Optimal. Leaf size=287 \[ \frac {b \left (35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right ) \left (1-c^2 x^2\right )}{35 c^7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \left (1-c^2 x^2\right )^2}{105 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b e^2 \left (7 c^2 d+5 e\right ) \left (1-c^2 x^2\right )^3}{175 c^7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^3 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+d^3 x \left (a+b \cosh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right ) \]
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Rubi [A]
time = 0.25, antiderivative size = 287, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {200, 5908, 12,
1624, 1813, 1864} \begin {gather*} d^3 x \left (a+b \cosh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3 b e^2 \left (1-c^2 x^2\right )^3 \left (7 c^2 d+5 e\right )}{175 c^7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b e^3 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b e \left (1-c^2 x^2\right )^2 \left (35 c^4 d^2+42 c^2 d e+15 e^2\right )}{105 c^7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (1-c^2 x^2\right ) \left (35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right )}{35 c^7 \sqrt {c x-1} \sqrt {c x+1}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 200
Rule 1624
Rule 1813
Rule 1864
Rule 5908
Rubi steps
\begin {align*} \int \left (d+e x^2\right )^3 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{35 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{35} (b c) \int \frac {x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right )}{\sqrt {-1+c^2 x^2}} \, dx}{35 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \frac {35 d^3+35 d^2 e x+21 d e^2 x^2+5 e^3 x^3}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{70 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=d^3 x \left (a+b \cosh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3}{c^6 \sqrt {-1+c^2 x}}+\frac {e \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \sqrt {-1+c^2 x}}{c^6}+\frac {3 e^2 \left (7 c^2 d+5 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac {5 e^3 \left (-1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{70 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b \left (35 c^6 d^3+35 c^4 d^2 e+21 c^2 d e^2+5 e^3\right ) \left (1-c^2 x^2\right )}{35 c^7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \left (1-c^2 x^2\right )^2}{105 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {3 b e^2 \left (7 c^2 d+5 e\right ) \left (1-c^2 x^2\right )^3}{175 c^7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^3 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+d^3 x \left (a+b \cosh ^{-1}(c x)\right )+d^2 e x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {3}{5} d e^2 x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^3 x^7 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}
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Mathematica [A]
time = 0.20, size = 193, normalized size = 0.67 \begin {gather*} a \left (d^3 x+d^2 e x^3+\frac {3}{5} d e^2 x^5+\frac {e^3 x^7}{7}\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (240 e^3+24 c^2 e^2 \left (49 d+5 e x^2\right )+2 c^4 e \left (1225 d^2+294 d e x^2+45 e^2 x^4\right )+c^6 \left (3675 d^3+1225 d^2 e x^2+441 d e^2 x^4+75 e^3 x^6\right )\right )}{3675 c^7}+\frac {1}{35} b x \left (35 d^3+35 d^2 e x^2+21 d e^2 x^4+5 e^3 x^6\right ) \cosh ^{-1}(c x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.85, size = 235, normalized size = 0.82
method | result | size |
derivativedivides | \(\frac {\frac {a \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b \left (\mathrm {arccosh}\left (c x \right ) d^{3} c^{7} x +\mathrm {arccosh}\left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \,\mathrm {arccosh}\left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\mathrm {arccosh}\left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} e^{3} x^{4}+3675 d^{3} c^{6}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} e^{3} x^{2}+1176 c^{2} d \,e^{2}+240 e^{3}\right )}{3675}\right )}{c^{6}}}{c}\) | \(235\) |
default | \(\frac {\frac {a \left (d^{3} c^{7} x +d^{2} c^{7} e \,x^{3}+\frac {3}{5} d \,c^{7} e^{2} x^{5}+\frac {1}{7} e^{3} c^{7} x^{7}\right )}{c^{6}}+\frac {b \left (\mathrm {arccosh}\left (c x \right ) d^{3} c^{7} x +\mathrm {arccosh}\left (c x \right ) d^{2} c^{7} e \,x^{3}+\frac {3 \,\mathrm {arccosh}\left (c x \right ) d \,c^{7} e^{2} x^{5}}{5}+\frac {\mathrm {arccosh}\left (c x \right ) e^{3} c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (75 c^{6} e^{3} x^{6}+441 c^{6} d \,e^{2} x^{4}+1225 c^{6} d^{2} e \,x^{2}+90 c^{4} e^{3} x^{4}+3675 d^{3} c^{6}+588 c^{4} d \,e^{2} x^{2}+2450 c^{4} d^{2} e +120 c^{2} e^{3} x^{2}+1176 c^{2} d \,e^{2}+240 e^{3}\right )}{3675}\right )}{c^{6}}}{c}\) | \(235\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 285, normalized size = 0.99 \begin {gather*} \frac {1}{7} \, a x^{7} e^{3} + \frac {3}{5} \, a d x^{5} e^{2} + a d^{2} x^{3} e + a d^{3} x + \frac {1}{3} \, {\left (3 \, x^{3} \operatorname {arcosh}\left (c x\right ) - c {\left (\frac {\sqrt {c^{2} x^{2} - 1} x^{2}}{c^{2}} + \frac {2 \, \sqrt {c^{2} x^{2} - 1}}{c^{4}}\right )}\right )} b d^{2} e + \frac {{\left (c x \operatorname {arcosh}\left (c x\right ) - \sqrt {c^{2} x^{2} - 1}\right )} b d^{3}}{c} + \frac {1}{25} \, {\left (15 \, x^{5} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {3 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{2}} + \frac {4 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1}}{c^{6}}\right )} c\right )} b d e^{2} + \frac {1}{245} \, {\left (35 \, x^{7} \operatorname {arcosh}\left (c x\right ) - {\left (\frac {5 \, \sqrt {c^{2} x^{2} - 1} x^{6}}{c^{2}} + \frac {6 \, \sqrt {c^{2} x^{2} - 1} x^{4}}{c^{4}} + \frac {8 \, \sqrt {c^{2} x^{2} - 1} x^{2}}{c^{6}} + \frac {16 \, \sqrt {c^{2} x^{2} - 1}}{c^{8}}\right )} c\right )} b e^{3} \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. 611 vs.
