Optimal. Leaf size=226 \[ -\frac {16 b d^3 \sqrt {1+c^2 x^2}}{1155 c^5}-\frac {8 b d^3 \left (1+c^2 x^2\right )^{3/2}}{3465 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{5/2}}{1925 c^5}-\frac {b d^3 \left (1+c^2 x^2\right )^{7/2}}{1617 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{9/2}}{297 c^5}-\frac {b d^3 \left (1+c^2 x^2\right )^{11/2}}{121 c^5}+\frac {1}{5} d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{7} c^2 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{11} c^6 d^3 x^{11} \left (a+b \sinh ^{-1}(c x)\right ) \]
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Rubi [A]
time = 0.19, antiderivative size = 226, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {276, 5803, 12,
1813, 1634} \begin {gather*} \frac {1}{11} c^6 d^3 x^{11} \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{7} c^2 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{5} d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )-\frac {b d^3 \left (c^2 x^2+1\right )^{11/2}}{121 c^5}+\frac {4 b d^3 \left (c^2 x^2+1\right )^{9/2}}{297 c^5}-\frac {b d^3 \left (c^2 x^2+1\right )^{7/2}}{1617 c^5}-\frac {2 b d^3 \left (c^2 x^2+1\right )^{5/2}}{1925 c^5}-\frac {8 b d^3 \left (c^2 x^2+1\right )^{3/2}}{3465 c^5}-\frac {16 b d^3 \sqrt {c^2 x^2+1}}{1155 c^5} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 276
Rule 1634
Rule 1813
Rule 5803
Rubi steps
\begin {align*} \int x^4 \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right ) \, dx &=\frac {1}{5} d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{7} c^2 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{11} c^6 d^3 x^{11} \left (a+b \sinh ^{-1}(c x)\right )-(b c) \int \frac {d^3 x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right )}{1155 \sqrt {1+c^2 x^2}} \, dx\\ &=\frac {1}{5} d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{7} c^2 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{11} c^6 d^3 x^{11} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (b c d^3\right ) \int \frac {x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right )}{\sqrt {1+c^2 x^2}} \, dx}{1155}\\ &=\frac {1}{5} d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{7} c^2 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{11} c^6 d^3 x^{11} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (b c d^3\right ) \text {Subst}\left (\int \frac {x^2 \left (231+495 c^2 x+385 c^4 x^2+105 c^6 x^3\right )}{\sqrt {1+c^2 x}} \, dx,x,x^2\right )}{2310}\\ &=\frac {1}{5} d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{7} c^2 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{11} c^6 d^3 x^{11} \left (a+b \sinh ^{-1}(c x)\right )-\frac {\left (b c d^3\right ) \text {Subst}\left (\int \left (\frac {16}{c^4 \sqrt {1+c^2 x}}+\frac {8 \sqrt {1+c^2 x}}{c^4}+\frac {6 \left (1+c^2 x\right )^{3/2}}{c^4}+\frac {5 \left (1+c^2 x\right )^{5/2}}{c^4}-\frac {140 \left (1+c^2 x\right )^{7/2}}{c^4}+\frac {105 \left (1+c^2 x\right )^{9/2}}{c^4}\right ) \, dx,x,x^2\right )}{2310}\\ &=-\frac {16 b d^3 \sqrt {1+c^2 x^2}}{1155 c^5}-\frac {8 b d^3 \left (1+c^2 x^2\right )^{3/2}}{3465 c^5}-\frac {2 b d^3 \left (1+c^2 x^2\right )^{5/2}}{1925 c^5}-\frac {b d^3 \left (1+c^2 x^2\right )^{7/2}}{1617 c^5}+\frac {4 b d^3 \left (1+c^2 x^2\right )^{9/2}}{297 c^5}-\frac {b d^3 \left (1+c^2 x^2\right )^{11/2}}{121 c^5}+\frac {1}{5} d^3 x^5 \left (a+b \sinh ^{-1}(c x)\right )+\frac {3}{7} c^2 d^3 x^7 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{3} c^4 d^3 x^9 \left (a+b \sinh ^{-1}(c x)\right )+\frac {1}{11} c^6 d^3 x^{11} \left (a+b \sinh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 143, normalized size = 0.63 \begin {gather*} \frac {d^3 \left (3465 a c^5 x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right )-b \sqrt {1+c^2 x^2} \left (50488-25244 c^2 x^2+18933 c^4 x^4+117625 c^6 x^6+111475 c^8 x^8+33075 c^{10} x^{10}\right )+3465 b c^5 x^5 \left (231+495 c^2 x^2+385 c^4 x^4+105 c^6 x^6\right ) \sinh ^{-1}(c x)\right )}{4002075 c^5} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.01, size = 206, normalized size = 0.91
method | result | size |
derivativedivides | \(\frac {d^{3} a \left (\frac {1}{11} c^{11} x^{11}+\frac {1}{3} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{3} b \left (\frac {\arcsinh \left (c x \right ) c^{11} x^{11}}{11}+\frac {\arcsinh \left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{10} x^{10} \sqrt {c^{2} x^{2}+1}}{121}-\frac {91 c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{3267}-\frac {4705 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {6311 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}\right )}{c^{5}}\) | \(206\) |
default | \(\frac {d^{3} a \left (\frac {1}{11} c^{11} x^{11}+\frac {1}{3} c^{9} x^{9}+\frac {3}{7} c^{7} x^{7}+\frac {1}{5} c^{5} x^{5}\right )+d^{3} b \left (\frac {\arcsinh \left (c x \right ) c^{11} x^{11}}{11}+\frac {\arcsinh \left (c x \right ) c^{9} x^{9}}{3}+\frac {3 \arcsinh \left (c x \right ) c^{7} x^{7}}{7}+\frac {\arcsinh \left (c x \right ) c^{5} x^{5}}{5}-\frac {c^{10} x^{10} \sqrt {c^{2} x^{2}+1}}{121}-\frac {91 c^{8} x^{8} \sqrt {c^{2} x^{2}+1}}{3267}-\frac {4705 c^{6} x^{6} \sqrt {c^{2} x^{2}+1}}{160083}-\frac {6311 c^{4} x^{4} \sqrt {c^{2} x^{2}+1}}{1334025}+\frac {25244 c^{2} x^{2} \sqrt {c^{2} x^{2}+1}}{4002075}-\frac {50488 \sqrt {c^{2} x^{2}+1}}{4002075}\right )}{c^{5}}\) | \(206\) |
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. 465 vs.
