Optimal. Leaf size=97 \[ -\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{9 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d} \]
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Rubi [A]
time = 0.04, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5996, 12, 5883,
102, 75} \begin {gather*} \frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d}-\frac {b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2}{9 d}-\frac {2 b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1}}{9 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 75
Rule 102
Rule 5883
Rule 5996
Rubi steps
\begin {align*} \int (c e+d e x)^2 \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int e^2 x^2 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int x^2 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d}\\ &=-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d}-\frac {\left (b e^2\right ) \text {Subst}\left (\int \frac {2 x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d}\\ &=-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d}-\frac {\left (2 b e^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d}\\ &=-\frac {2 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x}}{9 d}-\frac {b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 71, normalized size = 0.73 \begin {gather*} \frac {e^2 \left (-\frac {1}{9} b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (2+c^2+2 c d x+d^2 x^2\right )+\frac {1}{3} (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )\right )}{d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.08, size = 67, normalized size = 0.69
method | result | size |
derivativedivides | \(\frac {\frac {e^{2} \left (d x +c \right )^{3} a}{3}+b \,e^{2} \left (\frac {\left (d x +c \right )^{3} \mathrm {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(67\) |
default | \(\frac {\frac {e^{2} \left (d x +c \right )^{3} a}{3}+b \,e^{2} \left (\frac {\left (d x +c \right )^{3} \mathrm {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(67\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 443 vs.
\(2 (80) = 160\).
time = 0.28, size = 443, normalized size = 4.57 \begin {gather*} \frac {1}{3} \, a d^{2} x^{3} e^{2} + a c d x^{2} e^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \operatorname {arcosh}\left (d x + c\right ) - d {\left (\frac {3 \, c^{2} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x}{d^{2}} - \frac {{\left (c^{2} - 1\right )} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{3}} - \frac {3 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c}{d^{3}}\right )}\right )} b c d e^{2} + \frac {1}{18} \, {\left (6 \, x^{3} \operatorname {arcosh}\left (d x + c\right ) - d {\left (\frac {2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x^{2}}{d^{2}} - \frac {15 \, c^{3} \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{4}} - \frac {5 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c x}{d^{3}} + \frac {9 \, {\left (c^{2} - 1\right )} c \log \left (2 \, d^{2} x + 2 \, c d + 2 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} d\right )}{d^{4}} + \frac {15 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c^{2}}{d^{4}} - \frac {4 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (c^{2} - 1\right )}}{d^{4}}\right )}\right )} b d^{2} e^{2} + a c^{2} x e^{2} + \frac {{\left ({\left (d x + c\right )} \operatorname {arcosh}\left (d x + c\right ) - \sqrt {{\left (d x + c\right )}^{2} - 1}\right )} b c^{2} e^{2}}{d} \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. 352 vs.
\(2 (80) = 160\).
time = 0.37, size = 352, normalized size = 3.63 \begin {gather*} \frac {3 \, {\left (a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x\right )} \cosh \left (1\right )^{2} + 6 \, {\left (a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x\right )} \cosh \left (1\right ) \sinh \left (1\right ) + 3 \, {\left (a d^{3} x^{3} + 3 \, a c d^{2} x^{2} + 3 \, a c^{2} d x\right )} \sinh \left (1\right )^{2} + 3 \, {\left ({\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (b d^{3} x^{3} + 3 \, b c d^{2} x^{2} + 3 \, b c^{2} d x + b c^{3}\right )} \sinh \left (1\right )^{2}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + 2 \, b\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + 2 \, b\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (b d^{2} x^{2} + 2 \, b c d x + b c^{2} + 2 \, b\right )} \sinh \left (1\right )^{2}\right )}}{9 \, 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. 258 vs.
\(2 (88) = 176\).
time = 0.26, size = 258, normalized size = 2.66 \begin {gather*} \begin {cases} a c^{2} e^{2} x + a c d e^{2} x^{2} + \frac {a d^{2} e^{2} x^{3}}{3} + \frac {b c^{3} e^{2} \operatorname {acosh}{\left (c + d x \right )}}{3 d} + b c^{2} e^{2} x \operatorname {acosh}{\left (c + d x \right )} - \frac {b c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{9 d} + b c d e^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )} - \frac {2 b c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{9} + \frac {b d^{2} e^{2} x^{3} \operatorname {acosh}{\left (c + d x \right )}}{3} - \frac {b d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{9} - \frac {2 b e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{9 d} & \text {for}\: d \neq 0 \\c^{2} e^{2} x \left (a + b \operatorname {acosh}{\left (c \right )}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 419 vs.
\(2 (83) = 166\).
time = 0.77, size = 419, normalized size = 4.32 \begin {gather*} \frac {1}{3} \, a d^{2} e^{2} x^{3} + a c d e^{2} x^{2} - {\left (d {\left (\frac {c \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d {\left | d \right |}} + \frac {\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{d^{2}}\right )} - x \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )\right )} b c^{2} e^{2} + \frac {1}{2} \, {\left (2 \, x^{2} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (\frac {x}{d^{2}} - \frac {3 \, c}{d^{3}}\right )} - \frac {{\left (2 \, c^{2} + 1\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{2} {\left | d \right |}}\right )} d\right )} b c d e^{2} + \frac {1}{18} \, {\left (6 \, x^{3} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (\sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (x {\left (\frac {2 \, x}{d^{2}} - \frac {5 \, c}{d^{3}}\right )} + \frac {11 \, c^{2} d + 4 \, d}{d^{5}}\right )} + \frac {3 \, {\left (2 \, c^{3} + 3 \, c\right )} \log \left ({\left | -c d - {\left (x {\left | d \right |} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right )} {\left | d \right |} \right |}\right )}{d^{3} {\left | d \right |}}\right )} d\right )} b d^{2} e^{2} + a c^{2} e^{2} x \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 (c\,e+d\,e\,x\right )}^2\,\left (a+b\,\mathrm {acosh}\left (c+d\,x\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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