Optimal. Leaf size=142 \[ -\frac {5 b x \sqrt {1-c x}}{112 c^6 \sqrt {\frac {1}{1+c x}}}-\frac {5 b x^3 \sqrt {1-c x}}{168 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^5 \sqrt {1-c x}}{42 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {5 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \text {ArcSin}(c x)}{112 c^7} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 142, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {6418, 102, 12,
92, 41, 222} \begin {gather*} \frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {5 b \sqrt {\frac {1}{c x+1}} \sqrt {c x+1} \text {ArcSin}(c x)}{112 c^7}-\frac {5 b x \sqrt {1-c x}}{112 c^6 \sqrt {\frac {1}{c x+1}}}-\frac {5 b x^3 \sqrt {1-c x}}{168 c^4 \sqrt {\frac {1}{c x+1}}}-\frac {b x^5 \sqrt {1-c x}}{42 c^2 \sqrt {\frac {1}{c x+1}}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 41
Rule 92
Rule 102
Rule 222
Rule 6418
Rubi steps
\begin {align*} \int x^6 \left (a+b \text {sech}^{-1}(c x)\right ) \, dx &=\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {1}{7} \left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^6}{\sqrt {1-c x} \sqrt {1+c x}} \, dx\\ &=-\frac {b x^5 \sqrt {1-c x}}{42 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {\left (b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int -\frac {5 x^4}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{42 c^2}\\ &=-\frac {b x^5 \sqrt {1-c x}}{42 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (5 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^4}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{42 c^2}\\ &=-\frac {5 b x^3 \sqrt {1-c x}}{168 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^5 \sqrt {1-c x}}{42 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )-\frac {\left (5 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int -\frac {3 x^2}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{168 c^4}\\ &=-\frac {5 b x^3 \sqrt {1-c x}}{168 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^5 \sqrt {1-c x}}{42 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (5 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {x^2}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{56 c^4}\\ &=-\frac {5 b x \sqrt {1-c x}}{112 c^6 \sqrt {\frac {1}{1+c x}}}-\frac {5 b x^3 \sqrt {1-c x}}{168 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^5 \sqrt {1-c x}}{42 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (5 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c x} \sqrt {1+c x}} \, dx}{112 c^6}\\ &=-\frac {5 b x \sqrt {1-c x}}{112 c^6 \sqrt {\frac {1}{1+c x}}}-\frac {5 b x^3 \sqrt {1-c x}}{168 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^5 \sqrt {1-c x}}{42 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {\left (5 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x}\right ) \int \frac {1}{\sqrt {1-c^2 x^2}} \, dx}{112 c^6}\\ &=-\frac {5 b x \sqrt {1-c x}}{112 c^6 \sqrt {\frac {1}{1+c x}}}-\frac {5 b x^3 \sqrt {1-c x}}{168 c^4 \sqrt {\frac {1}{1+c x}}}-\frac {b x^5 \sqrt {1-c x}}{42 c^2 \sqrt {\frac {1}{1+c x}}}+\frac {1}{7} x^7 \left (a+b \text {sech}^{-1}(c x)\right )+\frac {5 b \sqrt {\frac {1}{1+c x}} \sqrt {1+c x} \sin ^{-1}(c x)}{112 c^7}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.14, size = 143, normalized size = 1.01 \begin {gather*} \frac {a x^7}{7}+b \sqrt {\frac {1-c x}{1+c x}} \left (-\frac {5 x}{112 c^6}-\frac {5 x^2}{112 c^5}-\frac {5 x^3}{168 c^4}-\frac {5 x^4}{168 c^3}-\frac {x^5}{42 c^2}-\frac {x^6}{42 c}\right )+\frac {1}{7} b x^7 \text {sech}^{-1}(c x)+\frac {5 i b \log \left (-2 i c x+2 \sqrt {\frac {1-c x}{1+c x}} (1+c x)\right )}{112 c^7} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.18, size = 138, normalized size = 0.97
method | result | size |
derivativedivides | \(\frac {\frac {c^{7} x^{7} a}{7}+b \left (\frac {c^{7} x^{7} \mathrm {arcsech}\left (c x \right )}{7}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-8 \sqrt {-c^{2} x^{2}+1}\, c^{5} x^{5}-10 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-15 c x \sqrt {-c^{2} x^{2}+1}+15 \arcsin \left (c x \right )\right )}{336 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{7}}\) | \(138\) |
default | \(\frac {\frac {c^{7} x^{7} a}{7}+b \left (\frac {c^{7} x^{7} \mathrm {arcsech}\left (c x \right )}{7}+\frac {\sqrt {-\frac {c x -1}{c x}}\, c x \sqrt {\frac {c x +1}{c x}}\, \left (-8 \sqrt {-c^{2} x^{2}+1}\, c^{5} x^{5}-10 c^{3} x^{3} \sqrt {-c^{2} x^{2}+1}-15 c x \sqrt {-c^{2} x^{2}+1}+15 \arcsin \left (c x \right )\right )}{336 \sqrt {-c^{2} x^{2}+1}}\right )}{c^{7}}\) | \(138\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.47, size = 135, normalized size = 0.95 \begin {gather*} \frac {1}{7} \, a x^{7} + \frac {1}{336} \, {\left (48 \, x^{7} \operatorname {arsech}\left (c x\right ) - \frac {\frac {15 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {5}{2}} + 40 \, {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{\frac {3}{2}} + 33 \, \sqrt {\frac {1}{c^{2} x^{2}} - 1}}{c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{3} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )}^{2} + 3 \, c^{6} {\left (\frac {1}{c^{2} x^{2}} - 1\right )} + c^{6}} + \frac {15 \, \arctan \left (\sqrt {\frac {1}{c^{2} x^{2}} - 1}\right )}{c^{6}}}{c}\right )} b \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 183, normalized size = 1.29 \begin {gather*} \frac {48 \, a c^{7} x^{7} - 48 \, b c^{7} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{x}\right ) - 30 \, b \arctan \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} - 1}{c x}\right ) + 48 \, {\left (b c^{7} x^{7} - b c^{7}\right )} \log \left (\frac {c x \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}} + 1}{c x}\right ) - {\left (8 \, b c^{6} x^{6} + 10 \, b c^{4} x^{4} + 15 \, b c^{2} x^{2}\right )} \sqrt {-\frac {c^{2} x^{2} - 1}{c^{2} x^{2}}}}{336 \, c^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^{6} \left (a + b \operatorname {asech}{\left (c x \right )}\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int x^6\,\left (a+b\,\mathrm {acosh}\left (\frac {1}{c\,x}\right )\right ) \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________