Optimal. Leaf size=77 \[ \frac {2 b d \sqrt {1-c^2 x^2}}{3 c}+\frac {b d \left (1-c^2 x^2\right )^{3/2}}{9 c}+d x (a+b \text {ArcSin}(c x))-\frac {1}{3} c^2 d x^3 (a+b \text {ArcSin}(c x)) \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 77, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4739, 12, 455,
45} \begin {gather*} -\frac {1}{3} c^2 d x^3 (a+b \text {ArcSin}(c x))+d x (a+b \text {ArcSin}(c x))+\frac {b d \left (1-c^2 x^2\right )^{3/2}}{9 c}+\frac {2 b d \sqrt {1-c^2 x^2}}{3 c} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 12
Rule 45
Rule 455
Rule 4739
Rubi steps
\begin {align*} \int \left (d-c^2 d x^2\right ) \left (a+b \sin ^{-1}(c x)\right ) \, dx &=d x \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} c^2 d x^3 \left (a+b \sin ^{-1}(c x)\right )-(b c) \int \frac {d x \left (1-\frac {c^2 x^2}{3}\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=d x \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} c^2 d x^3 \left (a+b \sin ^{-1}(c x)\right )-(b c d) \int \frac {x \left (1-\frac {c^2 x^2}{3}\right )}{\sqrt {1-c^2 x^2}} \, dx\\ &=d x \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} c^2 d x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{2} (b c d) \text {Subst}\left (\int \frac {1-\frac {c^2 x}{3}}{\sqrt {1-c^2 x}} \, dx,x,x^2\right )\\ &=d x \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} c^2 d x^3 \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{2} (b c d) \text {Subst}\left (\int \left (\frac {2}{3 \sqrt {1-c^2 x}}+\frac {1}{3} \sqrt {1-c^2 x}\right ) \, dx,x,x^2\right )\\ &=\frac {2 b d \sqrt {1-c^2 x^2}}{3 c}+\frac {b d \left (1-c^2 x^2\right )^{3/2}}{9 c}+d x \left (a+b \sin ^{-1}(c x)\right )-\frac {1}{3} c^2 d x^3 \left (a+b \sin ^{-1}(c x)\right )\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.06, size = 88, normalized size = 1.14 \begin {gather*} a d x-\frac {1}{3} a c^2 d x^3+\frac {7 b d \sqrt {1-c^2 x^2}}{9 c}-\frac {1}{9} b c d x^2 \sqrt {1-c^2 x^2}+b d x \text {ArcSin}(c x)-\frac {1}{3} b c^2 d x^3 \text {ArcSin}(c x) \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.00, size = 82, normalized size = 1.06
method | result | size |
derivativedivides | \(\frac {-d a \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-c x \arcsin \left (c x \right )+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c}\) | \(82\) |
default | \(\frac {-d a \left (\frac {1}{3} c^{3} x^{3}-c x \right )-d b \left (\frac {c^{3} x^{3} \arcsin \left (c x \right )}{3}-c x \arcsin \left (c x \right )+\frac {c^{2} x^{2} \sqrt {-c^{2} x^{2}+1}}{9}-\frac {7 \sqrt {-c^{2} x^{2}+1}}{9}\right )}{c}\) | \(82\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.49, size = 97, normalized size = 1.26 \begin {gather*} -\frac {1}{3} \, a c^{2} d x^{3} - \frac {1}{9} \, {\left (3 \, x^{3} \arcsin \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 c^{2} d + a d x + \frac {{\left (c x \arcsin \left (c x\right ) + \sqrt {-c^{2} x^{2} + 1}\right )} b d}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 2.29, size = 71, normalized size = 0.92 \begin {gather*} -\frac {3 \, a c^{3} d x^{3} - 9 \, a c d x + 3 \, {\left (b c^{3} d x^{3} - 3 \, b c d x\right )} \arcsin \left (c x\right ) + {\left (b c^{2} d x^{2} - 7 \, b d\right )} \sqrt {-c^{2} x^{2} + 1}}{9 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [A]
time = 0.14, size = 90, normalized size = 1.17 \begin {gather*} \begin {cases} - \frac {a c^{2} d x^{3}}{3} + a d x - \frac {b c^{2} d x^{3} \operatorname {asin}{\left (c x \right )}}{3} - \frac {b c d x^{2} \sqrt {- c^{2} x^{2} + 1}}{9} + b d x \operatorname {asin}{\left (c x \right )} + \frac {7 b d \sqrt {- c^{2} x^{2} + 1}}{9 c} & \text {for}\: c \neq 0 \\a d x & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.41, size = 80, normalized size = 1.04 \begin {gather*} -\frac {1}{3} \, a c^{2} d x^{3} - \frac {1}{3} \, {\left (c^{2} x^{2} - 1\right )} b d x \arcsin \left (c x\right ) + \frac {2}{3} \, b d x \arcsin \left (c x\right ) + a d x + \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {3}{2}} b d}{9 \, c} + \frac {2 \, \sqrt {-c^{2} x^{2} + 1} b d}{3 \, c} \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*} \left \{\begin {array}{cl} \frac {b\,d\,\left (\sqrt {1-c^2\,x^2}+c\,x\,\mathrm {asin}\left (c\,x\right )\right )}{c}-b\,c^2\,d\,\left (\frac {\sqrt {\frac {1}{c^2}-x^2}\,\left (\frac {2}{c^2}+x^2\right )}{9}+\frac {x^3\,\mathrm {asin}\left (c\,x\right )}{3}\right )-\frac {a\,d\,x\,\left (c^2\,x^2-3\right )}{3} & \text {\ if\ \ }0<c\\ \int \left (a+b\,\mathrm {asin}\left (c\,x\right )\right )\,\left (d-c^2\,d\,x^2\right ) \,d x & \text {\ if\ \ }\neg 0<c \end {array}\right . \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________