\(2 (194) = 388\).
time = 0.27, size = 465, normalized size = 2.06 \begin {gather*} \frac {1}{11} \, a c^{6} d^{3} x^{11} + \frac {1}{3} \, a c^{4} d^{3} x^{9} + \frac {3}{7} \, a c^{2} d^{3} x^{7} + \frac {1}{7623} \, {\left (693 \, x^{11} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {63 \, \sqrt {c^{2} x^{2} + 1} x^{10}}{c^{2}} - \frac {70 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{4}} + \frac {80 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{6}} - \frac {96 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{10}} - \frac {256 \, \sqrt {c^{2} x^{2} + 1}}{c^{12}}\right )} c\right )} b c^{6} d^{3} + \frac {1}{945} \, {\left (315 \, x^{9} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {35 \, \sqrt {c^{2} x^{2} + 1} x^{8}}{c^{2}} - \frac {40 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{4}} + \frac {48 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{6}} - \frac {64 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{8}} + \frac {128 \, \sqrt {c^{2} x^{2} + 1}}{c^{10}}\right )} c\right )} b c^{4} d^{3} + \frac {1}{5} \, a d^{3} x^{5} + \frac {3}{245} \, {\left (35 \, x^{7} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} + 1} x^{6}}{c^{2}} - \frac {6 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{6}} - \frac {16 \, \sqrt {c^{2} x^{2} + 1}}{c^{8}}\right )} c\right )} b c^{2} d^{3} + \frac {1}{75} \, {\left (15 \, x^{5} \operatorname {arsinh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} + 1} x^{4}}{c^{2}} - \frac {4 \, \sqrt {c^{2} x^{2} + 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} + 1}}{c^{6}}\right )} c\right )} b d^{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 201, normalized size = 0.89 \begin {gather*} \frac {363825 \, a c^{11} d^{3} x^{11} + 1334025 \, a c^{9} d^{3} x^{9} + 1715175 \, a c^{7} d^{3} x^{7} + 800415 \, a c^{5} d^{3} x^{5} + 3465 \, {\left (105 \, b c^{11} d^{3} x^{11} + 385 \, b c^{9} d^{3} x^{9} + 495 \, b c^{7} d^{3} x^{7} + 231 \, b c^{5} d^{3} x^{5}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - {\left (33075 \, b c^{10} d^{3} x^{10} + 111475 \, b c^{8} d^{3} x^{8} + 117625 \, b c^{6} d^{3} x^{6} + 18933 \, b c^{4} d^{3} x^{4} - 25244 \, b c^{2} d^{3} x^{2} + 50488 \, b d^{3}\right )} \sqrt {c^{2} x^{2} + 1}}{4002075 \, c^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 3.25, size = 289, normalized size = 1.28 \begin {gather*} \begin {cases} \frac {a c^{6} d^{3} x^{11}}{11} + \frac {a c^{4} d^{3} x^{9}}{3} + \frac {3 a c^{2} d^{3} x^{7}}{7} + \frac {a d^{3} x^{5}}{5} + \frac {b c^{6} d^{3} x^{11} \operatorname {asinh}{\left (c x \right )}}{11} - \frac {b c^{5} d^{3} x^{10} \sqrt {c^{2} x^{2} + 1}}{121} + \frac {b c^{4} d^{3} x^{9} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {91 b c^{3} d^{3} x^{8} \sqrt {c^{2} x^{2} + 1}}{3267} + \frac {3 b c^{2} d^{3} x^{7} \operatorname {asinh}{\left (c x \right )}}{7} - \frac {4705 b c d^{3} x^{6} \sqrt {c^{2} x^{2} + 1}}{160083} + \frac {b d^{3} x^{5} \operatorname {asinh}{\left (c x \right )}}{5} - \frac {6311 b d^{3} x^{4} \sqrt {c^{2} x^{2} + 1}}{1334025 c} + \frac {25244 b d^{3} x^{2} \sqrt {c^{2} x^{2} + 1}}{4002075 c^{3}} - \frac {50488 b d^{3} \sqrt {c^{2} x^{2} + 1}}{4002075 c^{5}} & \text {for}\: c \neq 0 \\\frac {a d^{3} x^{5}}{5} & \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: RuntimeError} \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^4\,\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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