x^3 \left (6+8 c^2 x^2+3 c^4 x^4\right )+b^2 c x \left (-657+219 c^2 x^2+344 c^4 x^4+108 c^6 x^6\right )-6 a b \sqrt {1+c^2 x^2} \left (-219+146 c^2 x^2+344 c^4 x^4+144 c^6 x^6\right )\right )+6 b \left (-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 a \left (-73+768 c^4 x^4+1024 c^6 x^6+384 c^8 x^8\right )\right ) \sinh ^{-1}(c x)+9 b^2 \left (-73+768 c^4 x^4+1024 c^6 x^6+384 c^8 x^8\right ) \sinh ^{-1}(c x)^2\right )}{27648 c^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int x^{3} \left (c^{2} d \,x^{2}+d \right )^{2} \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 762 vs.
\(2 (264) = 528\).
time = 0.36, size = 762, normalized size = 2.57 \begin {gather*} \frac {1}{8} \, b^{2} c^{4} d^{2} x^{8} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{8} \, a^{2} c^{4} d^{2} x^{8} + \frac {1}{3} \, b^{2} c^{2} d^{2} x^{6} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{3} \, a^{2} c^{2} d^{2} x^{6} + \frac {1}{4} \, b^{2} d^{2} x^{4} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{1536} \, {\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 )} a b c^{4} d^{2} + \frac {1}{9216} \, {\left ({\left (\frac {36 \, x^{8}}{c^{2}} - \frac {56 \, x^{6}}{c^{4}} + \frac {105 \, x^{4}}{c^{6}} - \frac {315 \, x^{2}}{c^{8}} + \frac {315 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{10}}\right )} c^{2} - 6 \, {\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 \operatorname {arsinh}\left (c x\right )\right )} b^{2} c^{4} d^{2} + \frac {1}{4} \, a^{2} d^{2} x^{4} + \frac {1}{72} \, {\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 )} a b c^{2} d^{2} + \frac {1}{432} \, {\left ({\left (\frac {8 \, x^{6}}{c^{2}} - \frac {15 \, x^{4}}{c^{4}} + \frac {45 \, x^{2}}{c^{6}} - \frac {45 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{8}}\right )} c^{2} - 6 \, {\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 \operatorname {arsinh}\left (c x\right )\right )} b^{2} c^{2} d^{2} + \frac {1}{16} \, {\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 )} a b d^{2} + \frac {1}{32} \, {\left ({\left (\frac {x^{4}}{c^{2}} - \frac {3 \, x^{2}}{c^{4}} + \frac {3 \, \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2}}{c^{6}}\right )} c^{2} - 2 \, {\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 \operatorname {arsinh}\left (c x\right )\right )} b^{2} d^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 348, normalized size = 1.18 \begin {gather*} \frac {108 \, {\left (32 \, a^{2} + b^{2}\right )} c^{8} d^{2} x^{8} + 8 \, {\left (1152 \, a^{2} + 43 \, b^{2}\right )} c^{6} d^{2} x^{6} + 3 \, {\left (2304 \, a^{2} + 73 \, b^{2}\right )} c^{4} d^{2} x^{4} - 657 \, b^{2} c^{2} d^{2} x^{2} + 9 \, {\left (384 \, b^{2} c^{8} d^{2} x^{8} + 1024 \, b^{2} c^{6} d^{2} x^{6} + 768 \, b^{2} c^{4} d^{2} x^{4} - 73 \, b^{2} d^{2}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (1152 \, a b c^{8} d^{2} x^{8} + 3072 \, a b c^{6} d^{2} x^{6} + 2304 \, a b c^{4} d^{2} x^{4} - 219 \, a b d^{2} - {\left (144 \, b^{2} c^{7} d^{2} x^{7} + 344 \, b^{2} c^{5} d^{2} x^{5} + 146 \, b^{2} c^{3} d^{2} x^{3} - 219 \, b^{2} c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (144 \, a b c^{7} d^{2} x^{7} + 344 \, a b c^{5} d^{2} x^{5} + 146 \, a b c^{3} d^{2} x^{3} - 219 \, a b c d^{2} x\right )} \sqrt {c^{2} x^{2} + 1}}{27648 \, c^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 2.02, size = 515, normalized size = 1.74 \begin {gather*} \begin {cases} \frac {a^{2} c^{4} d^{2} x^{8}}{8} + \frac {a^{2} c^{2} d^{2} x^{6}}{3} + \frac {a^{2} d^{2} x^{4}}{4} + \frac {a b c^{4} d^{2} x^{8} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {a b c^{3} d^{2} x^{7} \sqrt {c^{2} x^{2} + 1}}{32} + \frac {2 a b c^{2} d^{2} x^{6} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {43 a b c d^{2} x^{5} \sqrt {c^{2} x^{2} + 1}}{576} + \frac {a b d^{2} x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {73 a b d^{2} x^{3} \sqrt {c^{2} x^{2} + 1}}{2304 c} + \frac {73 a b d^{2} x \sqrt {c^{2} x^{2} + 1}}{1536 c^{3}} - \frac {73 a b d^{2} \operatorname {asinh}{\left (c x \right )}}{1536 c^{4}} + \frac {b^{2} c^{4} d^{2} x^{8} \operatorname {asinh}^{2}{\left (c x \right )}}{8} + \frac {b^{2} c^{4} d^{2} x^{8}}{256} - \frac {b^{2} c^{3} d^{2} x^{7} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{32} + \frac {b^{2} c^{2} d^{2} x^{6} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {43 b^{2} c^{2} d^{2} x^{6}}{3456} - \frac {43 b^{2} c d^{2} x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{576} + \frac {b^{2} d^{2} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {73 b^{2} d^{2} x^{4}}{9216} - \frac {73 b^{2} d^{2} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{2304 c} - \frac {73 b^{2} d^{2} x^{2}}{3072 c^{2}} + \frac {73 b^{2} d^{2} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{1536 c^{3}} - \frac {73 b^{2} d^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{3072 c^{4}} & \text {for}\: c \neq 0 \\\frac {a^{2} 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 = -0.00 \begin {gather*} \int x^3\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\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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