Optimal. Leaf size=150 \[ \frac {4}{9} b^2 e^2 x+\frac {2 b^2 e^2 (c+d x)^3}{27 d}-\frac {4 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}-\frac {2 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d} \]
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Rubi [A]
time = 0.23, antiderivative size = 150, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 7, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {5996, 12, 5883,
5939, 5915, 8, 30} \begin {gather*} \frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {2 b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}-\frac {4 b e^2 \sqrt {c+d x-1} \sqrt {c+d x+1} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}+\frac {2 b^2 e^2 (c+d x)^3}{27 d}+\frac {4}{9} b^2 e^2 x \end {gather*}
Antiderivative was successfully verified.
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Rule 8
Rule 12
Rule 30
Rule 5883
Rule 5915
Rule 5939
Rule 5996
Rubi steps
\begin {align*} \int (c e+d e x)^2 \left (a+b \cosh ^{-1}(c+d x)\right )^2 \, dx &=\frac {\text {Subst}\left (\int e^2 x^2 \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 \text {Subst}\left (\int x^2 \left (a+b \cosh ^{-1}(x)\right )^2 \, dx,x,c+d x\right )}{d}\\ &=\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {\left (2 b e^2\right ) \text {Subst}\left (\int \frac {x^3 \left (a+b \cosh ^{-1}(x)\right )}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{3 d}\\ &=-\frac {2 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}-\frac {\left (4 b e^2\right ) \text {Subst}\left (\int \frac {x \left (a+b \cosh ^{-1}(x)\right )}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{9 d}+\frac {\left (2 b^2 e^2\right ) \text {Subst}\left (\int x^2 \, dx,x,c+d x\right )}{9 d}\\ &=\frac {2 b^2 e^2 (c+d x)^3}{27 d}-\frac {4 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}-\frac {2 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}+\frac {\left (4 b^2 e^2\right ) \text {Subst}(\int 1 \, dx,x,c+d x)}{9 d}\\ &=\frac {4}{9} b^2 e^2 x+\frac {2 b^2 e^2 (c+d x)^3}{27 d}-\frac {4 b e^2 \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}-\frac {2 b e^2 \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x} \left (a+b \cosh ^{-1}(c+d x)\right )}{9 d}+\frac {e^2 (c+d x)^3 \left (a+b \cosh ^{-1}(c+d x)\right )^2}{3 d}\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 168, normalized size = 1.12 \begin {gather*} \frac {e^2 \left (12 b^2 (c+d x)+\left (9 a^2+2 b^2\right ) (c+d x)^3+6 a b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (-2-(c+d x)^2\right )+6 b \left (3 a (c+d x)^3-2 b \sqrt {-1+c+d x} \sqrt {1+c+d x}-b \sqrt {-1+c+d x} (c+d x)^2 \sqrt {1+c+d x}\right ) \cosh ^{-1}(c+d x)+9 b^2 (c+d x)^3 \cosh ^{-1}(c+d x)^2\right )}{27 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. \(289\) vs.
\(2(132)=264\).
time = 8.61, size = 290, normalized size = 1.93
method | result | size |
default | \(\frac {e^{2} \left (d x +c \right )^{3} a^{2}}{3 d}+\frac {e^{2} b^{2} \left (9 \mathrm {arccosh}\left (d x +c \right )^{2} x^{3} d^{3}+27 \mathrm {arccosh}\left (d x +c \right )^{2} x^{2} c \,d^{2}-6 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, x^{2} d^{2}+27 \mathrm {arccosh}\left (d x +c \right )^{2} x \,c^{2} d -12 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, x c d +2 d^{3} x^{3}+9 \mathrm {arccosh}\left (d x +c \right )^{2} c^{3}-6 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c +1}\, \sqrt {d x +c -1}\, c^{2}+6 x^{2} c \,d^{2}+6 c^{2} d x -12 \,\mathrm {arccosh}\left (d x +c \right ) \sqrt {d x +c -1}\, \sqrt {d x +c +1}+2 c^{3}+12 d x +12 c \right )}{27 d}+\frac {2 a b \,e^{2} \left (\frac {\left (d x +c \right )^{3} \mathrm {arccosh}\left (d x +c \right )}{3}-\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (\left (d x +c \right )^{2}+2\right )}{9}\right )}{d}\) | \(290\) |
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. 784 vs.
