Optimal. Leaf size=260 \[ \frac {b \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \left (1-c^2 x^2\right )}{105 c^7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \left (35 c^4 d^2+84 c^2 d e+45 e^2\right ) \left (1-c^2 x^2\right )^2}{315 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e \left (14 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^3}{175 c^7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^2 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right ) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.22, antiderivative size = 260, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {276, 5958, 12,
534, 1265, 785} \begin {gather*} \frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )+\frac {b e \left (1-c^2 x^2\right )^3 \left (14 c^2 d+15 e\right )}{175 c^7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b e^2 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt {c x-1} \sqrt {c x+1}}-\frac {b \left (1-c^2 x^2\right )^2 \left (35 c^4 d^2+84 c^2 d e+45 e^2\right )}{315 c^7 \sqrt {c x-1} \sqrt {c x+1}}+\frac {b \left (1-c^2 x^2\right ) \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}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 276
Rule 534
Rule 785
Rule 1265
Rule 5958
Rubi steps
\begin {align*} \int x^2 \left (d+e x^2\right )^2 \left (a+b \cosh ^{-1}(c x)\right ) \, dx &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-(b c) \int \frac {x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{105 \sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {1}{105} (b c) \int \frac {x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt {-1+c x} \sqrt {1+c x}} \, dx\\ &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )-\frac {\left (b c \sqrt {-1+c^2 x^2}\right ) \int \frac {x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )}{\sqrt {-1+c^2 x^2}} \, dx}{105 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 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 {x \left (35 d^2+42 d e x+15 e^2 x^2\right )}{\sqrt {-1+c^2 x}} \, dx,x,x^2\right )}{210 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 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^4 d^2+42 c^2 d e+15 e^2}{c^6 \sqrt {-1+c^2 x}}+\frac {\left (35 c^4 d^2+84 c^2 d e+45 e^2\right ) \sqrt {-1+c^2 x}}{c^6}+\frac {3 e \left (14 c^2 d+15 e\right ) \left (-1+c^2 x\right )^{3/2}}{c^6}+\frac {15 e^2 \left (-1+c^2 x\right )^{5/2}}{c^6}\right ) \, dx,x,x^2\right )}{210 \sqrt {-1+c x} \sqrt {1+c x}}\\ &=\frac {b \left (35 c^4 d^2+42 c^2 d e+15 e^2\right ) \left (1-c^2 x^2\right )}{105 c^7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b \left (35 c^4 d^2+84 c^2 d e+45 e^2\right ) \left (1-c^2 x^2\right )^2}{315 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {b e \left (14 c^2 d+15 e\right ) \left (1-c^2 x^2\right )^3}{175 c^7 \sqrt {-1+c x} \sqrt {1+c x}}-\frac {b e^2 \left (1-c^2 x^2\right )^4}{49 c^7 \sqrt {-1+c x} \sqrt {1+c x}}+\frac {1}{3} d^2 x^3 \left (a+b \cosh ^{-1}(c x)\right )+\frac {2}{5} d e x^5 \left (a+b \cosh ^{-1}(c x)\right )+\frac {1}{7} e^2 x^7 \left (a+b \cosh ^{-1}(c x)\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.14, size = 163, normalized size = 0.63 \begin {gather*} \frac {105 a x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right )-\frac {b \sqrt {-1+c x} \sqrt {1+c x} \left (720 e^2+24 c^2 e \left (98 d+15 e x^2\right )+2 c^4 \left (1225 d^2+588 d e x^2+135 e^2 x^4\right )+c^6 \left (1225 d^2 x^2+882 d e x^4+225 e^2 x^6\right )\right )}{c^7}+105 b x^3 \left (35 d^2+42 d e x^2+15 e^2 x^4\right ) \cosh ^{-1}(c x)}{11025} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 2.92, size = 195, normalized size = 0.75
method | result | size |
