Optimal. Leaf size=278 \[ -\frac {x^3 \left (1-c^2 x^2\right )^2}{b c (a+b \text {ArcSin}(c x))}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{64 b^2 c^4}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b^2 c^4}-\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (\frac {5 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b^2 c^4}-\frac {7 \cos \left (\frac {7 a}{b}\right ) \text {CosIntegral}\left (\frac {7 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b^2 c^4}+\frac {3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{64 b^2 c^4}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b^2 c^4}-\frac {5 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b^2 c^4}-\frac {7 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b^2 c^4} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.58, antiderivative size = 278, normalized size of antiderivative = 1.00, number of steps
used = 28, number of rules used = 6, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.214, Rules used = {4799, 4809,
4491, 3384, 3380, 3383} \begin {gather*} \frac {3 \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{64 b^2 c^4}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b^2 c^4}-\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (\frac {5 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b^2 c^4}-\frac {7 \cos \left (\frac {7 a}{b}\right ) \text {CosIntegral}\left (\frac {7 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b^2 c^4}+\frac {3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{64 b^2 c^4}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b^2 c^4}-\frac {5 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b^2 c^4}-\frac {7 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b^2 c^4}-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c (a+b \text {ArcSin}(c x))} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 4799
Rule 4809
Rubi steps
\begin {align*} \int \frac {x^3 \left (1-c^2 x^2\right )^{3/2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {3 \int \frac {x^2 \left (1-c^2 x^2\right )}{a+b \sin ^{-1}(c x)} \, dx}{b c}-\frac {(7 c) \int \frac {x^4 \left (1-c^2 x^2\right )}{a+b \sin ^{-1}(c x)} \, dx}{b}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {3 \text {Subst}\left (\int \frac {\cos ^3(x) \sin ^2(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}-\frac {7 \text {Subst}\left (\int \frac {\cos ^3(x) \sin ^4(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {3 \text {Subst}\left (\int \left (\frac {\cos (x)}{8 (a+b x)}-\frac {\cos (3 x)}{16 (a+b x)}-\frac {\cos (5 x)}{16 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}-\frac {7 \text {Subst}\left (\int \left (\frac {3 \cos (x)}{64 (a+b x)}-\frac {3 \cos (3 x)}{64 (a+b x)}-\frac {\cos (5 x)}{64 (a+b x)}+\frac {\cos (7 x)}{64 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^4}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {7 \text {Subst}\left (\int \frac {\cos (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {7 \text {Subst}\left (\int \frac {\cos (7 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {3 \text {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^4}-\frac {3 \text {Subst}\left (\int \frac {\cos (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^4}-\frac {21 \text {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac {21 \text {Subst}\left (\int \frac {\cos (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac {3 \text {Subst}\left (\int \frac {\cos (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}-\frac {\left (21 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac {\left (3 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}-\frac {\left (3 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^4}+\frac {\left (21 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac {\left (7 \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {\left (3 \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^4}-\frac {\left (7 \cos \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {\left (21 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac {\left (3 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^4}-\frac {\left (3 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^4}+\frac {\left (21 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}+\frac {\left (7 \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}-\frac {\left (3 \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^4}-\frac {\left (7 \sin \left (\frac {7 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {7 a}{b}+7 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 b c^4}\\ &=-\frac {x^3 \left (1-c^2 x^2\right )^2}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {3 \cos \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {Ci}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac {7 \cos \left (\frac {7 a}{b}\right ) \text {Ci}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac {3 \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{64 b^2 c^4}+\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac {5 \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b^2 c^4}-\frac {7 \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b^2 c^4}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.74, size = 399, normalized size = 1.44 \begin {gather*} -\frac {64 b c^3 x^3-128 b c^5 x^5+64 b c^7 x^7-3 (a+b \text {ArcSin}(c x)) \cos \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a}{b}+\text {ArcSin}(c x)\right )-9 (a+b \text {ArcSin}(c x)) \cos \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+5 a \cos \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (5 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+5 b \text {ArcSin}(c x) \cos \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (5 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+7 a \cos \left (\frac {7 a}{b}\right ) \text {CosIntegral}\left (7 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+7 b \text {ArcSin}(c x) \cos \left (\frac {7 a}{b}\right ) \text {CosIntegral}\left (7 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )-3 a \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c x)\right )-3 b \text {ArcSin}(c x) \sin \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c x)\right )-9 a \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )-9 b \text {ArcSin}(c x) \sin \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+5 a \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+5 b \text {ArcSin}(c x) \sin \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+7 a \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+7 b \text {ArcSin}(c x) \sin \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )}{64 b^2 c^4 (a+b \text {ArcSin}(c x))} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.12, size = 455, normalized size = 1.64
method | result | size |
default | \(-\frac {7 \arcsin \left (c x \right ) \cosineIntegral \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) b -9 \arcsin \left (c x \right ) \sinIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) b -9 \arcsin \left (c x \right ) \cosineIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) b +5 \arcsin \left (c x \right ) \sinIntegral \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) b +5 \arcsin \left (c x \right ) \cosineIntegral \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) b -3 \arcsin \left (c x \right ) \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) b -3 \arcsin \left (c x \right ) \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) b +7 \arcsin \left (c x \right ) \sinIntegral \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) b +7 \cosineIntegral \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right ) a -9 \sinIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right ) a -9 \cosineIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right ) a +5 \sinIntegral \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right ) a +5 \cosineIntegral \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right ) a -3 \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right ) a -3 \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right ) a +7 \sinIntegral \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right ) a +3 x b c -\sin \left (7 \arcsin \left (c x \right )\right ) b +3 \sin \left (3 \arcsin \left (c x \right )\right ) b -\sin \left (5 \arcsin \left (c x \right )\right ) b}{64 c^{4} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\) | \(455\) |
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^{3} \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {3}{2}}}{\left (a + b \operatorname {asin}{\left (c x \right )}\right )^{2}}\, 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. 2065 vs.
\(2 (261) = 522\).
time = 0.54, size = 2065, normalized size = 7.43 \begin {gather*} \text {Too large to display} \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^3\,{\left (1-c^2\,x^2\right )}^{3/2}}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________