Optimal. Leaf size=57 \[ 2 b^2 x-\frac {2 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d} \]
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Rubi [A]
time = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5858, 5772,
5798, 8} \begin {gather*} -\frac {2 b \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}+2 b^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 5772
Rule 5798
Rule 5858
Rubi steps
\begin {align*} \int \left (a+b \sinh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}-\frac {(2 b) \text {Subst}\left (\int \frac {x \left (a+b \sinh ^{-1}(x)\right )}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {2 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}+\frac {\left (2 b^2\right ) \text {Subst}(\int 1 \, dx,x,c+d x)}{d}\\ &=2 b^2 x-\frac {2 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 87, normalized size = 1.53 \begin {gather*} \frac {\left (a^2+2 b^2\right ) (c+d x)-2 a b \sqrt {1+(c+d x)^2}+2 b \left (a c+a d x-b \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)+b^2 (c+d x) \sinh ^{-1}(c+d x)^2}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.17, size = 90, normalized size = 1.58
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a^{2}+b^{2} \left (\arcsinh \left (d x +c \right )^{2} \left (d x +c \right )-2 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )+2 a b \left (\left (d x +c \right ) \arcsinh \left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(90\) |
default | \(\frac {\left (d x +c \right ) a^{2}+b^{2} \left (\arcsinh \left (d x +c \right )^{2} \left (d x +c \right )-2 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )+2 a b \left (\left (d x +c \right ) \arcsinh \left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(90\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 141 vs.
\(2 (55) = 110\).
time = 0.36, size = 141, normalized size = 2.47 \begin {gather*} \frac {{\left (a^{2} + 2 \, b^{2}\right )} d x + {\left (b^{2} d x + b^{2} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} - 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} a b + 2 \, {\left (a b d x + a b c - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} b^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )}{d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 143 vs.
\(2 (51) = 102\).
time = 0.11, size = 143, normalized size = 2.51 \begin {gather*} \begin {cases} a^{2} x + \frac {2 a b c \operatorname {asinh}{\left (c + d x \right )}}{d} + 2 a b x \operatorname {asinh}{\left (c + d x \right )} - \frac {2 a b \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} + \frac {b^{2} c \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} + b^{2} x \operatorname {asinh}^{2}{\left (c + d x \right )} + 2 b^{2} x - \frac {2 b^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{2} & \text {otherwise} \end {cases} \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 {\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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