Optimal. Leaf size=115 \[ 24 b^4 x-\frac {24 b^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{d}+\frac {12 b^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}-\frac {4 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d} \]
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Rubi [A]
time = 0.11, antiderivative size = 115, 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, 8} \begin {gather*} -\frac {24 b^3 \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )}{d}+\frac {12 b^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}-\frac {4 b \sqrt {(c+d x)^2+1} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d}+24 b^4 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 )^4 \, dx &=\frac {\text {Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d}-\frac {(4 b) \text {Subst}\left (\int \frac {x \left (a+b \sinh ^{-1}(x)\right )^3}{\sqrt {1+x^2}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {4 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d}+\frac {\left (12 b^2\right ) \text {Subst}\left (\int \left (a+b \sinh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {12 b^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}-\frac {4 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d}-\frac {\left (24 b^3\right ) \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 {24 b^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{d}+\frac {12 b^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}-\frac {4 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d}+\frac {\left (24 b^4\right ) \text {Subst}(\int 1 \, dx,x,c+d x)}{d}\\ &=24 b^4 x-\frac {24 b^3 \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )}{d}+\frac {12 b^2 (c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^2}{d}-\frac {4 b \sqrt {1+(c+d x)^2} \left (a+b \sinh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \sinh ^{-1}(c+d x)\right )^4}{d}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 226, normalized size = 1.97 \begin {gather*} \frac {\left (a^4+12 a^2 b^2+24 b^4\right ) (c+d x)-4 a b \left (a^2+6 b^2\right ) \sqrt {1+(c+d x)^2}-4 b \left (-a^3 (c+d x)-6 a b^2 (c+d x)+3 a^2 b \sqrt {1+(c+d x)^2}+6 b^3 \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)+6 b^2 \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)^2-4 b^3 \left (-a (c+d x)+b \sqrt {1+(c+d x)^2}\right ) \sinh ^{-1}(c+d x)^3+b^4 (c+d x) \sinh ^{-1}(c+d x)^4}{d} \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. \(244\) vs.
\(2(111)=222\).
time = 2.69, size = 245, normalized size = 2.13
method | result | size |
derivativedivides | \(\frac {\left (d x +c \right ) a^{4}+b^{4} \left (\arcsinh \left (d x +c \right )^{4} \left (d x +c \right )-4 \arcsinh \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+12 \arcsinh \left (d x +c \right )^{2} \left (d x +c \right )-24 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+24 d x +24 c \right )+4 a \,b^{3} \left (\arcsinh \left (d x +c \right )^{3} \left (d x +c \right )-3 \arcsinh \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+6 \left (d x +c \right ) \arcsinh \left (d x +c \right )-6 \sqrt {1+\left (d x +c \right )^{2}}\right )+6 a^{2} b^{2} \left (\arcsinh \left (d x +c \right )^{2} \left (d x +c \right )-2 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )+4 a^{3} b \left (\left (d x +c \right ) \arcsinh \left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(245\) |
default | \(\frac {\left (d x +c \right ) a^{4}+b^{4} \left (\arcsinh \left (d x +c \right )^{4} \left (d x +c \right )-4 \arcsinh \left (d x +c \right )^{3} \sqrt {1+\left (d x +c \right )^{2}}+12 \arcsinh \left (d x +c \right )^{2} \left (d x +c \right )-24 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+24 d x +24 c \right )+4 a \,b^{3} \left (\arcsinh \left (d x +c \right )^{3} \left (d x +c \right )-3 \arcsinh \left (d x +c \right )^{2} \sqrt {1+\left (d x +c \right )^{2}}+6 \left (d x +c \right ) \arcsinh \left (d x +c \right )-6 \sqrt {1+\left (d x +c \right )^{2}}\right )+6 a^{2} b^{2} \left (\arcsinh \left (d x +c \right )^{2} \left (d x +c \right )-2 \arcsinh \left (d x +c \right ) \sqrt {1+\left (d x +c \right )^{2}}+2 d x +2 c \right )+4 a^{3} b \left (\left (d x +c \right ) \arcsinh \left (d x +c \right )-\sqrt {1+\left (d x +c \right )^{2}}\right )}{d}\) | \(245\) |
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. 344 vs.
\(2 (111) = 222\).
time = 0.38, size = 344, normalized size = 2.99 \begin {gather*} \frac {{\left (b^{4} d x + b^{4} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{4} + 4 \, {\left (a b^{3} d x + a b^{3} c - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} b^{4}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{3} + {\left (a^{4} + 12 \, a^{2} b^{2} + 24 \, b^{4}\right )} d x - 6 \, {\left (2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1} a b^{3} - {\left (a^{2} b^{2} + 2 \, b^{4}\right )} d x - {\left (a^{2} b^{2} + 2 \, b^{4}\right )} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )^{2} + 4 \, {\left ({\left (a^{3} b + 6 \, a b^{3}\right )} d x + {\left (a^{3} b + 6 \, a b^{3}\right )} c - 3 \, {\left (a^{2} b^{2} + 2 \, b^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}\right ) - 4 \, {\left (a^{3} b + 6 \, a b^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} + 1}}{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. 444 vs.
\(2 (105) = 210\).
time = 0.34, size = 444, normalized size = 3.86 \begin {gather*} \begin {cases} a^{4} x + \frac {4 a^{3} b c \operatorname {asinh}{\left (c + d x \right )}}{d} + 4 a^{3} b x \operatorname {asinh}{\left (c + d x \right )} - \frac {4 a^{3} b \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} + \frac {6 a^{2} b^{2} c \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} + 6 a^{2} b^{2} x \operatorname {asinh}^{2}{\left (c + d x \right )} + 12 a^{2} b^{2} x - \frac {12 a^{2} b^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}{\left (c + d x \right )}}{d} + \frac {4 a b^{3} c \operatorname {asinh}^{3}{\left (c + d x \right )}}{d} + \frac {24 a b^{3} c \operatorname {asinh}{\left (c + d x \right )}}{d} + 4 a b^{3} x \operatorname {asinh}^{3}{\left (c + d x \right )} + 24 a b^{3} x \operatorname {asinh}{\left (c + d x \right )} - \frac {12 a b^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} - \frac {24 a b^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1}}{d} + \frac {b^{4} c \operatorname {asinh}^{4}{\left (c + d x \right )}}{d} + \frac {12 b^{4} c \operatorname {asinh}^{2}{\left (c + d x \right )}}{d} + b^{4} x \operatorname {asinh}^{4}{\left (c + d x \right )} + 12 b^{4} x \operatorname {asinh}^{2}{\left (c + d x \right )} + 24 b^{4} x - \frac {4 b^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} + 1} \operatorname {asinh}^{3}{\left (c + d x \right )}}{d} - \frac {24 b^{4} \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 )^{4} & \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 )}^4 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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