Optimal. Leaf size=390 \[ -\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {ArcSin}(c x)}}-\frac {5 d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c}-\frac {5 d^2 \sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {d^2 \sqrt {\frac {5 \pi }{2}} \cos \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {5 d^2 \sqrt {\frac {\pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 b^{3/2} c}+\frac {5 d^2 \sqrt {\frac {3 \pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^{3/2} c}+\frac {d^2 \sqrt {\frac {5 \pi }{2}} \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{4 b^{3/2} c} \]
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Rubi [A]
time = 0.52, antiderivative size = 390, normalized size of antiderivative = 1.00, number of steps
used = 19, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {4751, 4809,
4491, 3387, 3386, 3432, 3385, 3433} \begin {gather*} \frac {5 \sqrt {\frac {\pi }{2}} d^2 \sin \left (\frac {a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c}+\frac {5 \sqrt {\frac {3 \pi }{2}} d^2 \sin \left (\frac {3 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {\sqrt {\frac {5 \pi }{2}} d^2 \sin \left (\frac {5 a}{b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {5 \sqrt {\frac {\pi }{2}} d^2 \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c}-\frac {5 \sqrt {\frac {3 \pi }{2}} d^2 \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {\sqrt {\frac {5 \pi }{2}} d^2 \cos \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \text {ArcSin}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \text {ArcSin}(c x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3385
Rule 3386
Rule 3387
Rule 3432
Rule 3433
Rule 4491
Rule 4751
Rule 4809
Rubi steps
\begin {align*} \int \frac {\left (d-c^2 d x^2\right )^2}{\left (a+b \sin ^{-1}(c x)\right )^{3/2}} \, dx &=-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}-\frac {\left (10 c d^2\right ) \int \frac {x \left (1-c^2 x^2\right )^{3/2}}{\sqrt {a+b \sin ^{-1}(c x)}} \, dx}{b}\\ &=-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}-\frac {\left (10 d^2\right ) \text {Subst}\left (\int \frac {\cos ^4(x) \sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{b c}\\ &=-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}-\frac {\left (10 d^2\right ) \text {Subst}\left (\int \left (\frac {\sin (x)}{8 \sqrt {a+b x}}+\frac {3 \sin (3 x)}{16 \sqrt {a+b x}}+\frac {\sin (5 x)}{16 \sqrt {a+b x}}\right ) \, dx,x,\sin ^{-1}(c x)\right )}{b c}\\ &=-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}-\frac {\left (5 d^2\right ) \text {Subst}\left (\int \frac {\sin (5 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c}-\frac {\left (5 d^2\right ) \text {Subst}\left (\int \frac {\sin (x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c}-\frac {\left (15 d^2\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c}\\ &=-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}-\frac {\left (5 d^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c}-\frac {\left (15 d^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c}-\frac {\left (5 d^2 \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}+5 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c}+\frac {\left (5 d^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{4 b c}+\frac {\left (15 d^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c}+\frac {\left (5 d^2 \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \frac {\cos \left (\frac {5 a}{b}+5 x\right )}{\sqrt {a+b x}} \, dx,x,\sin ^{-1}(c x)\right )}{8 b c}\\ &=-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}-\frac {\left (5 d^2 \cos \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 b^2 c}-\frac {\left (15 d^2 \cos \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{4 b^2 c}-\frac {\left (5 d^2 \cos \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \sin \left (\frac {5 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{4 b^2 c}+\frac {\left (5 d^2 \sin \left (\frac {a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{2 b^2 c}+\frac {\left (15 d^2 \sin \left (\frac {3 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {3 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{4 b^2 c}+\frac {\left (5 d^2 \sin \left (\frac {5 a}{b}\right )\right ) \text {Subst}\left (\int \cos \left (\frac {5 x^2}{b}\right ) \, dx,x,\sqrt {a+b \sin ^{-1}(c x)}\right )}{4 b^2 c}\\ &=-\frac {2 d^2 \left (1-c^2 x^2\right )^{5/2}}{b c \sqrt {a+b \sin ^{-1}(c x)}}-\frac {5 d^2 \sqrt {\frac {\pi }{2}} \cos \left (\frac {a}{b}\right ) S\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{2 b^{3/2} c}-\frac {5 d^2 \sqrt {\frac {3 \pi }{2}} \cos \left (\frac {3 a}{b}\right ) S\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}-\frac {d^2 \sqrt {\frac {5 \pi }{2}} \cos \left (\frac {5 a}{b}\right ) S\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right )}{4 b^{3/2} c}+\frac {5 d^2 \sqrt {\frac {\pi }{2}} C\left (\frac {\sqrt {\frac {2}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {a}{b}\right )}{2 b^{3/2} c}+\frac {5 d^2 \sqrt {\frac {3 \pi }{2}} C\left (\frac {\sqrt {\frac {6}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {3 a}{b}\right )}{4 b^{3/2} c}+\frac {d^2 \sqrt {\frac {5 \pi }{2}} C\left (\frac {\sqrt {\frac {10}{\pi }} \sqrt {a+b \sin ^{-1}(c x)}}{\sqrt {b}}\right ) \sin \left (\frac {5 a}{b}\right )}{4 b^{3/2} c}\\ \end {align*}
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Mathematica [C] Result contains complex when optimal does not.
