Optimal. Leaf size=100 \[ 6 a b^2 x-\frac {6 b^3 \sqrt {1+(c+d x)^2}}{d}+\frac {6 b^3 (c+d x) \sinh ^{-1}(c+d x)}{d}-\frac {3 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d} \]
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Rubi [A]
time = 0.08, antiderivative size = 100, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 4, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {5858, 5772,
5798, 267} \begin {gather*} 6 a b^2 x-\frac {3 b \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}-\frac {6 b^3 \sqrt {(c+d x)^2+1}}{d}+\frac {6 b^3 (c+d x) \sinh ^{-1}(c+d x)}{d} \end {gather*}
Antiderivative was successfully verified.
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Rule 267
Rule 5772
Rule 5798
Rule 5858
Rubi steps
\begin {align*} \int \left (a+b \sinh ^{-1}(c+d x)\right )^3 \, dx &=\frac {\text {Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right )^3 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}-\frac {(3 b) \text {Subst}\left (\int \frac {x \left (a+b \sinh ^{-1}(x)\right )^2}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {3 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {\left (6 b^2\right ) \text {Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=6 a b^2 x-\frac {3 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {\left (6 b^3\right ) \text {Subst}\left (\int \sinh ^{-1}(x) \, dx,x,c+d x\right )}{d}\\ &=6 a b^2 x+\frac {6 b^3 (c+d x) \sinh ^{-1}(c+d x)}{d}-\frac {3 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}-\frac {\left (6 b^3\right ) \text {Subst}\left (\int \frac {x}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d}\\ &=6 a b^2 x-\frac {6 b^3 \sqrt {1+(c+d x)^2}}{d}+\frac {6 b^3 (c+d x) \sinh ^{-1}(c+d x)}{d}-\frac {3 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 147, normalized size = 1.47 \begin {gather*} \frac {a \left (a^2+6 b^2\right ) (c+d x)-3 b \left (a^2+2 b^2\right ) \sqrt {1+(c+d x)^2}-3 b \left (-a^2 (c+d x)-2 b^2 (c+d x)+2 a b \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)-3 b^2 \left (-a (c+d x)+b \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)^2+b^3 (c+d x) \sinh ^{-1}(c+d x)^3}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 2.29, size = 160, normalized size = 1.60
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\arcsinh \left (d x +c \right )^{3} \left (d x +c \right )-3 \arcsinh \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+6 \left (d x +c \right ) \arcsinh \left (d x +c \right )-6 \sqrt {1+\left (d x +c \right )^{2}}\right )+3 a \,b^{2} \left (\arcsinh \left (d x +c \right )^{2} \left (d x +c \right )-2 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )+3 a^{2} b \left (\left (d x +c \right ) \arcsinh \left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(160\) |
default | \(\frac {\left (d x +c \right ) a^{3}+b^{3} \left (\arcsinh \left (d x +c \right )^{3} \left (d x +c \right )-3 \arcsinh \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+6 \left (d x +c \right ) \arcsinh \left (d x +c \right )-6 \sqrt {1+\left (d x +c \right )^{2}}\right )+3 a \,b^{2} \left (\arcsinh \left (d x +c \right )^{2} \left (d x +c \right )-2 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )+3 a^{2} b \left (\left (d x +c \right ) \arcsinh \left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(160\) |
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. 239 vs.
\(2 (96) = 192\).
time = 0.41, size = 239, normalized size = 2.39 \begin {gather*} \frac {{\left (b^{3} d x + b^{3} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + {\left (a^{3} + 6 \, a b^{2}\right )} d x + 3 \, {\left (a b^{2} d x + a b^{2} c - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} b^{3}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} - 3 \, {\left (2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} a b^{2} - {\left (a^{2} b + 2 \, b^{3}\right )} d x - {\left (a^{2} b + 2 \, b^{3}\right )} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} {\left (a^{2} b + 2 \, b^{3}\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. 282 vs.
\(2 (92) = 184\).
time = 0.18, size = 282, normalized size = 2.82 \begin {gather*} \begin {cases} a^{3} x + \frac {3 a^{2} b c \operatorname {asinh}{\left (c + d x \right )}}{d} + 3 a^{2} b x \operatorname {asinh}{\left (c + d x \right )} - \frac {3 a^{2} b \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} + \frac {3 a b^{2} c \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} + 3 a b^{2} x \operatorname {asinh}^{2}{\left (c + d x \right )} + 6 a b^{2} x - \frac {6 a b^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{d} + \frac {b^{3} c \operatorname {asinh}^{3}{\left (c + d x \right )}}{d} + \frac {6 b^{3} c \operatorname {asinh}{\left (c + d x \right )}}{d} + b^{3} x \operatorname {asinh}^{3}{\left (c + d x \right )} + 6 b^{3} x \operatorname {asinh}{\left (c + d x \right )} - \frac {3 b^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} - \frac {6 b^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {asinh}{\left (c \right )}\right )^{3} & \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 = -0.01 \begin {gather*} \int {\left (a+b\,\mathrm {asinh}\left (c+d\,x\right )\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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