Optimal. Leaf size=125 \[ \frac {14}{9} b^2 d x+\frac {2}{27} b^2 c^2 d x^3-\frac {4 b d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {2 b d \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac {2}{3} d x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} d x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \]
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Rubi [A]
time = 0.10, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {5786, 5772,
5798, 8} \begin {gather*} \frac {1}{3} d x \left (c^2 x^2+1\right ) \left (a+b \sinh ^{-1}(c x)\right )^2-\frac {2 b d \left (c^2 x^2+1\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}-\frac {4 b d \sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}+\frac {2}{3} d x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {2}{27} b^2 c^2 d x^3+\frac {14}{9} b^2 d x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 5772
Rule 5786
Rule 5798
Rubi steps
\begin {align*} \int \left (d+c^2 d x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx &=\frac {1}{3} d x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} (2 d) \int \left (a+b \sinh ^{-1}(c x)\right )^2 \, dx-\frac {1}{3} (2 b c d) \int x \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right ) \, dx\\ &=-\frac {2 b d \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac {2}{3} d x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} d x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{9} \left (2 b^2 d\right ) \int \left (1+c^2 x^2\right ) \, dx-\frac {1}{3} (4 b c d) \int \frac {x \left (a+b \sinh ^{-1}(c x)\right )}{\sqrt {1+c^2 x^2}} \, dx\\ &=\frac {2}{9} b^2 d x+\frac {2}{27} b^2 c^2 d x^3-\frac {4 b d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {2 b d \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac {2}{3} d x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} d x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} \left (4 b^2 d\right ) \int 1 \, dx\\ &=\frac {14}{9} b^2 d x+\frac {2}{27} b^2 c^2 d x^3-\frac {4 b d \sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )}{3 c}-\frac {2 b d \left (1+c^2 x^2\right )^{3/2} \left (a+b \sinh ^{-1}(c x)\right )}{9 c}+\frac {2}{3} d x \left (a+b \sinh ^{-1}(c x)\right )^2+\frac {1}{3} d x \left (1+c^2 x^2\right ) \left (a+b \sinh ^{-1}(c x)\right )^2\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 135, normalized size = 1.08 \begin {gather*} \frac {d \left (9 a^2 c x \left (3+c^2 x^2\right )-6 a b \sqrt {1+c^2 x^2} \left (7+c^2 x^2\right )+2 b^2 c x \left (21+c^2 x^2\right )-6 b \left (-3 a c x \left (3+c^2 x^2\right )+b \sqrt {1+c^2 x^2} \left (7+c^2 x^2\right )\right ) \sinh ^{-1}(c x)+9 b^2 c x \left (3+c^2 x^2\right ) \sinh ^{-1}(c x)^2\right )}{27 c} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 180.00, size = 0, normalized size = 0.00 \[\int \left (c^{2} d \,x^{2}+d \right ) \left (a +b \arcsinh \left (c x \right )\right )^{2}\, dx\]
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. 230 vs.
\(2 (109) = 218\).
time = 0.29, size = 230, normalized size = 1.84 \begin {gather*} \frac {1}{3} \, b^{2} c^{2} d x^{3} \operatorname {arsinh}\left (c x\right )^{2} + \frac {1}{3} \, a^{2} c^{2} d x^{3} + \frac {2}{9} \, {\left (3 \, x^{3} \operatorname {arsinh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )}\right )} a b c^{2} d - \frac {2}{27} \, {\left (3 \, c {\left (\frac {\sqrt {c^{2} x^{2} + 1} x^{2}}{c^{2}} - \frac {2 \, \sqrt {c^{2} x^{2} + 1}}{c^{4}}\right )} \operatorname {arsinh}\left (c x\right ) - \frac {c^{2} x^{3} - 6 \, x}{c^{2}}\right )} b^{2} c^{2} d + b^{2} d x \operatorname {arsinh}\left (c x\right )^{2} + 2 \, b^{2} d {\left (x - \frac {\sqrt {c^{2} x^{2} + 1} \operatorname {arsinh}\left (c x\right )}{c}\right )} + a^{2} d x + \frac {2 \, {\left (c x \operatorname {arsinh}\left (c x\right ) - \sqrt {c^{2} x^{2} + 1}\right )} a b d}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.36, size = 178, normalized size = 1.42 \begin {gather*} \frac {{\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{3} d x^{3} + 3 \, {\left (9 \, a^{2} + 14 \, b^{2}\right )} c d x + 9 \, {\left (b^{2} c^{3} d x^{3} + 3 \, b^{2} c d x\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right )^{2} + 6 \, {\left (3 \, a b c^{3} d x^{3} + 9 \, a b c d x - {\left (b^{2} c^{2} d x^{2} + 7 \, b^{2} d\right )} \sqrt {c^{2} x^{2} + 1}\right )} \log \left (c x + \sqrt {c^{2} x^{2} + 1}\right ) - 6 \, {\left (a b c^{2} d x^{2} + 7 \, a b d\right )} \sqrt {c^{2} x^{2} + 1}}{27 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 0.24, size = 224, normalized size = 1.79 \begin {gather*} \begin {cases} \frac {a^{2} c^{2} d x^{3}}{3} + a^{2} d x + \frac {2 a b c^{2} d x^{3} \operatorname {asinh}{\left (c x \right )}}{3} - \frac {2 a b c d x^{2} \sqrt {c^{2} x^{2} + 1}}{9} + 2 a b d x \operatorname {asinh}{\left (c x \right )} - \frac {14 a b d \sqrt {c^{2} x^{2} + 1}}{9 c} + \frac {b^{2} c^{2} d x^{3} \operatorname {asinh}^{2}{\left (c x \right )}}{3} + \frac {2 b^{2} c^{2} d x^{3}}{27} - \frac {2 b^{2} c d x^{2} \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9} + b^{2} d x \operatorname {asinh}^{2}{\left (c x \right )} + \frac {14 b^{2} d x}{9} - \frac {14 b^{2} d \sqrt {c^{2} x^{2} + 1} \operatorname {asinh}{\left (c x \right )}}{9 c} & \text {for}\: c \neq 0 \\a^{2} d x & \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 {\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2\,\left (d\,c^2\,x^2+d\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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