\(2 (127) = 254\).
time = 0.38, size = 784, normalized size = 5.23 \begin {gather*} \frac {{\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c d^{2} x^{2} + 3 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{2} + 4 \, b^{2}\right )} d x\right )} \cosh \left (1\right )^{2} + 9 \, {\left ({\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} c^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (b^{2} d^{3} x^{3} + 3 \, b^{2} c d^{2} x^{2} + 3 \, b^{2} c^{2} d x + b^{2} 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 )^{2} + 2 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c d^{2} x^{2} + 3 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{2} + 4 \, b^{2}\right )} d x\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} d^{3} x^{3} + 3 \, {\left (9 \, a^{2} + 2 \, b^{2}\right )} c d^{2} x^{2} + 3 \, {\left ({\left (9 \, a^{2} + 2 \, b^{2}\right )} c^{2} + 4 \, b^{2}\right )} d x\right )} \sinh \left (1\right )^{2} + 6 \, {\left (3 \, {\left (a b d^{3} x^{3} + 3 \, a b c d^{2} x^{2} + 3 \, a b c^{2} d x + a b c^{3}\right )} \cosh \left (1\right )^{2} + 6 \, {\left (a b d^{3} x^{3} + 3 \, a b c d^{2} x^{2} + 3 \, a b c^{2} d x + a b c^{3}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + 3 \, {\left (a b d^{3} x^{3} + 3 \, a b c d^{2} x^{2} + 3 \, a b c^{2} d x + a b c^{3}\right )} \sinh \left (1\right )^{2} - \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, b^{2}\right )} \cosh \left (1\right )^{2} + 2 \, {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, b^{2}\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (b^{2} d^{2} x^{2} + 2 \, b^{2} c d x + b^{2} c^{2} + 2 \, b^{2}\right )} \sinh \left (1\right )^{2}\right )}\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 6 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left ({\left (a b d^{2} x^{2} + 2 \, a b c d x + a b c^{2} + 2 \, a b\right )} \cosh \left (1\right )^{2} + 2 \, {\left (a b d^{2} x^{2} + 2 \, a b c d x + a b c^{2} + 2 \, a b\right )} \cosh \left (1\right ) \sinh \left (1\right ) + {\left (a b d^{2} x^{2} + 2 \, a b c d x + a b c^{2} + 2 \, a b\right )} \sinh \left (1\right )^{2}\right )}}{27 \, 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. 610 vs.
\(2 (143) = 286\).
time = 0.38, size = 610, normalized size = 4.07 \begin {gather*} \begin {cases} a^{2} c^{2} e^{2} x + a^{2} c d e^{2} x^{2} + \frac {a^{2} d^{2} e^{2} x^{3}}{3} + \frac {2 a b c^{3} e^{2} \operatorname {acosh}{\left (c + d x \right )}}{3 d} + 2 a b c^{2} e^{2} x \operatorname {acosh}{\left (c + d x \right )} - \frac {2 a b c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{9 d} + 2 a b c d e^{2} x^{2} \operatorname {acosh}{\left (c + d x \right )} - \frac {4 a b c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{9} + \frac {2 a b d^{2} e^{2} x^{3} \operatorname {acosh}{\left (c + d x \right )}}{3} - \frac {2 a b d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{9} - \frac {4 a b e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{9 d} + \frac {b^{2} c^{3} e^{2} \operatorname {acosh}^{2}{\left (c + d x \right )}}{3 d} + b^{2} c^{2} e^{2} x \operatorname {acosh}^{2}{\left (c + d x \right )} + \frac {2 b^{2} c^{2} e^{2} x}{9} - \frac {2 b^{2} c^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{9 d} + b^{2} c d e^{2} x^{2} \operatorname {acosh}^{2}{\left (c + d x \right )} + \frac {2 b^{2} c d e^{2} x^{2}}{9} - \frac {4 b^{2} c e^{2} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{9} + \frac {b^{2} d^{2} e^{2} x^{3} \operatorname {acosh}^{2}{\left (c + d x \right )}}{3} + \frac {2 b^{2} d^{2} e^{2} x^{3}}{27} - \frac {2 b^{2} d e^{2} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{9} + \frac {4 b^{2} e^{2} x}{9} - \frac {4 b^{2} e^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1} \operatorname {acosh}{\left (c + d x \right )}}{9 d} & \text {for}\: d \neq 0 \\c^{2} e^{2} x \left (a + b \operatorname {acosh}{\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 = -0.01 \begin {gather*} \int {\left (c\,e+d\,e\,x\right )}^2\,{\left (a+b\,\mathrm {acosh}\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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