Optimal. Leaf size=282 \[ -\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \text {ArcSin}(c x))}+\frac {\text {CosIntegral}\left (\frac {2 (a+b \text {ArcSin}(c x))}{b}\right ) \sin \left (\frac {2 a}{b}\right )}{16 b^2 c^3}-\frac {\text {CosIntegral}\left (\frac {4 (a+b \text {ArcSin}(c x))}{b}\right ) \sin \left (\frac {4 a}{b}\right )}{8 b^2 c^3}-\frac {3 \text {CosIntegral}\left (\frac {6 (a+b \text {ArcSin}(c x))}{b}\right ) \sin \left (\frac {6 a}{b}\right )}{16 b^2 c^3}-\frac {\text {CosIntegral}\left (\frac {8 (a+b \text {ArcSin}(c x))}{b}\right ) \sin \left (\frac {8 a}{b}\right )}{16 b^2 c^3}-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \text {ArcSin}(c x))}{b}\right )}{8 b^2 c^3}+\frac {3 \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {8 a}{b}\right ) \text {Si}\left (\frac {8 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b^2 c^3} \]
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Rubi [A]
time = 0.55, antiderivative size = 282, 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 {\sin \left (\frac {2 a}{b}\right ) \text {CosIntegral}\left (\frac {2 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b^2 c^3}-\frac {\sin \left (\frac {4 a}{b}\right ) \text {CosIntegral}\left (\frac {4 (a+b \text {ArcSin}(c x))}{b}\right )}{8 b^2 c^3}-\frac {3 \sin \left (\frac {6 a}{b}\right ) \text {CosIntegral}\left (\frac {6 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b^2 c^3}-\frac {\sin \left (\frac {8 a}{b}\right ) \text {CosIntegral}\left (\frac {8 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b^2 c^3}-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 (a+b \text {ArcSin}(c x))}{b}\right )}{8 b^2 c^3}+\frac {3 \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {8 a}{b}\right ) \text {Si}\left (\frac {8 (a+b \text {ArcSin}(c x))}{b}\right )}{16 b^2 c^3}-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c (a+b \text {ArcSin}(c x))} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 4799
Rule 4809
Rubi steps
\begin {align*} \int \frac {x^2 \left (1-c^2 x^2\right )^{5/2}}{\left (a+b \sin ^{-1}(c x)\right )^2} \, dx &=-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {2 \int \frac {x \left (1-c^2 x^2\right )^2}{a+b \sin ^{-1}(c x)} \, dx}{b c}-\frac {(8 c) \int \frac {x^3 \left (1-c^2 x^2\right )^2}{a+b \sin ^{-1}(c x)} \, dx}{b}\\ &=-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {2 \text {Subst}\left (\int \frac {\cos ^5(x) \sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}-\frac {8 \text {Subst}\left (\int \frac {\cos ^5(x) \sin ^3(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}\\ &=-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {2 \text {Subst}\left (\int \left (\frac {5 \sin (2 x)}{32 (a+b x)}+\frac {\sin (4 x)}{8 (a+b x)}+\frac {\sin (6 x)}{32 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}-\frac {8 \text {Subst}\left (\int \left (\frac {3 \sin (2 x)}{64 (a+b x)}+\frac {\sin (4 x)}{64 (a+b x)}-\frac {\sin (6 x)}{64 (a+b x)}-\frac {\sin (8 x)}{128 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c^3}\\ &=-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {\text {Subst}\left (\int \frac {\sin (6 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^3}+\frac {\text {Subst}\left (\int \frac {\sin (8 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^3}-\frac {\text {Subst}\left (\int \frac {\sin (4 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}+\frac {\text {Subst}\left (\int \frac {\sin (6 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}+\frac {\text {Subst}\left (\int \frac {\sin (4 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c^3}+\frac {5 \text {Subst}\left (\int \frac {\sin (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^3}-\frac {3 \text {Subst}\left (\int \frac {\sin (2 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c^3}\\ &=-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {\left (5 \cos \left (\frac {2 a}{b}\right )\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 )}{16 b c^3}-\frac {\left (3 \cos \left (\frac {2 a}{b}\right )\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 )}{8 b c^3}-\frac {\cos \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 )}{8 b c^3}+\frac {\cos \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 )}{4 b c^3}+\frac {\cos \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 )}{16 b c^3}+\frac {\cos \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 )}{8 b c^3}+\frac {\cos \left (\frac {8 a}{b}\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {8 a}{b}+8 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^3}-\frac {\left (5 \sin \left (\frac {2 a}{b}\right )\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 )}{16 b c^3}+\frac {\left (3 \sin \left (\frac {2 a}{b}\right )\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 )}{8 b c^3}+\frac {\sin \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 )}{8 b c^3}-\frac {\sin \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 )}{4 b c^3}-\frac {\sin \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 )}{16 b c^3}-\frac {\sin \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 )}{8 b c^3}-\frac {\sin \left (\frac {8 a}{b}\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {8 a}{b}+8 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{16 b c^3}\\ &=-\frac {x^2 \left (1-c^2 x^2\right )^3}{b c \left (a+b \sin ^{-1}(c x)\right )}+\frac {\text {Ci}\left (\frac {2 a}{b}+2 \sin ^{-1}(c x)\right ) \sin \left (\frac {2 a}{b}\right )}{16 b^2 c^3}-\frac {\text {Ci}\left (\frac {4 a}{b}+4 \sin ^{-1}(c x)\right ) \sin \left (\frac {4 a}{b}\right )}{8 b^2 c^3}-\frac {3 \text {Ci}\left (\frac {6 a}{b}+6 \sin ^{-1}(c x)\right ) \sin \left (\frac {6 a}{b}\right )}{16 b^2 c^3}-\frac {\text {Ci}\left (\frac {8 a}{b}+8 \sin ^{-1}(c x)\right ) \sin \left (\frac {8 a}{b}\right )}{16 b^2 c^3}-\frac {\cos \left (\frac {2 a}{b}\right ) \text {Si}\left (\frac {2 a}{b}+2 \sin ^{-1}(c x)\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {4 a}{b}\right ) \text {Si}\left (\frac {4 a}{b}+4 \sin ^{-1}(c x)\right )}{8 b^2 c^3}+\frac {3 \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (\frac {6 a}{b}+6 \sin ^{-1}(c x)\right )}{16 b^2 c^3}+\frac {\cos \left (\frac {8 a}{b}\right ) \text {Si}\left (\frac {8 a}{b}+8 \sin ^{-1}(c x)\right )}{16 b^2 c^3}\\ \end {align*}
