Optimal. Leaf size=104 \[ \frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{12 d e^5 (c+d x)^3}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{6 d e^5 (c+d x)}-\frac {a+b \cosh ^{-1}(c+d x)}{4 d e^5 (c+d x)^4} \]
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Rubi [A]
time = 0.05, antiderivative size = 104, 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,
105, 97} \begin {gather*} -\frac {a+b \cosh ^{-1}(c+d x)}{4 d e^5 (c+d x)^4}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{6 d e^5 (c+d x)}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{12 d e^5 (c+d x)^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 97
Rule 105
Rule 5883
Rule 5996
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^5} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{e^5 x^5} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{x^5} \, dx,x,c+d x\right )}{d e^5}\\ &=-\frac {a+b \cosh ^{-1}(c+d x)}{4 d e^5 (c+d x)^4}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^4 \sqrt {1+x}} \, dx,x,c+d x\right )}{4 d e^5}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{12 d e^5 (c+d x)^3}-\frac {a+b \cosh ^{-1}(c+d x)}{4 d e^5 (c+d x)^4}+\frac {b \text {Subst}\left (\int \frac {2}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx,x,c+d x\right )}{12 d e^5}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{12 d e^5 (c+d x)^3}-\frac {a+b \cosh ^{-1}(c+d x)}{4 d e^5 (c+d x)^4}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^2 \sqrt {1+x}} \, dx,x,c+d x\right )}{6 d e^5}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{12 d e^5 (c+d x)^3}+\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{6 d e^5 (c+d x)}-\frac {a+b \cosh ^{-1}(c+d x)}{4 d e^5 (c+d x)^4}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 86, normalized size = 0.83 \begin {gather*} \frac {-3 a+b \sqrt {-1+c+d x} \sqrt {1+c+d x} \left (c+2 c^3+d x+6 c^2 d x+6 c d^2 x^2+2 d^3 x^3\right )-3 b \cosh ^{-1}(c+d x)}{12 d e^5 (c+d x)^4} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 8.35, size = 76, normalized size = 0.73
method | result | size |
derivativedivides | \(\frac {-\frac {a}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b \left (-\frac {\mathrm {arccosh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{2}+1\right )}{12 \left (d x +c \right )^{3}}\right )}{e^{5}}}{d}\) | \(76\) |
default | \(\frac {-\frac {a}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b \left (-\frac {\mathrm {arccosh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{2}+1\right )}{12 \left (d x +c \right )^{3}}\right )}{e^{5}}}{d}\) | \(76\) |
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. 246 vs.
\(2 (87) = 174\).
time = 0.30, size = 246, normalized size = 2.37 \begin {gather*} \frac {1}{12} \, b {\left (\frac {{\left (2 \, d^{4} x^{4} + 8 \, c d^{3} x^{3} + 2 \, c^{4} + {\left (12 \, c^{2} d^{2} - d^{2}\right )} x^{2} - c^{2} + 2 \, {\left (4 \, c^{3} d - c d\right )} x - 1\right )} d}{{\left (d^{5} x^{3} e^{5} + 3 \, c d^{4} x^{2} e^{5} + 3 \, c^{2} d^{3} x e^{5} + c^{3} d^{2} e^{5}\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}} - \frac {3 \, \operatorname {arcosh}\left (d x + c\right )}{d^{5} x^{4} e^{5} + 4 \, c d^{4} x^{3} e^{5} + 6 \, c^{2} d^{3} x^{2} e^{5} + 4 \, c^{3} d^{2} x e^{5} + c^{4} d e^{5}}\right )} - \frac {a}{4 \, {\left (d^{5} x^{4} e^{5} + 4 \, c d^{4} x^{3} e^{5} + 6 \, c^{2} d^{3} x^{2} e^{5} + 4 \, c^{3} d^{2} x e^{5} + c^{4} d e^{5}\right )}} \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. 475 vs.
\(2 (87) = 174\).
time = 0.38, size = 475, normalized size = 4.57 \begin {gather*} \frac {3 \, a d^{4} x^{4} + 12 \, a c d^{3} x^{3} + 18 \, a c^{2} d^{2} x^{2} + 12 \, a c^{3} d x - 3 \, b c^{4} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + {\left (2 \, b c^{4} d^{3} x^{3} + 6 \, b c^{5} d^{2} x^{2} + 2 \, b c^{7} + b c^{5} + {\left (6 \, b c^{6} + b c^{4}\right )} d x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{12 \, {\left ({\left (c^{4} d^{5} x^{4} + 4 \, c^{5} d^{4} x^{3} + 6 \, c^{6} d^{3} x^{2} + 4 \, c^{7} d^{2} x + c^{8} d\right )} \cosh \left (1\right )^{5} + 5 \, {\left (c^{4} d^{5} x^{4} + 4 \, c^{5} d^{4} x^{3} + 6 \, c^{6} d^{3} x^{2} + 4 \, c^{7} d^{2} x + c^{8} d\right )} \cosh \left (1\right )^{4} \sinh \left (1\right ) + 10 \, {\left (c^{4} d^{5} x^{4} + 4 \, c^{5} d^{4} x^{3} + 6 \, c^{6} d^{3} x^{2} + 4 \, c^{7} d^{2} x + c^{8} d\right )} \cosh \left (1\right )^{3} \sinh \left (1\right )^{2} + 10 \, {\left (c^{4} d^{5} x^{4} + 4 \, c^{5} d^{4} x^{3} + 6 \, c^{6} d^{3} x^{2} + 4 \, c^{7} d^{2} x + c^{8} d\right )} \cosh \left (1\right )^{2} \sinh \left (1\right )^{3} + 5 \, {\left (c^{4} d^{5} x^{4} + 4 \, c^{5} d^{4} x^{3} + 6 \, c^{6} d^{3} x^{2} + 4 \, c^{7} d^{2} x + c^{8} d\right )} \cosh \left (1\right ) \sinh \left (1\right )^{4} + {\left (c^{4} d^{5} x^{4} + 4 \, c^{5} d^{4} x^{3} + 6 \, c^{6} d^{3} x^{2} + 4 \, c^{7} d^{2} x + c^{8} d\right )} \sinh \left (1\right )^{5}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {a}{c^{5} + 5 c^{4} d x + 10 c^{3} d^{2} x^{2} + 10 c^{2} d^{3} x^{3} + 5 c d^{4} x^{4} + d^{5} x^{5}}\, dx + \int \frac {b \operatorname {acosh}{\left (c + d x \right )}}{c^{5} + 5 c^{4} d x + 10 c^{3} d^{2} x^{2} + 10 c^{2} d^{3} x^{3} + 5 c d^{4} x^{4} + d^{5} x^{5}}\, dx}{e^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^5} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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