Optimal. Leaf size=119 \[ -\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{32 d}-\frac {b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{16 d}-\frac {3 b e^3 \cosh ^{-1}(c+d x)}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d} \]
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Rubi [A]
time = 0.05, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5996, 12, 5883,
102, 92, 54} \begin {gather*} \frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)^3}{16 d}-\frac {3 b e^3 \sqrt {c+d x-1} \sqrt {c+d x+1} (c+d x)}{32 d}-\frac {3 b e^3 \cosh ^{-1}(c+d x)}{32 d} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 54
Rule 92
Rule 102
Rule 5883
Rule 5996
Rubi steps
\begin {align*} \int (c e+d e x)^3 \left (a+b \cosh ^{-1}(c+d x)\right ) \, dx &=\frac {\text {Subst}\left (\int e^3 x^3 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 \text {Subst}\left (\int x^3 \left (a+b \cosh ^{-1}(x)\right ) \, dx,x,c+d x\right )}{d}\\ &=\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {\left (b e^3\right ) \text {Subst}\left (\int \frac {x^4}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{4 d}\\ &=-\frac {b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{16 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {\left (b e^3\right ) \text {Subst}\left (\int \frac {3 x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{16 d}\\ &=-\frac {b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{16 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{16 d}\\ &=-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{32 d}-\frac {b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{16 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}-\frac {\left (3 b e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} \sqrt {1+x}} \, dx,x,c+d x\right )}{32 d}\\ &=-\frac {3 b e^3 \sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}}{32 d}-\frac {b e^3 \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}}{16 d}-\frac {3 b e^3 \cosh ^{-1}(c+d x)}{32 d}+\frac {e^3 (c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )}{4 d}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 115, normalized size = 0.97 \begin {gather*} \frac {e^3 \left (-\frac {1}{4} b \sqrt {-1+c+d x} (c+d x)^3 \sqrt {1+c+d x}+(c+d x)^4 \left (a+b \cosh ^{-1}(c+d x)\right )-\frac {3}{8} b \left (\sqrt {-1+c+d x} (c+d x) \sqrt {1+c+d x}+2 \tanh ^{-1}\left (\sqrt {\frac {-1+c+d x}{1+c+d x}}\right )\right )\right )}{4 d} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.03, size = 144, normalized size = 1.21
method | result | size |
derivativedivides | \(\frac {\frac {e^{3} \left (d x +c \right )^{4} a}{4}+\frac {b \,e^{3} \left (d x +c \right )^{4} \mathrm {arccosh}\left (d x +c \right )}{4}-\frac {b \,e^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}}{16}-\frac {3 b \,e^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{32}-\frac {3 b \,e^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )}{32 \sqrt {\left (d x +c \right )^{2}-1}}}{d}\) | \(144\) |
default | \(\frac {\frac {e^{3} \left (d x +c \right )^{4} a}{4}+\frac {b \,e^{3} \left (d x +c \right )^{4} \mathrm {arccosh}\left (d x +c \right )}{4}-\frac {b \,e^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )^{3}}{16}-\frac {3 b \,e^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (d x +c \right )}{32}-\frac {3 b \,e^{3} \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \ln \left (d x +c +\sqrt {\left (d x +c \right )^{2}-1}\right )}{32 \sqrt {\left (d x +c \right )^{2}-1}}}{d}\) | \(144\) |
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. 789 vs.
\(2 (99) = 198\).
time = 0.28, size = 789, normalized size = 6.63 \begin {gather*} \frac {1}{4} \, a d^{3} x^{4} e^{3} + a c d^{2} x^{3} e^{3} + \frac {3}{2} \, a c^{2} d x^{2} e^{3} + \frac {3}{4} \, {\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^{2} d e^{3} + \frac {1}{6} \, {\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 c d^{2} e^{3} + \frac {1}{96} \, {\left (24 \, x^{4} \operatorname {arcosh}\left (d x + c\right ) - {\left (\frac {6 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} x^{3}}{d^{2}} - \frac {14 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c x^{2}}{d^{3}} + \frac {105 \, c^{4} \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^{5}} + \frac {35 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c^{2} x}{d^{4}} - \frac {90 \, {\left (c^{2} - 1\right )} 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^{5}} - \frac {105 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} c^{3}}{d^{5}} - \frac {9 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (c^{2} - 1\right )} x}{d^{4}} + \frac {9 \, {\left (c^{2} - 1\right )}^{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^{5}} + \frac {55 \, \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1} {\left (c^{2} - 1\right )} c}{d^{5}}\right )} d\right )} b d^{3} e^{3} + a c^{3} x e^{3} + \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^{3} e^{3}}{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. 655 vs.
