Optimal. Leaf size=47 \[ \frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {d+c^2 d x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 47, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.040, Rules used = {5783}
\begin {gather*} \frac {\sqrt {c^2 x^2+1} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {c^2 d x^2+d}} \end {gather*}
Antiderivative was successfully verified.
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Rule 5783
Rubi steps
\begin {align*} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {d+c^2 d x^2}} \, dx &=\frac {\sqrt {1+c^2 x^2} \int \frac {\left (a+b \sinh ^{-1}(c x)\right )^2}{\sqrt {1+c^2 x^2}} \, dx}{\sqrt {d+c^2 d x^2}}\\ &=\frac {\sqrt {1+c^2 x^2} \left (a+b \sinh ^{-1}(c x)\right )^3}{3 b c \sqrt {d+c^2 d x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 62, normalized size = 1.32 \begin {gather*} \frac {\sqrt {1+c^2 x^2} \sinh ^{-1}(c x) \left (3 a^2+3 a b \sinh ^{-1}(c x)+b^2 \sinh ^{-1}(c x)^2\right )}{3 c \sqrt {d+c^2 d x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(119\) vs.
\(2(41)=82\).
time = 0.57, size = 120, normalized size = 2.55
method | result | size |
default | \(\frac {a^{2} \ln \left (\frac {x \,c^{2} d}{\sqrt {c^{2} d}}+\sqrt {c^{2} d \,x^{2}+d}\right )}{\sqrt {c^{2} d}}+\frac {b^{2} \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{3}}{3 \sqrt {c^{2} x^{2}+1}\, c d}+\frac {a b \sqrt {d \left (c^{2} x^{2}+1\right )}\, \arcsinh \left (c x \right )^{2}}{\sqrt {c^{2} x^{2}+1}\, c d}\) | \(120\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 47, normalized size = 1.00 \begin {gather*} \frac {b^{2} \operatorname {arsinh}\left (c x\right )^{3}}{3 \, c \sqrt {d}} + \frac {a b \operatorname {arsinh}\left (c x\right )^{2}}{c \sqrt {d}} + \frac {a^{2} \operatorname {arsinh}\left (c x\right )}{c \sqrt {d}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (a + b \operatorname {asinh}{\left (c x \right )}\right )^{2}}{\sqrt {d \left (c^{2} x^{2} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 = -0.02 \begin {gather*} \int \frac {{\left (a+b\,\mathrm {asinh}\left (c\,x\right )\right )}^2}{\sqrt {d\,c^2\,x^2+d}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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