Optimal. Leaf size=221 \[ \frac {\sqrt {-2 d x^2-d^2 x^4}}{3 b d x \left (a+b \text {ArcCos}\left (1+d x^2\right )\right )^{3/2}}+\frac {x}{3 b^2 \sqrt {a+b \text {ArcCos}\left (1+d x^2\right )}}+\frac {2 \left (\frac {1}{b}\right )^{5/2} \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) \text {FresnelC}\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcCos}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {1}{2} \text {ArcCos}\left (1+d x^2\right )\right )}{3 d x}+\frac {2 \left (\frac {1}{b}\right )^{5/2} \sqrt {\pi } S\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \text {ArcCos}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {a}{2 b}\right ) \sin \left (\frac {1}{2} \text {ArcCos}\left (1+d x^2\right )\right )}{3 d x} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.04, antiderivative size = 221, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {4913, 4904}
\begin {gather*} \frac {x}{3 b^2 \sqrt {a+b \text {ArcCos}\left (d x^2+1\right )}}+\frac {\sqrt {-d^2 x^4-2 d x^2}}{3 b d x \left (a+b \text {ArcCos}\left (d x^2+1\right )\right )^{3/2}}+\frac {2 \sqrt {\pi } \left (\frac {1}{b}\right )^{5/2} \cos \left (\frac {a}{2 b}\right ) \sin \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 )}{3 d x}+\frac {2 \sqrt {\pi } \left (\frac {1}{b}\right )^{5/2} \sin \left (\frac {a}{2 b}\right ) \sin \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 )}{3 d x} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 4904
Rule 4913
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^{5/2}} \, dx &=\frac {\sqrt {-2 d x^2-d^2 x^4}}{3 b d x \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^{3/2}}+\frac {x}{3 b^2 \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )}}-\frac {\int \frac {1}{\sqrt {a+b \cos ^{-1}\left (1+d x^2\right )}} \, dx}{3 b^2}\\ &=\frac {\sqrt {-2 d x^2-d^2 x^4}}{3 b d x \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^{3/2}}+\frac {x}{3 b^2 \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )}}+\frac {2 \left (\frac {1}{b}\right )^{5/2} \sqrt {\pi } \cos \left (\frac {a}{2 b}\right ) C\left (\frac {\sqrt {\frac {1}{b}} \sqrt {a+b \cos ^{-1}\left (1+d x^2\right )}}{\sqrt {\pi }}\right ) \sin \left (\frac {1}{2} \cos ^{-1}\left (1+d x^2\right )\right )}{3 d x}+\frac {2 \left (\frac {1}{b}\right )^{5/2} \sqrt {\pi } 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 ) \sin \left (\frac {1}{2} \cos ^{-1}\left (1+d x^2\right )\right )}{3 d x}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.52, size = 234, normalized size = 1.06 \begin {gather*} \frac {2 \sin \left (\frac {1}{2} \text {ArcCos}\left (1+d x^2\right )\right ) \left (b \cos \left (\frac {1}{2} \text {ArcCos}\left (1+d x^2\right )\right )+\sqrt {\frac {1}{b}} \sqrt {\pi } \left (a+b \text {ArcCos}\left (1+d x^2\right )\right )^{3/2} \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 } \left (a+b \text {ArcCos}\left (1+d x^2\right )\right )^{3/2} 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 )-a \sin \left (\frac {1}{2} \text {ArcCos}\left (1+d x^2\right )\right )-b \text {ArcCos}\left (1+d x^2\right ) \sin \left (\frac {1}{2} \text {ArcCos}\left (1+d x^2\right )\right )\right )}{3 b^2 d x \left (a+b \text {ArcCos}\left (1+d x^2\right )\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \arccos \left (d \,x^{2}+1\right )\right )^{\frac {5}{2}}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
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 {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {acos}\left (d\,x^2+1\right )\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________