\(2 (253) = 506\).
time = 0.36, size = 611, normalized size = 2.13 \begin {gather*} \frac {525 \, a c^{7} x^{7} \cosh \left (1\right )^{3} + 525 \, a c^{7} x^{7} \sinh \left (1\right )^{3} + 2205 \, a c^{7} d x^{5} \cosh \left (1\right )^{2} + 3675 \, a c^{7} d^{2} x^{3} \cosh \left (1\right ) + 3675 \, a c^{7} d^{3} x + 315 \, {\left (5 \, a c^{7} x^{7} \cosh \left (1\right ) + 7 \, a c^{7} d x^{5}\right )} \sinh \left (1\right )^{2} + 105 \, {\left (5 \, b c^{7} x^{7} \cosh \left (1\right )^{3} + 5 \, b c^{7} x^{7} \sinh \left (1\right )^{3} + 21 \, b c^{7} d x^{5} \cosh \left (1\right )^{2} + 35 \, b c^{7} d^{2} x^{3} \cosh \left (1\right ) + 35 \, b c^{7} d^{3} x + 3 \, {\left (5 \, b c^{7} x^{7} \cosh \left (1\right ) + 7 \, b c^{7} d x^{5}\right )} \sinh \left (1\right )^{2} + {\left (15 \, b c^{7} x^{7} \cosh \left (1\right )^{2} + 42 \, b c^{7} d x^{5} \cosh \left (1\right ) + 35 \, b c^{7} d^{2} x^{3}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 105 \, {\left (15 \, a c^{7} x^{7} \cosh \left (1\right )^{2} + 42 \, a c^{7} d x^{5} \cosh \left (1\right ) + 35 \, a c^{7} d^{2} x^{3}\right )} \sinh \left (1\right ) - {\left (3675 \, b c^{6} d^{3} + 15 \, {\left (5 \, b c^{6} x^{6} + 6 \, b c^{4} x^{4} + 8 \, b c^{2} x^{2} + 16 \, b\right )} \cosh \left (1\right )^{3} + 15 \, {\left (5 \, b c^{6} x^{6} + 6 \, b c^{4} x^{4} + 8 \, b c^{2} x^{2} + 16 \, b\right )} \sinh \left (1\right )^{3} + 147 \, {\left (3 \, b c^{6} d x^{4} + 4 \, b c^{4} d x^{2} + 8 \, b c^{2} d\right )} \cosh \left (1\right )^{2} + 3 \, {\left (147 \, b c^{6} d x^{4} + 196 \, b c^{4} d x^{2} + 392 \, b c^{2} d + 15 \, {\left (5 \, b c^{6} x^{6} + 6 \, b c^{4} x^{4} + 8 \, b c^{2} x^{2} + 16 \, b\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )^{2} + 1225 \, {\left (b c^{6} d^{2} x^{2} + 2 \, b c^{4} d^{2}\right )} \cosh \left (1\right ) + {\left (1225 \, b c^{6} d^{2} x^{2} + 2450 \, b c^{4} d^{2} + 45 \, {\left (5 \, b c^{6} x^{6} + 6 \, b c^{4} x^{4} + 8 \, b c^{2} x^{2} + 16 \, b\right )} \cosh \left (1\right )^{2} + 294 \, {\left (3 \, b c^{6} d x^{4} + 4 \, b c^{4} d x^{2} + 8 \, b c^{2} d\right )} \cosh \left (1\right )\right )} \sinh \left (1\right )\right )} \sqrt {c^{2} x^{2} - 1}}{3675 \, c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.74, size = 396, normalized size = 1.38 \begin {gather*} \begin {cases} a d^{3} x + a d^{2} e x^{3} + \frac {3 a d e^{2} x^{5}}{5} + \frac {a e^{3} x^{7}}{7} + b d^{3} x \operatorname {acosh}{\left (c x \right )} + b d^{2} e x^{3} \operatorname {acosh}{\left (c x \right )} + \frac {3 b d e^{2} x^{5} \operatorname {acosh}{\left (c x \right )}}{5} + \frac {b e^{3} x^{7} \operatorname {acosh}{\left (c x \right )}}{7} - \frac {b d^{3} \sqrt {c^{2} x^{2} - 1}}{c} - \frac {b d^{2} e x^{2} \sqrt {c^{2} x^{2} - 1}}{3 c} - \frac {3 b d e^{2} x^{4} \sqrt {c^{2} x^{2} - 1}}{25 c} - \frac {b e^{3} x^{6} \sqrt {c^{2} x^{2} - 1}}{49 c} - \frac {2 b d^{2} e \sqrt {c^{2} x^{2} - 1}}{3 c^{3}} - \frac {4 b d e^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{25 c^{3}} - \frac {6 b e^{3} x^{4} \sqrt {c^{2} x^{2} - 1}}{245 c^{3}} - \frac {8 b d e^{2} \sqrt {c^{2} x^{2} - 1}}{25 c^{5}} - \frac {8 b e^{3} x^{2} \sqrt {c^{2} x^{2} - 1}}{245 c^{5}} - \frac {16 b e^{3} \sqrt {c^{2} x^{2} - 1}}{245 c^{7}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (d^{3} x + d^{2} e x^{3} + \frac {3 d e^{2} x^{5}}{5} + \frac {e^{3} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \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: RuntimeError} \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 = -0.00 \begin {gather*} \int \left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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