Optimal. Leaf size=129 \[ -\frac {24 b^3 \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{d}+\frac {12 b^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}-\frac {4 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^4}{d}+24 b^4 x \]
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Rubi [A] time = 0.28, antiderivative size = 129, 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 = {5864, 5654, 5718, 8} \[ -\frac {24 b^3 \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{d}+\frac {12 b^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}-\frac {4 b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^4}{d}+24 b^4 x \]
Antiderivative was successfully verified.
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Rule 8
Rule 5654
Rule 5718
Rule 5864
Rubi steps
\begin {align*} \int \left (a+b \cosh ^{-1}(c+d x)\right )^4 \, dx &=\frac {\operatorname {Subst}\left (\int \left (a+b \cosh ^{-1}(x)\right )^4 \, dx,x,c+d x\right )}{d}\\ &=\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^4}{d}-\frac {(4 b) \operatorname {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}(x)\right )^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {4 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^4}{d}+\frac {\left (12 b^2\right ) \operatorname {Subst}\left (\int \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {12 b^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}-\frac {4 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^4}{d}-\frac {\left (24 b^3\right ) \operatorname {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}(x)\right )}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{d}\\ &=-\frac {24 b^3 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{d}+\frac {12 b^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}-\frac {4 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^4}{d}+\frac {\left (24 b^4\right ) \operatorname {Subst}(\int 1 \, dx,x,c+d x)}{d}\\ &=24 b^4 x-\frac {24 b^3 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{d}+\frac {12 b^2 (c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^2}{d}-\frac {4 b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )^3}{d}+\frac {(c+d x) \left (a+b \cosh ^{-1}(c+d x)\right )^4}{d}\\ \end {align*}
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Mathematica [B] time = 0.27, size = 261, normalized size = 2.02 \[ \frac {-4 a b \left (a^2+6 b^2\right ) \sqrt {c+d x-1} \sqrt {c+d x+1}+6 b^2 \cosh ^{-1}(c+d x)^2 \left (a^2 (c+d x)-2 a b \sqrt {c+d x-1} \sqrt {c+d x+1}+2 b^2 (c+d x)\right )+\left (a^4+12 a^2 b^2+24 b^4\right ) (c+d x)-4 b \cosh ^{-1}(c+d x) \left (-\left (a^3 (c+d x)\right )+3 a^2 b \sqrt {c+d x-1} \sqrt {c+d x+1}-6 a b^2 (c+d x)+6 b^3 \sqrt {c+d x-1} \sqrt {c+d x+1}\right )-4 b^3 \cosh ^{-1}(c+d x)^3 \left (b \sqrt {c+d x-1} \sqrt {c+d x+1}-a (c+d x)\right )+b^4 (c+d x) \cosh ^{-1}(c+d x)^4}{d} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.69, size = 344, normalized size = 2.67 \[ \frac {{\left (b^{4} d x + b^{4} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{4} + 4 \, {\left (a b^{3} d x + a b^{3} c - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} b^{4}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{3} + {\left (a^{4} + 12 \, a^{2} b^{2} + 24 \, b^{4}\right )} d x - 6 \, {\left (2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} a b^{3} - {\left (a^{2} b^{2} + 2 \, b^{4}\right )} d x - {\left (a^{2} b^{2} + 2 \, b^{4}\right )} c\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )^{2} + 4 \, {\left ({\left (a^{3} b + 6 \, a b^{3}\right )} d x + {\left (a^{3} b + 6 \, a b^{3}\right )} c - 3 \, {\left (a^{2} b^{2} + 2 \, b^{4}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 4 \, {\left (a^{3} b + 6 \, a b^{3}\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d} \]
