Optimal. Leaf size=261 \[ -\frac {b d^3 x \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{32 c}-\frac {7 b d^3 x \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{192 c}-\frac {35 b d^3 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{768 c}-\frac {35 b d^3 x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{512 c}+\frac {d^3 \left (c^2 x^2+1\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}-\frac {35 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{1024 c^2}+\frac {35 b^2 c^2 d^3 x^4}{3072}+\frac {b^2 d^3 \left (c^2 x^2+1\right )^4}{256 c^2}+\frac {7 b^2 d^3 \left (c^2 x^2+1\right )^3}{1152 c^2}+\frac {175 b^2 d^3 x^2}{3072} \]
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Rubi [A] time = 0.26, antiderivative size = 261, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5717, 5684, 5682, 5675, 30, 14, 261} \[ -\frac {b d^3 x \left (c^2 x^2+1\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{32 c}-\frac {7 b d^3 x \left (c^2 x^2+1\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{192 c}-\frac {35 b d^3 x \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{768 c}-\frac {35 b d^3 x \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{512 c}+\frac {d^3 \left (c^2 x^2+1\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}-\frac {35 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{1024 c^2}+\frac {35 b^2 c^2 d^3 x^4}{3072}+\frac {b^2 d^3 \left (c^2 x^2+1\right )^4}{256 c^2}+\frac {7 b^2 d^3 \left (c^2 x^2+1\right )^3}{1152 c^2}+\frac {175 b^2 d^3 x^2}{3072} \]
Antiderivative was successfully verified.
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Rule 14
Rule 30
Rule 261
Rule 5675
Rule 5682
Rule 5684
Rule 5717
Rubi steps
\begin {align*} \int x \left (d+c^2 d x^2\right )^3 \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}-\frac {\left (b d^3\right ) \int \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{4 c}\\ &=-\frac {b d^3 x \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{32 c}+\frac {d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}+\frac {1}{32} \left (b^2 d^3\right ) \int x \left (1+c^2 x^2\right )^3 \, dx-\frac {\left (7 b d^3\right ) \int \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{32 c}\\ &=\frac {b^2 d^3 \left (1+c^2 x^2\right )^4}{256 c^2}-\frac {7 b d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{192 c}-\frac {b d^3 x \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{32 c}+\frac {d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}+\frac {1}{192} \left (7 b^2 d^3\right ) \int x \left (1+c^2 x^2\right )^2 \, dx-\frac {\left (35 b d^3\right ) \int \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{192 c}\\ &=\frac {7 b^2 d^3 \left (1+c^2 x^2\right )^3}{1152 c^2}+\frac {b^2 d^3 \left (1+c^2 x^2\right )^4}{256 c^2}-\frac {35 b d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{768 c}-\frac {7 b d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{192 c}-\frac {b d^3 x \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{32 c}+\frac {d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}+\frac {1}{768} \left (35 b^2 d^3\right ) \int x \left (1+c^2 x^2\right ) \, dx-\frac {\left (35 b d^3\right ) \int \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx}{256 c}\\ &=\frac {7 b^2 d^3 \left (1+c^2 x^2\right )^3}{1152 c^2}+\frac {b^2 d^3 \left (1+c^2 x^2\right )^4}{256 c^2}-\frac {35 b d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{512 c}-\frac {35 b d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{768 c}-\frac {7 b d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{192 c}-\frac {b d^3 x \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{32 c}+\frac {d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}+\frac {1}{768} \left (35 b^2 d^3\right ) \int \left (x+c^2 x^3\right ) \, dx+\frac {1}{512} \left (35 b^2 d^3\right ) \int x \, dx-\frac {\left (35 b d^3\right ) \int \frac {a+b \sinh ^{-1}(c x)}{\sqrt {1+c^2 x^2}} \, dx}{512 c}\\ &=\frac {175 b^2 d^3 x^2}{3072}+\frac {35 b^2 c^2 d^3 x^4}{3072}+\frac {7 b^2 d^3 \left (1+c^2 x^2\right )^3}{1152 c^2}+\frac {b^2 d^3 \left (1+c^2 x^2\right )^4}{256 c^2}-\frac {35 b d^3 x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{512 c}-\frac {35 b d^3 x \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{768 c}-\frac {7 b d^3 x \left (1+c^2 x^2\right )^{5/2} \left (a+b \sinh ^{-1}(c x)\right )}{192 c}-\frac {b d^3 x \left (1+c^2 x^2\right )^{7/2} \left (a+b \sinh ^{-1}(c x)\right )}{32 c}-\frac {35 d^3 \left (a+b \sinh ^{-1}(c x)\right )^2}{1024 c^2}+\frac {d^3 \left (1+c^2 x^2\right )^4 \left (a+b \sinh ^{-1}(c x)\right )^2}{8 c^2}\\ \end {align*}
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Mathematica [A] time = 0.66, size = 256, normalized size = 0.98 \[ \frac {d^3 \left (c x \left (1152 a^2 c x \left (c^6 x^6+4 c^4 x^4+6 c^2 x^2+4\right )-6 a b \sqrt {c^2 x^2+1} \left (48 c^6 x^6+200 c^4 x^4+326 c^2 x^2+279\right )+b^2 c x \left (36 c^6 x^6+200 c^4 x^4+489 c^2 x^2+837\right )\right )+6 b \sinh ^{-1}(c x) \left (3 a \left (128 c^8 x^8+512 c^6 x^6+768 c^4 x^4+512 c^2 x^2+93\right )-b c x \sqrt {c^2 x^2+1} \left (48 c^6 x^6+200 c^4 x^4+326 c^2 x^2+279\right )\right )+9 b^2 \left (128 c^8 x^8+512 c^6 x^6+768 c^4 x^4+512 c^2 x^2+93\right ) \sinh ^{-1}(c x)^2\right )}{9216 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.54, size = 383, normalized size = 1.47 \[ \frac {36 \, {\left (32 \, a^{2} + b^{2}\right )} c^{8} d^{3} x^{8} + 8 \, {\left (576 \, a^{2} + 25 \, b^{2}\right )} c^{6} d^{3} x^{6} + 3 \, {\left (2304 \, a^{2} + 163 \, b^{2}\right )} c^{4} d^{3} x^{4} + 9 \, {\left (512 \, a^{2} + 93 \, b^{2}\right )} c^{2} d^{3} x^{2} + 9 \, {\left (128 \, b^{2} c^{8} d^{3} x^{8} + 512 \, b^{2} c^{6} d^{3} x^{6} + 768 \, b^{2} c^{4} d^{3} x^{4} + 512 \, b^{2} c^{2} d^{3} x^{2} + 93 \, b^{2} d^{3}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (384 \, a b c^{8} d^{3} x^{8} + 1536 \, a b c^{6} d^{3} x^{6} + 2304 \, a b c^{4} d^{3} x^{4} + 1536 \, a b c^{2} d^{3} x^{2} + 279 \, a b d^{3} - {\left (48 \, b^{2} c^{7} d^{3} x^{7} + 200 \, b^{2} c^{5} d^{3} x^{5} + 326 \, b^{2} c^{3} d^{3} x^{3} + 279 \, b^{2} c d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (48 \, a b c^{7} d^{3} x^{7} + 200 \, a b c^{5} d^{3} x^{5} + 326 \, a b c^{3} d^{3} x^{3} + 279 \, a b c d^{3} x\right )} \sqrt {c^{2} x^{2} + 1}}{9216 \, c^{2}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 339, normalized size = 1.30 \[ \frac {d^{3} a^{2} \left (\frac {1}{8} c^{8} x^{8}+\frac {1}{2} c^{6} x^{6}+\frac {3}{4} c^{4} x^{4}+\frac {1}{2} c^{2} x^{2}\right )+d^{3} b^{2} \left (\frac {\arcsinh \left (c x \right )^{2} \left (c^{2} x^{2}+1\right )^{4}}{8}-\frac {\arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {7}{2}}}{32}-\frac {7 \arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {5}{2}}}{192}-\frac {35 \arcsinh \left (c x \right ) c x \left (c^{2} x^{2}+1\right )^{\frac {3}{2}}}{768}-\frac {35 \arcsinh \left (c x \right ) \sqrt {c^{2} x^{2}+1}\, c x}{512}-\frac {35 \arcsinh \left (c x \right )^{2}}{1024}+\frac {\left (c^{2} x^{2}+1\right )^{4}}{256}+\frac {7 \left (c^{2} x^{2}+1\right )^{3}}{1152}+\frac {35 \left (c^{2} x^{2}+1\right )^{2}}{3072}+\frac {35 c^{2} x^{2}}{1024}+\frac {35}{1024}\right )+2 d^{3} a b \left (\frac {\arcsinh \left (c x \right ) c^{8} x^{8}}{8}+\frac {\arcsinh \left (c x \right ) c^{6} x^{6}}{2}+\frac {3 \arcsinh \left (c x \right ) c^{4} x^{4}}{4}+\frac {\arcsinh \left (c x \right ) c^{2} x^{2}}{2}-\frac {c^{7} x^{7} \sqrt {c^{2} x^{2}+1}}{64}-\frac {25 c^{5} x^{5} \sqrt {c^{2} x^{2}+1}}{384}-\frac {163 c^{3} x^{3} \sqrt {c^{2} x^{2}+1}}{1536}-\frac {93 c x \sqrt {c^{2} x^{2}+1}}{1024}+\frac {93 \arcsinh \left (c x \right )}{1024}\right )}{c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.45, size = 925, normalized size = 3.54 \[ \text {result too large to display} \]
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 \[ \int x\,{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,{\left (d\,c^2\,x^2+d\right )}^3 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 24.28, size = 573, normalized size = 2.20 \[ \begin {cases} \frac {a^{2} c^{6} d^{3} x^{8}}{8} + \frac {a^{2} c^{4} d^{3} x^{6}}{2} + \frac {3 a^{2} c^{2} d^{3} x^{4}}{4} + \frac {a^{2} d^{3} x^{2}}{2} + \frac {a b c^{6} d^{3} x^{8} \operatorname {asinh}{\left (c x \right )}}{4} - \frac {a b c^{5} d^{3} x^{7} \sqrt {c^{2} x^{2} + 1}}{32} + a b c^{4} d^{3} x^{6} \operatorname {asinh}{\left (c x \right )} - \frac {25 a b c^{3} d^{3} x^{5} \sqrt {c^{2} x^{2} + 1}}{192} + \frac {3 a b c^{2} d^{3} x^{4} \operatorname {asinh}{\left (c x \right )}}{2} - \frac {163 a b c d^{3} x^{3} \sqrt {c^{2} x^{2} + 1}}{768} + a b d^{3} x^{2} \operatorname {asinh}{\left (c x \right )} - \frac {93 a b d^{3} x \sqrt {c^{2} x^{2} + 1}}{512 c} + \frac {93 a b d^{3} \operatorname {asinh}{\left (c x \right )}}{512 c^{2}} + \frac {b^{2} c^{6} d^{3} x^{8} \operatorname {asinh}^{2}{\left (c x \right )}}{8} + \frac {b^{2} c^{6} d^{3} x^{8}}{256} - \frac {b^{2} c^{5} d^{3} x^{7} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{32} + \frac {b^{2} c^{4} d^{3} x^{6} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {25 b^{2} c^{4} d^{3} x^{6}}{1152} - \frac {25 b^{2} c^{3} d^{3} x^{5} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{192} + \frac {3 b^{2} c^{2} d^{3} x^{4} \operatorname {asinh}^{2}{\left (c x \right )}}{4} + \frac {163 b^{2} c^{2} d^{3} x^{4}}{3072} - \frac {163 b^{2} c d^{3} x^{3} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{768} + \frac {b^{2} d^{3} x^{2} \operatorname {asinh}^{2}{\left (c x \right )}}{2} + \frac {93 b^{2} d^{3} x^{2}}{1024} - \frac {93 b^{2} d^{3} x \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{512 c} + \frac {93 b^{2} d^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{1024 c^{2}} & \text {for}\: c \neq 0 \\\frac {a^{2} d^{3} x^{2}}{2} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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