Optimal. Leaf size=137 \[ \frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{20 d e^6 (c+d x)^4}+\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{40 d e^6 (c+d x)^2}-\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {3 b \text {ArcTan}\left (\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{40 d e^6} \]
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Rubi [A]
time = 0.06, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {5996, 12, 5883,
105, 94, 209} \begin {gather*} -\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {3 b \text {ArcTan}\left (\sqrt {c+d x-1} \sqrt {c+d x+1}\right )}{40 d e^6}+\frac {3 b \sqrt {c+d x-1} \sqrt {c+d x+1}}{40 d e^6 (c+d x)^2}+\frac {b \sqrt {c+d x-1} \sqrt {c+d x+1}}{20 d e^6 (c+d x)^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 94
Rule 105
Rule 209
Rule 5883
Rule 5996
Rubi steps
\begin {align*} \int \frac {a+b \cosh ^{-1}(c+d x)}{(c e+d e x)^6} \, dx &=\frac {\text {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{e^6 x^6} \, dx,x,c+d x\right )}{d}\\ &=\frac {\text {Subst}\left (\int \frac {a+b \cosh ^{-1}(x)}{x^6} \, dx,x,c+d x\right )}{d e^6}\\ &=-\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {b \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^5 \sqrt {1+x}} \, dx,x,c+d x\right )}{5 d e^6}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{20 d e^6 (c+d x)^4}-\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {b \text {Subst}\left (\int \frac {3}{\sqrt {-1+x} x^3 \sqrt {1+x}} \, dx,x,c+d x\right )}{20 d e^6}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{20 d e^6 (c+d x)^4}-\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x^3 \sqrt {1+x}} \, dx,x,c+d x\right )}{20 d e^6}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{20 d e^6 (c+d x)^4}+\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{40 d e^6 (c+d x)^2}-\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{\sqrt {-1+x} x \sqrt {1+x}} \, dx,x,c+d x\right )}{40 d e^6}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{20 d e^6 (c+d x)^4}+\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{40 d e^6 (c+d x)^2}-\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {(3 b) \text {Subst}\left (\int \frac {1}{1+x^2} \, dx,x,\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{40 d e^6}\\ &=\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{20 d e^6 (c+d x)^4}+\frac {3 b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{40 d e^6 (c+d x)^2}-\frac {a+b \cosh ^{-1}(c+d x)}{5 d e^6 (c+d x)^5}+\frac {3 b \tan ^{-1}\left (\sqrt {-1+c+d x} \sqrt {1+c+d x}\right )}{40 d e^6}\\ \end {align*}
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Mathematica [A]
time = 0.16, size = 136, normalized size = 0.99 \begin {gather*} \frac {\frac {b \sqrt {-1+c+d x} \sqrt {1+c+d x}}{4 (c+d x)^4}-\frac {a+b \cosh ^{-1}(c+d x)}{(c+d x)^5}+\frac {3 b \left (\frac {(-1+c+d x) (1+c+d x)}{(c+d x)^2}+\sqrt {-1+(c+d x)^2} \text {ArcTan}\left (\sqrt {-1+(c+d x)^2}\right )\right )}{8 \sqrt {-1+c+d x} \sqrt {1+c+d x}}}{5 d e^6} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 4.00, size = 141, normalized size = 1.03
method | result | size |
derivativedivides | \(\frac {-\frac {a}{5 e^{6} \left (d x +c \right )^{5}}-\frac {b \,\mathrm {arccosh}\left (d x +c \right )}{5 e^{6} \left (d x +c \right )^{5}}-\frac {3 b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{40 e^{6} \sqrt {\left (d x +c \right )^{2}-1}}+\frac {3 b \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{40 e^{6} \left (d x +c \right )^{2}}+\frac {b \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{20 e^{6} \left (d x +c \right )^{4}}}{d}\) | \(141\) |
default | \(\frac {-\frac {a}{5 e^{6} \left (d x +c \right )^{5}}-\frac {b \,\mathrm {arccosh}\left (d x +c \right )}{5 e^{6} \left (d x +c \right )^{5}}-\frac {3 b \sqrt {d x +c -1}\, \sqrt {d x +c +1}\, \arctan \left (\frac {1}{\sqrt {\left (d x +c \right )^{2}-1}}\right )}{40 e^{6} \sqrt {\left (d x +c \right )^{2}-1}}+\frac {3 b \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{40 e^{6} \left (d x +c \right )^{2}}+\frac {b \sqrt {d x +c -1}\, \sqrt {d x +c +1}}{20 e^{6} \left (d x +c \right )^{4}}}{d}\) | \(141\) |
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. 803 vs.
