Optimal. Leaf size=267 \[ \frac {\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \left (1-a^2 x^2\right )}{35 a^7 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {d \left (35 a^4 c^2+42 a^2 c d+15 d^2\right ) \left (1-a^2 x^2\right )^2}{105 a^7 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 d^2 \left (7 a^2 c+5 d\right ) \left (1-a^2 x^2\right )^3}{175 a^7 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {d^3 \left (1-a^2 x^2\right )^4}{49 a^7 \sqrt {-1+a x} \sqrt {1+a x}}+c^3 x \cosh ^{-1}(a x)+c^2 d x^3 \cosh ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cosh ^{-1}(a x) \]
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Rubi [A]
time = 0.25, antiderivative size = 267, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {200, 5908, 12,
1624, 1813, 1864} \begin {gather*} \frac {3 d^2 \left (1-a^2 x^2\right )^3 \left (7 a^2 c+5 d\right )}{175 a^7 \sqrt {a x-1} \sqrt {a x+1}}-\frac {d^3 \left (1-a^2 x^2\right )^4}{49 a^7 \sqrt {a x-1} \sqrt {a x+1}}-\frac {d \left (1-a^2 x^2\right )^2 \left (35 a^4 c^2+42 a^2 c d+15 d^2\right )}{105 a^7 \sqrt {a x-1} \sqrt {a x+1}}+\frac {\left (1-a^2 x^2\right ) \left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right )}{35 a^7 \sqrt {a x-1} \sqrt {a x+1}}+c^3 x \cosh ^{-1}(a x)+c^2 d x^3 \cosh ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cosh ^{-1}(a x) \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 (c+d x^2\right )^3 \cosh ^{-1}(a x) \, dx &=c^3 x \cosh ^{-1}(a x)+c^2 d x^3 \cosh ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cosh ^{-1}(a x)-a \int \frac {x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right )}{35 \sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=c^3 x \cosh ^{-1}(a x)+c^2 d x^3 \cosh ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cosh ^{-1}(a x)-\frac {1}{35} a \int \frac {x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right )}{\sqrt {-1+a x} \sqrt {1+a x}} \, dx\\ &=c^3 x \cosh ^{-1}(a x)+c^2 d x^3 \cosh ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cosh ^{-1}(a x)-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \int \frac {x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right )}{\sqrt {-1+a^2 x^2}} \, dx}{35 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=c^3 x \cosh ^{-1}(a x)+c^2 d x^3 \cosh ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cosh ^{-1}(a x)-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {35 c^3+35 c^2 d x+21 c d^2 x^2+5 d^3 x^3}{\sqrt {-1+a^2 x}} \, dx,x,x^2\right )}{70 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=c^3 x \cosh ^{-1}(a x)+c^2 d x^3 \cosh ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cosh ^{-1}(a x)-\frac {\left (a \sqrt {-1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3}{a^6 \sqrt {-1+a^2 x}}+\frac {d \left (35 a^4 c^2+42 a^2 c d+15 d^2\right ) \sqrt {-1+a^2 x}}{a^6}+\frac {3 d^2 \left (7 a^2 c+5 d\right ) \left (-1+a^2 x\right )^{3/2}}{a^6}+\frac {5 d^3 \left (-1+a^2 x\right )^{5/2}}{a^6}\right ) \, dx,x,x^2\right )}{70 \sqrt {-1+a x} \sqrt {1+a x}}\\ &=\frac {\left (35 a^6 c^3+35 a^4 c^2 d+21 a^2 c d^2+5 d^3\right ) \left (1-a^2 x^2\right )}{35 a^7 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {d \left (35 a^4 c^2+42 a^2 c d+15 d^2\right ) \left (1-a^2 x^2\right )^2}{105 a^7 \sqrt {-1+a x} \sqrt {1+a x}}+\frac {3 d^2 \left (7 a^2 c+5 d\right ) \left (1-a^2 x^2\right )^3}{175 a^7 \sqrt {-1+a x} \sqrt {1+a x}}-\frac {d^3 \left (1-a^2 x^2\right )^4}{49 a^7 \sqrt {-1+a x} \sqrt {1+a x}}+c^3 x \cosh ^{-1}(a x)+c^2 d x^3 \cosh ^{-1}(a x)+\frac {3}{5} c d^2 x^5 \cosh ^{-1}(a x)+\frac {1}{7} d^3 x^7 \cosh ^{-1}(a x)\\ \end {align*}
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Mathematica [A]
time = 0.14, size = 154, normalized size = 0.58 \begin {gather*} -\frac {\sqrt {-1+a x} \sqrt {1+a x} \left (240 d^3+24 a^2 d^2 \left (49 c+5 d x^2\right )+2 a^4 d \left (1225 c^2+294 c d x^2+45 d^2 x^4\right )+a^6 \left (3675 c^3+1225 c^2 d x^2+441 c d^2 x^4+75 d^3 x^6\right )\right )}{3675 a^7}+\frac {1}{35} x \left (35 c^3+35 c^2 d x^2+21 c d^2 x^4+5 d^3 x^6\right ) \cosh ^{-1}(a x) \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.64, size = 176, normalized size = 0.66
method | result | size |
derivativedivides | \(\frac {\mathrm {arccosh}\left (a x \right ) c^{3} a x +a \,\mathrm {arccosh}\left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \,\mathrm {arccosh}\left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \,\mathrm {arccosh}\left (a x \right ) d^{3} x^{7}}{7}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (75 d^{3} a^{6} x^{6}+441 c \,a^{6} d^{2} x^{4}+1225 c^{2} a^{6} d \,x^{2}+90 d^{3} a^{4} x^{4}+3675 a^{6} c^{3}+588 c \,a^{4} d^{2} x^{2}+2450 a^{4} c^{2} d +120 d^{3} a^{2} x^{2}+1176 a^{2} c \,d^{2}+240 d^{3}\right )}{3675 a^{6}}}{a}\) | \(176\) |
default | \(\frac {\mathrm {arccosh}\left (a x \right ) c^{3} a x +a \,\mathrm {arccosh}\left (a x \right ) c^{2} d \,x^{3}+\frac {3 a \,\mathrm {arccosh}\left (a x \right ) c \,d^{2} x^{5}}{5}+\frac {a \,\mathrm {arccosh}\left (a x \right ) d^{3} x^{7}}{7}-\frac {\sqrt {a x -1}\, \sqrt {a x +1}\, \left (75 d^{3} a^{6} x^{6}+441 c \,a^{6} d^{2} x^{4}+1225 c^{2} a^{6} d \,x^{2}+90 d^{3} a^{4} x^{4}+3675 a^{6} c^{3}+588 c \,a^{4} d^{2} x^{2}+2450 a^{4} c^{2} d +120 d^{3} a^{2} x^{2}+1176 a^{2} c \,d^{2}+240 d^{3}\right )}{3675 a^{6}}}{a}\) | \(176\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 257, normalized size = 0.96 \begin {gather*} -\frac {1}{3675} \, {\left (\frac {75 \, \sqrt {a^{2} x^{2} - 1} d^{3} x^{6}}{a^{2}} + \frac {441 \, \sqrt {a^{2} x^{2} - 1} c d^{2} x^{4}}{a^{2}} + \frac {1225 \, \sqrt {a^{2} x^{2} - 1} c^{2} d x^{2}}{a^{2}} + \frac {90 \, \sqrt {a^{2} x^{2} - 1} d^{3} x^{4}}{a^{4}} + \frac {3675 \, \sqrt {a^{2} x^{2} - 1} c^{3}}{a^{2}} + \frac {588 \, \sqrt {a^{2} x^{2} - 1} c d^{2} x^{2}}{a^{4}} + \frac {2450 \, \sqrt {a^{2} x^{2} - 1} c^{2} d}{a^{4}} + \frac {120 \, \sqrt {a^{2} x^{2} - 1} d^{3} x^{2}}{a^{6}} + \frac {1176 \, \sqrt {a^{2} x^{2} - 1} c d^{2}}{a^{6}} + \frac {240 \, \sqrt {a^{2} x^{2} - 1} d^{3}}{a^{8}}\right )} a + \frac {1}{35} \, {\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \operatorname {arcosh}\left (a x\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.35, size = 179, normalized size = 0.67 \begin {gather*} \frac {105 \, {\left (5 \, a^{7} d^{3} x^{7} + 21 \, a^{7} c d^{2} x^{5} + 35 \, a^{7} c^{2} d x^{3} + 35 \, a^{7} c^{3} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - {\left (75 \, a^{6} d^{3} x^{6} + 3675 \, a^{6} c^{3} + 2450 \, a^{4} c^{2} d + 1176 \, a^{2} c d^{2} + 9 \, {\left (49 \, a^{6} c d^{2} + 10 \, a^{4} d^{3}\right )} x^{4} + 240 \, d^{3} + {\left (1225 \, a^{6} c^{2} d + 588 \, a^{4} c d^{2} + 120 \, a^{2} d^{3}\right )} x^{2}\right )} \sqrt {a^{2} x^{2} - 1}}{3675 \, a^{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 = 0.71, size = 328, normalized size = 1.23 \begin {gather*} \begin {cases} c^{3} x \operatorname {acosh}{\left (a x \right )} + c^{2} d x^{3} \operatorname {acosh}{\left (a x \right )} + \frac {3 c d^{2} x^{5} \operatorname {acosh}{\left (a x \right )}}{5} + \frac {d^{3} x^{7} \operatorname {acosh}{\left (a x \right )}}{7} - \frac {c^{3} \sqrt {a^{2} x^{2} - 1}}{a} - \frac {c^{2} d x^{2} \sqrt {a^{2} x^{2} - 1}}{3 a} - \frac {3 c d^{2} x^{4} \sqrt {a^{2} x^{2} - 1}}{25 a} - \frac {d^{3} x^{6} \sqrt {a^{2} x^{2} - 1}}{49 a} - \frac {2 c^{2} d \sqrt {a^{2} x^{2} - 1}}{3 a^{3}} - \frac {4 c d^{2} x^{2} \sqrt {a^{2} x^{2} - 1}}{25 a^{3}} - \frac {6 d^{3} x^{4} \sqrt {a^{2} x^{2} - 1}}{245 a^{3}} - \frac {8 c d^{2} \sqrt {a^{2} x^{2} - 1}}{25 a^{5}} - \frac {8 d^{3} x^{2} \sqrt {a^{2} x^{2} - 1}}{245 a^{5}} - \frac {16 d^{3} \sqrt {a^{2} x^{2} - 1}}{245 a^{7}} & \text {for}\: a \neq 0 \\\frac {i \pi \left (c^{3} x + c^{2} d x^{3} + \frac {3 c d^{2} x^{5}}{5} + \frac {d^{3} x^{7}}{7}\right )}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.42, size = 214, normalized size = 0.80 \begin {gather*} \frac {1}{35} \, {\left (5 \, d^{3} x^{7} + 21 \, c d^{2} x^{5} + 35 \, c^{2} d x^{3} + 35 \, c^{3} x\right )} \log \left (a x + \sqrt {a^{2} x^{2} - 1}\right ) - \frac {{\left (35 \, a^{6} c^{3} + 35 \, a^{4} c^{2} d + 21 \, a^{2} c d^{2} + 5 \, d^{3}\right )} \sqrt {a^{2} x^{2} - 1}}{35 \, a^{7}} - \frac {1225 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{4} c^{2} d + 441 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {5}{2}} a^{2} c d^{2} + 1470 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} a^{2} c d^{2} + 75 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {7}{2}} d^{3} + 315 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {5}{2}} d^{3} + 525 \, {\left (a^{2} x^{2} - 1\right )}^{\frac {3}{2}} d^{3}}{3675 \, a^{7}} \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 \mathrm {acosh}\left (a\,x\right )\,{\left (d\,x^2+c\right )}^3 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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