Optimal. Leaf size=206 \[ \frac {\cos \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 (a+b \text {ArcSin}(c x))}{b}\right )}{32 b c^3}-\frac {\cos \left (\frac {4 a}{b}\right ) \text {CosIntegral}\left (\frac {4 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b c^3}-\frac {\cos \left (\frac {6 a}{b}\right ) \text {CosIntegral}\left (\frac {6 (a+b \text {ArcSin}(c x))}{b}\right )}{32 b c^3}+\frac {\log (a+b \text {ArcSin}(c x))}{16 b c^3}+\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \text {ArcSin}(c x))}{b}\right )}{32 b c^3}-\frac {\sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b c^3}-\frac {\sin \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 (a+b \text {ArcSin}(c x))}{b}\right )}{32 b c^3} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.26, antiderivative size = 206, normalized size of antiderivative = 1.00, number of steps
used = 12, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {4809, 4491,
3384, 3380, 3383} \begin {gather*} \frac {\cos \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 (a+b \text {ArcSin}(c x))}{b}\right )}{32 b c^3}-\frac {\cos \left (\frac {4 a}{b}\right ) \text {CosIntegral}\left (\frac {4 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b c^3}-\frac {\cos \left (\frac {6 a}{b}\right ) \text {CosIntegral}\left (\frac {6 (a+b \text {ArcSin}(c x))}{b}\right )}{32 b c^3}+\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \text {ArcSin}(c x))}{b}\right )}{32 b c^3}-\frac {\sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b c^3}-\frac {\sin \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 (a+b \text {ArcSin}(c x))}{b}\right )}{32 b c^3}+\frac {\log (a+b \text {ArcSin}(c x))}{16 b c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 4809
Rubi steps
\begin {align*} \int \frac {x^2 \left (1-c^2 x^2\right )^{3/2}}{a+b \sin ^{-1}(c x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\cos ^4(x) \sin ^2(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{c^3}\\ &=\frac {\text {Subst}\left (\int \left (\frac {1}{16 (a+b x)}+\frac {\cos (2 x)}{32 (a+b x)}-\frac {\cos (4 x)}{16 (a+b x)}-\frac {\cos (6 x)}{32 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^3}\\ &=\frac {\log \left (a+b \sin ^{-1}(c x)\right )}{16 b c^3}+\frac {\text {Subst}\left (\int \frac {\cos (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^3}-\frac {\text {Subst}\left (\int \frac {\cos (6 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^3}-\frac {\text {Subst}\left (\int \frac {\cos (4 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^3}\\ &=\frac {\log \left (a+b \sin ^{-1}(c x)\right )}{16 b c^3}+\frac {\cos \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^3}-\frac {\cos \left (\frac {4 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^3}-\frac {\cos \left (\frac {6 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^3}+\frac {\sin \left (\frac {2 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {2 a}{b}+2 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^3}-\frac {\sin \left (\frac {4 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {4 a}{b}+4 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 c^3}-\frac {\sin \left (\frac {6 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {6 a}{b}+6 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{32 c^3}\\ &=\frac {\cos \left (\frac {2 a}{b}\right ) \text {Ci}\left (\frac {2 a}{b}+2 \sin ^{-1}(c x)\right )}{32 b c^3}-\frac {\cos \left (\frac {4 a}{b}\right ) \text {Ci}\left (\frac {4 a}{b}+4 \sin ^{-1}(c x)\right )}{16 b c^3}-\frac {\cos \left (\frac {6 a}{b}\right ) \text {Ci}\left (\frac {6 a}{b}+6 \sin ^{-1}(c x)\right )}{32 b c^3}+\frac {\log \left (a+b \sin ^{-1}(c x)\right )}{16 b c^3}+\frac {\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \sin ^{-1}(c x)\right )}{32 b c^3}-\frac {\sin \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 a}{b}+4 \sin ^{-1}(c x)\right )}{16 b c^3}-\frac {\sin \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 a}{b}+6 \sin ^{-1}(c x)\right )}{32 b c^3}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.42, size = 165, normalized size = 0.80 \begin {gather*} -\frac {-\cos \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (2 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+2 \cos \left (\frac {4 a}{b}\right ) \text {CosIntegral}\left (4 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+\cos \left (\frac {6 a}{b}\right ) \text {CosIntegral}\left (6 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+2 \log (a+b \text {ArcSin}(c x))-4 \log (8 (a+b \text {ArcSin}(c x)))-\sin \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+2 \sin \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+\sin \left (\frac {6 a}{b}\right ) \text {Si}\left (6 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )}{32 b c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.09, size = 157, normalized size = 0.76
method | result | size |
default | \(\frac {\sinIntegral \left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right )+\cosineIntegral \left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right )-\sinIntegral \left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \sin \left (\frac {6 a}{b}\right )-\cosineIntegral \left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \cos \left (\frac {6 a}{b}\right )-2 \sinIntegral \left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \sin \left (\frac {4 a}{b}\right )-2 \cosineIntegral \left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right )+2 \ln \left (a +b \arcsin \left (c x \right )\right )}{32 c^{3} b}\) | \(157\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [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]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {x^{2} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{a + b \operatorname {asin}{\left (c x \right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 473 vs.
\(2 (192) = 384\).
time = 0.46, size = 473, normalized size = 2.30 \begin {gather*} -\frac {\cos \left (\frac {a}{b}\right )^{6} \operatorname {Ci}\left (\frac {6 \, a}{b} + 6 \, \arcsin \left (c x\right )\right )}{b c^{3}} - \frac {\cos \left (\frac {a}{b}\right )^{5} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {6 \, a}{b} + 6 \, \arcsin \left (c x\right )\right )}{b c^{3}} + \frac {3 \, \cos \left (\frac {a}{b}\right )^{4} \operatorname {Ci}\left (\frac {6 \, a}{b} + 6 \, \arcsin \left (c x\right )\right )}{2 \, b c^{3}} - \frac {\cos \left (\frac {a}{b}\right )^{4} \operatorname {Ci}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{2 \, b c^{3}} + \frac {\cos \left (\frac {a}{b}\right )^{3} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {6 \, a}{b} + 6 \, \arcsin \left (c x\right )\right )}{b c^{3}} - \frac {\cos \left (\frac {a}{b}\right )^{3} \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{2 \, b c^{3}} - \frac {9 \, \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {6 \, a}{b} + 6 \, \arcsin \left (c x\right )\right )}{16 \, b c^{3}} + \frac {\cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{2 \, b c^{3}} + \frac {\cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{16 \, b c^{3}} - \frac {3 \, \cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {6 \, a}{b} + 6 \, \arcsin \left (c x\right )\right )}{16 \, b c^{3}} + \frac {\cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{4 \, b c^{3}} + \frac {\cos \left (\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{16 \, b c^{3}} + \frac {\operatorname {Ci}\left (\frac {6 \, a}{b} + 6 \, \arcsin \left (c x\right )\right )}{32 \, b c^{3}} - \frac {\operatorname {Ci}\left (\frac {4 \, a}{b} + 4 \, \arcsin \left (c x\right )\right )}{16 \, b c^{3}} - \frac {\operatorname {Ci}\left (\frac {2 \, a}{b} + 2 \, \arcsin \left (c x\right )\right )}{32 \, b c^{3}} + \frac {\log \left (b \arcsin \left (c x\right ) + a\right )}{16 \, b c^{3}} \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 \frac {x^2\,{\left (1-c^2\,x^2\right )}^{3/2}}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________