time = 1.27, size = 522, normalized size = 1.34 \begin {gather*} \frac {d^2 e^{-\frac {5 i (a+b \text {ArcSin}(c x))}{b}} \left (-e^{\frac {5 i a}{b}}-5 e^{\frac {5 i a}{b}+2 i \text {ArcSin}(c x)}-10 e^{\frac {5 i a}{b}+4 i \text {ArcSin}(c x)}-10 e^{\frac {5 i a}{b}+6 i \text {ArcSin}(c x)}-5 e^{\frac {5 i a}{b}+8 i \text {ArcSin}(c x)}-e^{\frac {5 i (a+2 b \text {ArcSin}(c x))}{b}}+10 e^{\frac {4 i a}{b}+5 i \text {ArcSin}(c x)} \sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {i (a+b \text {ArcSin}(c x))}{b}\right )+10 e^{\frac {6 i a}{b}+5 i \text {ArcSin}(c x)} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {i (a+b \text {ArcSin}(c x))}{b}\right )+5 \sqrt {3} e^{\frac {2 i a}{b}+5 i \text {ArcSin}(c x)} \sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )+5 \sqrt {3} e^{\frac {8 i a}{b}+5 i \text {ArcSin}(c x)} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {3 i (a+b \text {ArcSin}(c x))}{b}\right )+\sqrt {5} e^{5 i \text {ArcSin}(c x)} \sqrt {-\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},-\frac {5 i (a+b \text {ArcSin}(c x))}{b}\right )+\sqrt {5} e^{\frac {5 i (2 a+b \text {ArcSin}(c x))}{b}} \sqrt {\frac {i (a+b \text {ArcSin}(c x))}{b}} \text {Gamma}\left (\frac {1}{2},\frac {5 i (a+b \text {ArcSin}(c x))}{b}\right )\right )}{16 b c \sqrt {a+b \text {ArcSin}(c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.28, size = 451, normalized size = 1.16
method | result | size |
default | \(-\frac {d^{2} \left (-10 \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {a}{b}\right ) \mathrm {S}\left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}-10 \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {a}{b}\right ) \FresnelC \left (\frac {\sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {1}{b}}\, b}\right ) \sqrt {-\frac {1}{b}}\, \sqrt {\pi }\, \sqrt {2}-5 \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {3 a}{b}\right ) \mathrm {S}\left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {2}-5 \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {3 a}{b}\right ) \FresnelC \left (\frac {3 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {3}{b}}\, b}\right ) \sqrt {-\frac {3}{b}}\, \sqrt {\pi }\, \sqrt {2}-\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \cos \left (\frac {5 a}{b}\right ) \mathrm {S}\left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right )-\sqrt {2}\, \sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, \sqrt {a +b \arcsin \left (c x \right )}\, \sin \left (\frac {5 a}{b}\right ) \FresnelC \left (\frac {5 \sqrt {2}\, \sqrt {a +b \arcsin \left (c x \right )}}{\sqrt {\pi }\, \sqrt {-\frac {5}{b}}\, b}\right )+10 \cos \left (-\frac {a +b \arcsin \left (c x \right )}{b}+\frac {a}{b}\right )+5 \cos \left (-\frac {3 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {3 a}{b}\right )+\cos \left (-\frac {5 \left (a +b \arcsin \left (c x \right )\right )}{b}+\frac {5 a}{b}\right )\right )}{8 c b \sqrt {a +b \arcsin \left (c x \right )}}\) | \(451\) |
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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*} d^{2} \left (\int \left (- \frac {2 c^{2} x^{2}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\right )\, dx + \int \frac {c^{4} x^{4}}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx + \int \frac {1}{a \sqrt {a + b \operatorname {asin}{\left (c x \right )}} + b \sqrt {a + b \operatorname {asin}{\left (c x \right )}} \operatorname {asin}{\left (c x \right )}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [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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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (d-c^2\,d\,x^2\right )}^2}{{\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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