\(2 (113) = 226\).
time = 0.45, size = 803, normalized size = 5.86 \begin {gather*} -\frac {8 \, a c^{5} - 6 \, {\left (b c^{5} d^{5} x^{5} + 5 \, b c^{6} d^{4} x^{4} + 10 \, b c^{7} d^{3} x^{3} + 10 \, b c^{8} d^{2} x^{2} + 5 \, b c^{9} d x + b c^{10}\right )} \arctan \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 8 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x\right )} \log \left (d x + c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - 8 \, {\left (b d^{5} x^{5} + 5 \, b c d^{4} x^{4} + 10 \, b c^{2} d^{3} x^{3} + 10 \, b c^{3} d^{2} x^{2} + 5 \, b c^{4} d x + b c^{5}\right )} \log \left (-d x - c + \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}\right ) - {\left (3 \, b c^{5} d^{3} x^{3} + 9 \, b c^{6} d^{2} x^{2} + 3 \, b c^{8} + 2 \, b c^{6} + {\left (9 \, b c^{7} + 2 \, b c^{5}\right )} d x\right )} \sqrt {d^{2} x^{2} + 2 \, c d x + c^{2} - 1}}{40 \, {\left ({\left (c^{5} d^{6} x^{5} + 5 \, c^{6} d^{5} x^{4} + 10 \, c^{7} d^{4} x^{3} + 10 \, c^{8} d^{3} x^{2} + 5 \, c^{9} d^{2} x + c^{10} d\right )} \cosh \left (1\right )^{6} + 6 \, {\left (c^{5} d^{6} x^{5} + 5 \, c^{6} d^{5} x^{4} + 10 \, c^{7} d^{4} x^{3} + 10 \, c^{8} d^{3} x^{2} + 5 \, c^{9} d^{2} x + c^{10} d\right )} \cosh \left (1\right )^{5} \sinh \left (1\right ) + 15 \, {\left (c^{5} d^{6} x^{5} + 5 \, c^{6} d^{5} x^{4} + 10 \, c^{7} d^{4} x^{3} + 10 \, c^{8} d^{3} x^{2} + 5 \, c^{9} d^{2} x + c^{10} d\right )} \cosh \left (1\right )^{4} \sinh \left (1\right )^{2} + 20 \, {\left (c^{5} d^{6} x^{5} + 5 \, c^{6} d^{5} x^{4} + 10 \, c^{7} d^{4} x^{3} + 10 \, c^{8} d^{3} x^{2} + 5 \, c^{9} d^{2} x + c^{10} d\right )} \cosh \left (1\right )^{3} \sinh \left (1\right )^{3} + 15 \, {\left (c^{5} d^{6} x^{5} + 5 \, c^{6} d^{5} x^{4} + 10 \, c^{7} d^{4} x^{3} + 10 \, c^{8} d^{3} x^{2} + 5 \, c^{9} d^{2} x + c^{10} d\right )} \cosh \left (1\right )^{2} \sinh \left (1\right )^{4} + 6 \, {\left (c^{5} d^{6} x^{5} + 5 \, c^{6} d^{5} x^{4} + 10 \, c^{7} d^{4} x^{3} + 10 \, c^{8} d^{3} x^{2} + 5 \, c^{9} d^{2} x + c^{10} d\right )} \cosh \left (1\right ) \sinh \left (1\right )^{5} + {\left (c^{5} d^{6} x^{5} + 5 \, c^{6} d^{5} x^{4} + 10 \, c^{7} d^{4} x^{3} + 10 \, c^{8} d^{3} x^{2} + 5 \, c^{9} d^{2} x + c^{10} d\right )} \sinh \left (1\right )^{6}\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^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx + \int \frac {b \operatorname {acosh}{\left (c + d x \right )}}{c^{6} + 6 c^{5} d x + 15 c^{4} d^{2} x^{2} + 20 c^{3} d^{3} x^{3} + 15 c^{2} d^{4} x^{4} + 6 c d^{5} x^{5} + d^{6} x^{6}}\, dx}{e^{6}} \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 \frac {a+b\,\mathrm {acosh}\left (c+d\,x\right )}{{\left (c\,e+d\,e\,x\right )}^6} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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