derivativedivides | \(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arccosh}\left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \,\mathrm {arccosh}\left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\mathrm {arccosh}\left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (225 c^{6} e^{2} x^{6}+882 c^{6} d e \,x^{4}+1225 c^{6} d^{2} x^{2}+270 c^{4} e^{2} x^{4}+1176 c^{4} d e \,x^{2}+2450 c^{4} d^{2}+360 c^{2} e^{2} x^{2}+2352 c^{2} d e +720 e^{2}\right )}{11025}\right )}{c^{4}}}{c^{3}}\) | \(195\) |
default | \(\frac {\frac {a \left (\frac {1}{3} d^{2} c^{7} x^{3}+\frac {2}{5} d \,c^{7} e \,x^{5}+\frac {1}{7} e^{2} c^{7} x^{7}\right )}{c^{4}}+\frac {b \left (\frac {\mathrm {arccosh}\left (c x \right ) d^{2} c^{7} x^{3}}{3}+\frac {2 \,\mathrm {arccosh}\left (c x \right ) d \,c^{7} e \,x^{5}}{5}+\frac {\mathrm {arccosh}\left (c x \right ) e^{2} c^{7} x^{7}}{7}-\frac {\sqrt {c x -1}\, \sqrt {c x +1}\, \left (225 c^{6} e^{2} x^{6}+882 c^{6} d e \,x^{4}+1225 c^{6} d^{2} x^{2}+270 c^{4} e^{2} x^{4}+1176 c^{4} d e \,x^{2}+2450 c^{4} d^{2}+360 c^{2} e^{2} x^{2}+2352 c^{2} d e +720 e^{2}\right )}{11025}\right )}{c^{4}}}{c^{3}}\) | \(195\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.26, size = 247, normalized size = 0.95 \begin {gather*} \frac {1}{7} \, a x^{7} e^{2} + \frac {2}{5} \, a d x^{5} e + \frac {1}{3} \, a d^{2} x^{3} + \frac {1}{9} \, {\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} + \frac {2}{75} \, {\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 + \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^{2} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 385, normalized size = 1.48 \begin {gather*} \frac {1575 \, a c^{7} x^{7} \cosh \left (1\right )^{2} + 1575 \, a c^{7} x^{7} \sinh \left (1\right )^{2} + 4410 \, a c^{7} d x^{5} \cosh \left (1\right ) + 3675 \, a c^{7} d^{2} x^{3} + 105 \, {\left (15 \, b c^{7} x^{7} \cosh \left (1\right )^{2} + 15 \, b c^{7} x^{7} \sinh \left (1\right )^{2} + 42 \, b c^{7} d x^{5} \cosh \left (1\right ) + 35 \, b c^{7} d^{2} x^{3} + 6 \, {\left (5 \, b c^{7} x^{7} \cosh \left (1\right ) + 7 \, b c^{7} d x^{5}\right )} \sinh \left (1\right )\right )} \log \left (c x + \sqrt {c^{2} x^{2} - 1}\right ) + 630 \, {\left (5 \, a c^{7} x^{7} \cosh \left (1\right ) + 7 \, a c^{7} d x^{5}\right )} \sinh \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} + 45 \, {\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 )^{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 ) + 6 \, {\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 )\right )} \sqrt {c^{2} x^{2} - 1}}{11025 \, c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [C] Result contains complex when optimal does not.
time = 0.73, size = 340, normalized size = 1.31 \begin {gather*} \begin {cases} \frac {a d^{2} x^{3}}{3} + \frac {2 a d e x^{5}}{5} + \frac {a e^{2} x^{7}}{7} + \frac {b d^{2} x^{3} \operatorname {acosh}{\left (c x \right )}}{3} + \frac {2 b d e x^{5} \operatorname {acosh}{\left (c x \right )}}{5} + \frac {b e^{2} x^{7} \operatorname {acosh}{\left (c x \right )}}{7} - \frac {b d^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{9 c} - \frac {2 b d e x^{4} \sqrt {c^{2} x^{2} - 1}}{25 c} - \frac {b e^{2} x^{6} \sqrt {c^{2} x^{2} - 1}}{49 c} - \frac {2 b d^{2} \sqrt {c^{2} x^{2} - 1}}{9 c^{3}} - \frac {8 b d e x^{2} \sqrt {c^{2} x^{2} - 1}}{75 c^{3}} - \frac {6 b e^{2} x^{4} \sqrt {c^{2} x^{2} - 1}}{245 c^{3}} - \frac {16 b d e \sqrt {c^{2} x^{2} - 1}}{75 c^{5}} - \frac {8 b e^{2} x^{2} \sqrt {c^{2} x^{2} - 1}}{245 c^{5}} - \frac {16 b e^{2} \sqrt {c^{2} x^{2} - 1}}{245 c^{7}} & \text {for}\: c \neq 0 \\\left (a + \frac {i \pi b}{2}\right ) \left (\frac {d^{2} x^{3}}{3} + \frac {2 d e x^{5}}{5} + \frac {e^{2} x^{7}}{7}\right ) & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^2\,\left (a+b\,\mathrm {acosh}\left (c\,x\right )\right )\,{\left (e\,x^2+d\right )}^2 \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________