\(2 (99) = 198\).
time = 0.38, size = 655, normalized size = 5.50 \begin {gather*} \frac {8 \, {\left (a d^{4} x^{4} + 4 \, a c d^{3} x^{3} + 6 \, a c^{2} d^{2} x^{2} + 4 \, a c^{3} d x\right )} \cosh \left (1\right )^{3} + 24 \, {\left (a d^{4} x^{4} + 4 \, a c d^{3} x^{3} + 6 \, a c^{2} d^{2} x^{2} + 4 \, a c^{3} d x\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 24 \, {\left (a d^{4} x^{4} + 4 \, a c d^{3} x^{3} + 6 \, a c^{2} d^{2} x^{2} + 4 \, a c^{3} d x\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + 8 \, {\left (a d^{4} x^{4} + 4 \, a c d^{3} x^{3} + 6 \, a c^{2} d^{2} x^{2} + 4 \, a c^{3} d x\right )} \sinh \left (1\right )^{3} + {\left ({\left (8 \, b d^{4} x^{4} + 32 \, b c d^{3} x^{3} + 48 \, b c^{2} d^{2} x^{2} + 32 \, b c^{3} d x + 8 \, b c^{4} - 3 \, b\right )} \cosh \left (1\right )^{3} + 3 \, {\left (8 \, b d^{4} x^{4} + 32 \, b c d^{3} x^{3} + 48 \, b c^{2} d^{2} x^{2} + 32 \, b c^{3} d x + 8 \, b c^{4} - 3 \, b\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, {\left (8 \, b d^{4} x^{4} + 32 \, b c d^{3} x^{3} + 48 \, b c^{2} d^{2} x^{2} + 32 \, b c^{3} d x + 8 \, b c^{4} - 3 \, b\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + {\left (8 \, b d^{4} x^{4} + 32 \, b c d^{3} x^{3} + 48 \, b c^{2} d^{2} x^{2} + 32 \, b c^{3} d x + 8 \, b c^{4} - 3 \, b\right )} \sinh \left (1\right )^{3}\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 (2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 2 \, b c^{3} + 3 \, {\left (2 \, b c^{2} + b\right )} d x + 3 \, b c\right )} \cosh \left (1\right )^{3} + 3 \, {\left (2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 2 \, b c^{3} + 3 \, {\left (2 \, b c^{2} + b\right )} d x + 3 \, b c\right )} \cosh \left (1\right )^{2} \sinh \left (1\right ) + 3 \, {\left (2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 2 \, b c^{3} + 3 \, {\left (2 \, b c^{2} + b\right )} d x + 3 \, b c\right )} \cosh \left (1\right ) \sinh \left (1\right )^{2} + {\left (2 \, b d^{3} x^{3} + 6 \, b c d^{2} x^{2} + 2 \, b c^{3} + 3 \, {\left (2 \, b c^{2} + b\right )} d x + 3 \, b c\right )} \sinh \left (1\right )^{3}\right )}}{32 \, 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. 394 vs.
\(2 (110) = 220\).
time = 0.40, size = 394, normalized size = 3.31 \begin {gather*} \begin {cases} a c^{3} e^{3} x + \frac {3 a c^{2} d e^{3} x^{2}}{2} + a c d^{2} e^{3} x^{3} + \frac {a d^{3} e^{3} x^{4}}{4} + \frac {b c^{4} e^{3} \operatorname {acosh}{\left (c + d x \right )}}{4 d} + b c^{3} e^{3} x \operatorname {acosh}{\left (c + d x \right )} - \frac {b c^{3} e^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{16 d} + \frac {3 b c^{2} d e^{3} x^{2} \operatorname {acosh}{\left (c + d x \right )}}{2} - \frac {3 b c^{2} e^{3} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{16} + b c d^{2} e^{3} x^{3} \operatorname {acosh}{\left (c + d x \right )} - \frac {3 b c d e^{3} x^{2} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{16} - \frac {3 b c e^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{32 d} + \frac {b d^{3} e^{3} x^{4} \operatorname {acosh}{\left (c + d x \right )}}{4} - \frac {b d^{2} e^{3} x^{3} \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{16} - \frac {3 b e^{3} x \sqrt {c^{2} + 2 c d x + d^{2} x^{2} - 1}}{32} - \frac {3 b e^{3} \operatorname {acosh}{\left (c + d x \right )}}{32 d} & \text {for}\: d \neq 0 \\c^{3} e^{3} 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. 617 vs.
\(2 (103) = 206\).
time = 0.93, size = 617, normalized size = 5.18 \begin {gather*} \frac {1}{4} \, a d^{3} e^{3} x^{4} + a c d^{2} e^{3} x^{3} + \frac {3}{2} \, a c^{2} d e^{3} 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^{3} e^{3} + \frac {3}{4} \, {\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^{2} d e^{3} + \frac {1}{6} \, {\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 c d^{2} e^{3} + \frac {1}{96} \, {\left (24 \, x^{4} \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 ({\left (2 \, x {\left (\frac {3 \, x}{d^{2}} - \frac {7 \, c}{d^{3}}\right )} + \frac {26 \, c^{2} d^{3} + 9 \, d^{3}}{d^{7}}\right )} x - \frac {5 \, {\left (10 \, c^{3} d^{2} + 11 \, c d^{2}\right )}}{d^{7}}\right )} - \frac {3 \, {\left (8 \, c^{4} + 24 \, c^{2} + 3\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^{4} {\left | d \right |}}\right )} d\right )} b d^{3} e^{3} + a c^{3} e^{3} 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 )}^3\,\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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