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 \[ \int {\left (b \operatorname {arcosh}\left (d x + c\right ) + a\right )}^{4}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.06, size = 275, normalized size = 2.13 \[ \frac {\left (d x +c \right ) a^{4}+b^{4} \left (\left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )^{4}-4 \mathrm {arccosh}\left (d x +c \right )^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+12 \left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )^{2}-24 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+24 d x +24 c \right )+4 a \,b^{3} \left (\left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )^{3}-3 \mathrm {arccosh}\left (d x +c \right )^{2} \sqrt {d x +c -1}\, \sqrt {d x +c +1}+6 \left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )-6 \sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )+6 a^{2} b^{2} \left (\left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )^{2}-2 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 d x +2 c \right )+4 a^{3} b \left (\left (d x +c \right ) \mathrm {arccosh}\left (d x +c \right )-\sqrt {d x +c -1}\, \sqrt {d x +c +1}\right )}{d} \]
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 \[ b^{4} x \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{4} + a^{4} x + \frac {4 \, {\left ({\left (d x + c\right )} \operatorname {arcosh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} - 1}\right )} a^{3} b}{d} + \int \frac {2 \, {\left (2 \, {\left ({\left (c^{3} - c\right )} a b^{3} + {\left (a b^{3} d^{3} - b^{4} d^{3}\right )} x^{3} + {\left (3 \, a b^{3} c d^{2} - 2 \, b^{4} c d^{2}\right )} x^{2} + {\left ({\left (c^{2} - 1\right )} a b^{3} + {\left (a b^{3} d^{2} - b^{4} d^{2}\right )} x^{2} + {\left (2 \, a b^{3} c d - b^{4} c d\right )} x\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left ({\left (3 \, c^{2} d - d\right )} a b^{3} - {\left (c^{2} d - d\right )} b^{4}\right )} x\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{3} + 3 \, {\left (a^{2} b^{2} d^{3} x^{3} + 3 \, a^{2} b^{2} c d^{2} x^{2} + {\left (3 \, c^{2} d - d\right )} a^{2} b^{2} x + {\left (c^{3} - c\right )} a^{2} b^{2} + {\left (a^{2} b^{2} d^{2} x^{2} + 2 \, a^{2} b^{2} c d x + {\left (c^{2} - 1\right )} a^{2} b^{2}\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}\right )} \log \left (d x + \sqrt {d x + c + 1} \sqrt {d x + c - 1} + c\right )^{2}\right )}}{d^{3} x^{3} + 3 \, c d^{2} x^{2} + c^{3} + {\left (d^{2} x^{2} + 2 \, c d x + c^{2} - 1\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1} + {\left (3 \, c^{2} d - d\right )} x - c}\,{d x} \]
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 \[ \int {\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right )}^4 \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 1.60, size = 444, normalized size = 3.44 \[ \begin {cases} a^{4} x + \frac {4 a^{3} b c \operatorname {acosh}{\left (c + d x \right )}}{d} + 4 a^{3} b x \operatorname {acosh}{\left (c + d x \right )} - \frac {4 a^{3} b \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{d} + \frac {6 a^{2} b^{2} c \operatorname {acosh}^{2}{\left (c + d x \right )}}{d} + 6 a^{2} b^{2} x \operatorname {acosh}^{2}{\left (c + d x \right )} + 12 a^{2} b^{2} x - \frac {12 a^{2} b^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{d} + \frac {4 a b^{3} c \operatorname {acosh}^{3}{\left (c + d x \right )}}{d} + \frac {24 a b^{3} c \operatorname {acosh}{\left (c + d x \right )}}{d} + 4 a b^{3} x \operatorname {acosh}^{3}{\left (c + d x \right )} + 24 a b^{3} x \operatorname {acosh}{\left (c + d x \right )} - \frac {12 a b^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}^{2}{\left (c + d x \right )}}{d} - \frac {24 a b^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{d} + \frac {b^{4} c \operatorname {acosh}^{4}{\left (c + d x \right )}}{d} + \frac {12 b^{4} c \operatorname {acosh}^{2}{\left (c + d x \right )}}{d} + b^{4} x \operatorname {acosh}^{4}{\left (c + d x \right )} + 12 b^{4} x \operatorname {acosh}^{2}{\left (c + d x \right )} + 24 b^{4} x - \frac {4 b^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}^{3}{\left (c + d x \right )}}{d} - \frac {24 b^{4} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{d} & \text {for}\: d \neq 0 \\x \left (a + b \operatorname {acosh}{\relax (c )}\right )^{4} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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