Optimal. Leaf size=245 \[ -\frac {5 \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right ) \sin \left (\frac {a}{b}\right )}{64 b c^2}-\frac {9 \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right ) \sin \left (\frac {3 a}{b}\right )}{64 b c^2}-\frac {5 \text {CosIntegral}\left (\frac {5 (a+b \text {ArcSin}(c x))}{b}\right ) \sin \left (\frac {5 a}{b}\right )}{64 b c^2}-\frac {\text {CosIntegral}\left (\frac {7 (a+b \text {ArcSin}(c x))}{b}\right ) \sin \left (\frac {7 a}{b}\right )}{64 b c^2}+\frac {5 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{64 b c^2}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b c^2}+\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b c^2}+\frac {\cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b c^2} \]
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Rubi [A]
time = 0.29, antiderivative size = 245, normalized size of antiderivative = 1.00, number of steps
used = 15, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {4809, 4491,
3384, 3380, 3383} \begin {gather*} -\frac {5 \sin \left (\frac {a}{b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{64 b c^2}-\frac {9 \sin \left (\frac {3 a}{b}\right ) \text {CosIntegral}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b c^2}-\frac {5 \sin \left (\frac {5 a}{b}\right ) \text {CosIntegral}\left (\frac {5 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b c^2}-\frac {\sin \left (\frac {7 a}{b}\right ) \text {CosIntegral}\left (\frac {7 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b c^2}+\frac {5 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \text {ArcSin}(c x)}{b}\right )}{64 b c^2}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b c^2}+\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b c^2}+\frac {\cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 (a+b \text {ArcSin}(c x))}{b}\right )}{64 b c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 3380
Rule 3383
Rule 3384
Rule 4491
Rule 4809
Rubi steps
\begin {align*} \int \frac {x \left (1-c^2 x^2\right )^{5/2}}{a+b \sin ^{-1}(c x)} \, dx &=\frac {\text {Subst}\left (\int \frac {\cos ^6(x) \sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{c^2}\\ &=\frac {\text {Subst}\left (\int \left (\frac {5 \sin (x)}{64 (a+b x)}+\frac {9 \sin (3 x)}{64 (a+b x)}+\frac {5 \sin (5 x)}{64 (a+b x)}+\frac {\sin (7 x)}{64 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{c^2}\\ &=\frac {\text {Subst}\left (\int \frac {\sin (7 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^2}+\frac {5 \text {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^2}+\frac {5 \text {Subst}\left (\int \frac {\sin (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^2}+\frac {9 \text {Subst}\left (\int \frac {\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )}{64 c^2}\\ &=\frac {\left (5 \cos \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 c^2}+\frac {\left (9 \cos \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 c^2}+\frac {\left (5 \cos \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 c^2}+\frac {\cos \left (\frac {7 a}{b}\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 c^2}-\frac {\left (5 \sin \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 c^2}-\frac {\left (9 \sin \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 c^2}-\frac {\left (5 \sin \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 c^2}-\frac {\sin \left (\frac {7 a}{b}\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 c^2}\\ &=-\frac {5 \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{64 b c^2}-\frac {9 \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right ) \sin \left (\frac {3 a}{b}\right )}{64 b c^2}-\frac {5 \text {Ci}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right ) \sin \left (\frac {5 a}{b}\right )}{64 b c^2}-\frac {\text {Ci}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right ) \sin \left (\frac {7 a}{b}\right )}{64 b c^2}+\frac {5 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{64 b c^2}+\frac {9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{64 b c^2}+\frac {5 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{64 b c^2}+\frac {\cos \left (\frac {7 a}{b}\right ) \text {Si}\left (\frac {7 a}{b}+7 \sin ^{-1}(c x)\right )}{64 b c^2}\\ \end {align*}
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Mathematica [A]