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Mathematica [A]
time = 0.73, size = 414, normalized size = 1.47 \begin {gather*} \frac {-16 b c^2 x^2+48 b c^4 x^4-48 b c^6 x^6+16 b c^8 x^8+(a+b \text {ArcSin}(c x)) \text {CosIntegral}\left (2 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right ) \sin \left (\frac {2 a}{b}\right )-2 (a+b \text {ArcSin}(c x)) \text {CosIntegral}\left (4 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right ) \sin \left (\frac {4 a}{b}\right )-3 a \text {CosIntegral}\left (6 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right ) \sin \left (\frac {6 a}{b}\right )-3 b \text {ArcSin}(c x) \text {CosIntegral}\left (6 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right ) \sin \left (\frac {6 a}{b}\right )-a \text {CosIntegral}\left (8 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right ) \sin \left (\frac {8 a}{b}\right )-b \text {ArcSin}(c x) \text {CosIntegral}\left (8 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right ) \sin \left (\frac {8 a}{b}\right )-a \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )-b \text {ArcSin}(c x) \cos \left (\frac {2 a}{b}\right ) \text {Si}\left (2 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+2 a \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+2 b \text {ArcSin}(c x) \cos \left (\frac {4 a}{b}\right ) \text {Si}\left (4 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+3 a \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (6 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+3 b \text {ArcSin}(c x) \cos \left (\frac {6 a}{b}\right ) \text {Si}\left (6 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+a \cos \left (\frac {8 a}{b}\right ) \text {Si}\left (8 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+b \text {ArcSin}(c x) \cos \left (\frac {8 a}{b}\right ) \text {Si}\left (8 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )}{16 b^2 c^3 (a+b \text {ArcSin}(c x))} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.11, size = 478, normalized size = 1.70
method | result | size |
default | \(\frac {16 \arcsin \left (c x \right ) \sinIntegral \left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) b -16 \arcsin \left (c x \right ) \cosineIntegral \left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \sin \left (\frac {4 a}{b}\right ) b +8 \arcsin \left (c x \right ) \sinIntegral \left (8 \arcsin \left (c x \right )+\frac {8 a}{b}\right ) \cos \left (\frac {8 a}{b}\right ) b -8 \arcsin \left (c x \right ) \cosineIntegral \left (8 \arcsin \left (c x \right )+\frac {8 a}{b}\right ) \sin \left (\frac {8 a}{b}\right ) b -8 \arcsin \left (c x \right ) \sinIntegral \left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) b +8 \arcsin \left (c x \right ) \cosineIntegral \left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) b +24 \arcsin \left (c x \right ) \sinIntegral \left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \cos \left (\frac {6 a}{b}\right ) b -24 \arcsin \left (c x \right ) \cosineIntegral \left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \sin \left (\frac {6 a}{b}\right ) b +16 \sinIntegral \left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \cos \left (\frac {4 a}{b}\right ) a -16 \cosineIntegral \left (4 \arcsin \left (c x \right )+\frac {4 a}{b}\right ) \sin \left (\frac {4 a}{b}\right ) a +8 \sinIntegral \left (8 \arcsin \left (c x \right )+\frac {8 a}{b}\right ) \cos \left (\frac {8 a}{b}\right ) a -8 \cosineIntegral \left (8 \arcsin \left (c x \right )+\frac {8 a}{b}\right ) \sin \left (\frac {8 a}{b}\right ) a -8 \sinIntegral \left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \cos \left (\frac {2 a}{b}\right ) a +8 \cosineIntegral \left (2 \arcsin \left (c x \right )+\frac {2 a}{b}\right ) \sin \left (\frac {2 a}{b}\right ) a +24 \sinIntegral \left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \cos \left (\frac {6 a}{b}\right ) a -24 \cosineIntegral \left (6 \arcsin \left (c x \right )+\frac {6 a}{b}\right ) \sin \left (\frac {6 a}{b}\right ) a +4 \cos \left (4 \arcsin \left (c x \right )\right ) b +\cos \left (8 \arcsin \left (c x \right )\right ) b -4 \cos \left (2 \arcsin \left (c x \right )\right ) b +4 \cos \left (6 \arcsin \left (c x \right )\right ) b -5 b}{128 c^{3} \left (a +b \arcsin \left (c x \right )\right ) b^{2}}\) | \(478\) |
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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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 {5}{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.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2461 vs.
\(2 (264) = 528\).
time = 0.56, size = 2461, normalized size = 8.73 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {x^2\,{\left (1-c^2\,x^2\right )}^{5/2}}{{\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.
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