Optimal. Leaf size=190 \[ \frac {\sqrt {2 d x^2-d^2 x^4}}{b d x \sqrt {a+b \text {ArcCos}\left (-1+d x^2\right )}}-\frac {2 \left (\frac {1}{b}\right )^{3/2} \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \cos \left (\frac {1}{2} \text {ArcCos}\left (-1+d x^2\right )\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcCos}\left (-1+d x^2\right )}}{\sqrt {\pi }}\right )}{d x}-\frac {2 \left (\frac {1}{b}\right )^{3/2} \sqrt {\pi } \cos \left (\frac {1}{2} \text {ArcCos}\left (-1+d x^2\right )\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcCos}\left (-1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right )}{d x} \]
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Rubi [A]
time = 0.02, antiderivative size = 190, 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 = {4908}
\begin {gather*} \frac {\sqrt {2 d x^2-d^2 x^4}}{b d x \sqrt {a+b \text {ArcCos}\left (d x^2-1\right )}}-\frac {2 \sqrt {\pi } \left (\frac {1}{b}\right )^{3/2} \cos \left (\frac {a}{2 b}\right ) \cos \left (\frac {1}{2} \text {ArcCos}\left (d x^2-1\right )\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcCos}\left (d x^2-1\right )}}{\sqrt {\pi }}\right )}{d x}-\frac {2 \sqrt {\pi } \left (\frac {1}{b}\right )^{3/2} \sin \left (\frac {a}{2 b}\right ) \cos \left (\frac {1}{2} \text {ArcCos}\left (d x^2-1\right )\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcCos}\left (d x^2-1\right )}}{\sqrt {\pi }}\right )}{d x} \end {gather*}
Antiderivative was successfully verified.
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Rule 4908
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \cos ^{-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 \cos ^{-1}\left (-1+d x^2\right )}}-\frac {2 \left (\frac {1}{b}\right )^{3/2} \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \cos \left (\frac {1}{2} \cos ^{-1}\left (-1+d x^2\right )\right ) C\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (-1+d x^2\right )}}{\sqrt {\pi }}\right )}{d x}-\frac {2 \left (\frac {1}{b}\right )^{3/2} \sqrt {\pi } \cos \left (\frac {1}{2} \cos ^{-1}\left (-1+d x^2\right )\right ) S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (-1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right )}{d x}\\ \end {align*}
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Mathematica [A]
time = 0.24, size = 161, normalized size = 0.85 \begin {gather*} \frac {2 \cos \left (\frac {1}{2} \text {ArcCos}\left (-1+d x^2\right )\right ) \left (-\sqrt {\frac {1}{b}} \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcCos}\left (-1+d x^2\right )}}{\sqrt {\pi }}\right )-\sqrt {\frac {1}{b}} \sqrt {\pi } S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcCos}\left (-1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right )+\frac {\sin \left (\frac {1}{2} \text {ArcCos}\left (-1+d x^2\right )\right )}{\sqrt {a+b \text {ArcCos}\left (-1+d x^2\right )}}\right )}{b d x} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \arccos \left (d \,x^{2}-1\right )\right )^{\frac {3}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
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*} \int \frac {1}{\left (a + b \operatorname {acos}{\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 = -0.01 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {acos}\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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