Optimal. Leaf size=104 \[ -\frac {a+b \cosh ^{-1}(c+d x)}{4 d e^5 (c+d x)^4}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{6 d e^5 (c+d x)}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{12 d e^5 (c+d x)^3} \]
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Rubi [A] time = 0.07, 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 = {5866, 12, 5662, 103, 95} \[ -\frac {a+b \cosh ^{-1}(c+d x)}{4 d e^5 (c+d x)^4}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{6 d e^5 (c+d x)}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{12 d e^5 (c+d x)^3} \]
Antiderivative was successfully verified.
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Rule 12
Rule 95
Rule 103
Rule 5662
Rule 5866
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^5} \, dx &=\frac {\operatorname {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{e^5 x^5} \, dx,x,c+d x\right )}{d}\\ &=\frac {\operatorname {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 \operatorname {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 \operatorname {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 \operatorname {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 = 0.08, size = 86, normalized size = 0.83 \[ \frac {-3 a+b \sqrt {c+d x-1} \sqrt {c+d x+1} \left (2 c^3+6 c^2 d x+6 c d^2 x^2+c+2 d^3 x^3+d x\right )-3 b \cosh ^{-1}(c+d x)}{12 d e^5 (c+d x)^4} \]
Antiderivative was successfully verified.
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fricas [B] time = 0.51, size = 208, normalized size = 2.00 \[ \frac {3 \, a d^{4} x^{4} + 12 \, a c d^{3} x^{3} + 18 \, a c^{2} d^{2} x^{2} + 12 \, a c^{3} d x - 3 \, b c^{4} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) + {\left (2 \, b c^{4} d^{3} x^{3} + 6 \, b c^{5} d^{2} x^{2} + 2 \, b c^{7} + b c^{5} + {\left (6 \, b c^{6} + b c^{4}\right )} d x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{12 \, {\left (c^{4} d^{5} e^{5} x^{4} + 4 \, c^{5} d^{4} e^{5} x^{3} + 6 \, c^{6} d^{3} e^{5} x^{2} + 4 \, c^{7} d^{2} e^{5} x + c^{8} d e^{5}\right )}} \]
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 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 76, normalized size = 0.73 \[ \frac {-\frac {a}{4 e^{5} \left (d x +c \right )^{4}}+\frac {b \left (-\frac {\mathrm {arccosh}\left (d x +c \right )}{4 \left (d x +c \right )^{4}}+\frac {\sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \left (2 \left (d x +c \right )^{2}+1\right )}{12 \left (d x +c \right )^{3}}\right )}{e^{5}}}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.56, size = 260, normalized size = 2.50 \[ \frac {1}{12} \, b {\left (\frac {{\left (2 \, d^{4} x^{4} + 8 \, c d^{3} x^{3} + 2 \, c^{4} + {\left (12 \, c^{2} d^{2} - d^{2}\right )} x^{2} - c^{2} + 2 \, {\left (4 \, c^{3} d - c d\right )} x - 1\right )} d}{{\left (d^{5} e^{5} x^{3} + 3 \, c d^{4} e^{5} x^{2} + 3 \, c^{2} d^{3} e^{5} x + c^{3} d^{2} e^{5}\right )} \sqrt {d x + c + 1} \sqrt {d x + c - 1}} - \frac {3 \, \operatorname {arcosh}\left (d x + c\right )}{d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}}\right )} - \frac {a}{4 \, {\left (d^{5} e^{5} x^{4} + 4 \, c d^{4} e^{5} x^{3} + 6 \, c^{2} d^{3} e^{5} x^{2} + 4 \, c^{3} d^{2} e^{5} x + c^{4} d e^{5}\right )}} \]
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 \[ \int \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^5} \,d x \]
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 \[ \frac {\int \frac {a}{c^{5} + 5 c^{4} d x + 10 c^{3} d^{2} x^{2} + 10 c^{2} d^{3} x^{3} + 5 c d^{4} x^{4} + d^{5} x^{5}}\, dx + \int \frac {b \operatorname {acosh}{\left (c + d x \right )}}{c^{5} + 5 c^{4} d x + 10 c^{3} d^{2} x^{2} + 10 c^{2} d^{3} x^{3} + 5 c d^{4} x^{4} + d^{5} x^{5}}\, dx}{e^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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