Optimal. Leaf size=238 \[ -\frac {\sqrt {-2 d x^2-d^2 x^4}}{b d x \sqrt {a+b \text {ArcSin}\left (1+d x^2\right )}}+\frac {\left (\frac {1}{b}\right )^{3/2} \sqrt {\pi } x S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )}-\frac {\left (\frac {1}{b}\right )^{3/2} \sqrt {\pi } x \text {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )} \]
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Rubi [A]
time = 0.03, antiderivative size = 238, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {4906}
\begin {gather*} -\frac {\sqrt {-d^2 x^4-2 d x^2}}{b d x \sqrt {a+b \text {ArcSin}\left (d x^2+1\right )}}-\frac {\sqrt {\pi } \left (\frac {1}{b}\right )^{3/2} x \left (\sin \left (\frac {a}{2 b}\right )+\cos \left (\frac {a}{2 b}\right )\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\cos \left (\frac {1}{2} \text {ArcSin}\left (d x^2+1\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (d x^2+1\right )\right )}+\frac {\sqrt {\pi } \left (\frac {1}{b}\right )^{3/2} x \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}\left (d x^2+1\right )}}{\sqrt {\pi }}\right )}{\cos \left (\frac {1}{2} \text {ArcSin}\left (d x^2+1\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (d x^2+1\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 4906
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \sin ^{-1}\left (1+d x^2\right )\right )^{3/2}} \, dx &=-\frac {\sqrt {-2 d x^2-d^2 x^4}}{b d x \sqrt {a+b \sin ^{-1}\left (1+d x^2\right )}}+\frac {\left (\frac {1}{b}\right )^{3/2} \sqrt {\pi } x S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \sin ^{-1}\left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1+d x^2\right )\right )}-\frac {\left (\frac {1}{b}\right )^{3/2} \sqrt {\pi } x C\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \sin ^{-1}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \sin ^{-1}\left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \sin ^{-1}\left (1+d x^2\right )\right )}\\ \end {align*}
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Mathematica [A]
time = 0.42, size = 238, normalized size = 1.00 \begin {gather*} -\frac {\sqrt {-2 d x^2-d^2 x^4}}{b d x \sqrt {a+b \text {ArcSin}\left (1+d x^2\right )}}+\frac {\left (\frac {1}{b}\right )^{3/2} \sqrt {\pi } x S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )-\sin \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )}-\frac {\left (\frac {1}{b}\right )^{3/2} \sqrt {\pi } x \text {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcSin}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \left (\cos \left (\frac {a}{2 b}\right )+\sin \left (\frac {a}{2 b}\right )\right )}{\cos \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )-\sin \left (\frac {1}{2} \text {ArcSin}\left (1+d x^2\right )\right )} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \arcsin \left (d \,x^{2}+1\right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \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*} \int \frac {1}{\left (a + b \operatorname {asin}{\left (d x^{2} + 1 \right )}\right )^{\frac {3}{2}}}\, dx \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 {1}{{\left (a+b\,\mathrm {asin}\left (d\,x^2+1\right )\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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