time = 0.65, size = 180, normalized size = 0.73 \begin {gather*} \frac {-5 \text {CosIntegral}\left (\frac {a}{b}+\text {ArcSin}(c x)\right ) \sin \left (\frac {a}{b}\right )-9 \text {CosIntegral}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right ) \sin \left (\frac {3 a}{b}\right )-5 \text {CosIntegral}\left (5 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right ) \sin \left (\frac {5 a}{b}\right )-\text {CosIntegral}\left (7 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right ) \sin \left (\frac {7 a}{b}\right )+5 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\text {ArcSin}(c x)\right )+9 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (3 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+5 \cos \left (\frac {5 a}{b}\right ) \text {Si}\left (5 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )+\cos \left (\frac {7 a}{b}\right ) \text {Si}\left (7 \left (\frac {a}{b}+\text {ArcSin}(c x)\right )\right )}{64 b c^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 185, normalized size = 0.76
method | result | size |
default | \(\frac {\sinIntegral \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \cos \left (\frac {7 a}{b}\right )-\cosineIntegral \left (7 \arcsin \left (c x \right )+\frac {7 a}{b}\right ) \sin \left (\frac {7 a}{b}\right )+5 \sinIntegral \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \cos \left (\frac {5 a}{b}\right )-5 \cosineIntegral \left (5 \arcsin \left (c x \right )+\frac {5 a}{b}\right ) \sin \left (\frac {5 a}{b}\right )+9 \sinIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \cos \left (\frac {3 a}{b}\right )-9 \cosineIntegral \left (3 \arcsin \left (c x \right )+\frac {3 a}{b}\right ) \sin \left (\frac {3 a}{b}\right )+5 \sinIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \cos \left (\frac {a}{b}\right )-5 \cosineIntegral \left (\arcsin \left (c x \right )+\frac {a}{b}\right ) \sin \left (\frac {a}{b}\right )}{64 c^{2} b}\) | \(185\) |
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 \left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{a + b \operatorname {asin}{\left (c x \right )}}\, 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. 614 vs.
\(2 (229) = 458\).
time = 0.46, size = 614, normalized size = 2.51 \begin {gather*} -\frac {\cos \left (\frac {a}{b}\right )^{6} \operatorname {Ci}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{b c^{2}} + \frac {\cos \left (\frac {a}{b}\right )^{7} \operatorname {Si}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right )}{b c^{2}} + \frac {5 \, \cos \left (\frac {a}{b}\right )^{4} \operatorname {Ci}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, b c^{2}} - \frac {5 \, \cos \left (\frac {a}{b}\right )^{4} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{4 \, b c^{2}} - \frac {7 \, \cos \left (\frac {a}{b}\right )^{5} \operatorname {Si}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right )}{4 \, b c^{2}} + \frac {5 \, \cos \left (\frac {a}{b}\right )^{5} \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{4 \, b c^{2}} - \frac {3 \, \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{8 \, b c^{2}} + \frac {15 \, \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{16 \, b c^{2}} - \frac {9 \, \cos \left (\frac {a}{b}\right )^{2} \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{16 \, b c^{2}} + \frac {7 \, \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right )}{8 \, b c^{2}} - \frac {25 \, \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{16 \, b c^{2}} + \frac {9 \, \cos \left (\frac {a}{b}\right )^{3} \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{16 \, b c^{2}} + \frac {\operatorname {Ci}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{64 \, b c^{2}} - \frac {5 \, \operatorname {Ci}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{64 \, b c^{2}} + \frac {9 \, \operatorname {Ci}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{64 \, b c^{2}} - \frac {5 \, \operatorname {Ci}\left (\frac {a}{b} + \arcsin \left (c x\right )\right ) \sin \left (\frac {a}{b}\right )}{64 \, b c^{2}} - \frac {7 \, \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {7 \, a}{b} + 7 \, \arcsin \left (c x\right )\right )}{64 \, b c^{2}} + \frac {25 \, \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {5 \, a}{b} + 5 \, \arcsin \left (c x\right )\right )}{64 \, b c^{2}} - \frac {27 \, \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {3 \, a}{b} + 3 \, \arcsin \left (c x\right )\right )}{64 \, b c^{2}} + \frac {5 \, \cos \left (\frac {a}{b}\right ) \operatorname {Si}\left (\frac {a}{b} + \arcsin \left (c x\right )\right )}{64 \, b c^{2}} \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\,{\left (1-c^2\,x^2\right )}^{5/2}}{a+b\,\mathrm {asin}\left (c\,x\right )} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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