Optimal. Leaf size=151 \[ \frac {\sqrt {-2 d x^2-d^2 x^4}}{2 b d x \left (a+b \text {ArcCos}\left (1+d x^2\right )\right )}+\frac {x \text {CosIntegral}\left (\frac {a+b \text {ArcCos}\left (1+d x^2\right )}{2 b}\right ) \sin \left (\frac {a}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}-\frac {x \cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \text {ArcCos}\left (1+d x^2\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.02, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 14, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {4910}
\begin {gather*} \frac {x \sin \left (\frac {a}{2 b}\right ) \text {CosIntegral}\left (\frac {a+b \text {ArcCos}\left (d x^2+1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}-\frac {x \cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \text {ArcCos}\left (d x^2+1\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}+\frac {\sqrt {-d^2 x^4-2 d x^2}}{2 b d x \left (a+b \text {ArcCos}\left (d x^2+1\right )\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 4910
Rubi steps
\begin {align*} \int \frac {1}{\left (a+b \cos ^{-1}\left (1+d x^2\right )\right )^2} \, dx &=\frac {\sqrt {-2 d x^2-d^2 x^4}}{2 b d x \left (a+b \cos ^{-1}\left (1+d x^2\right )\right )}+\frac {x \text {Ci}\left (\frac {a+b \cos ^{-1}\left (1+d x^2\right )}{2 b}\right ) \sin \left (\frac {a}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}-\frac {x \cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \cos ^{-1}\left (1+d x^2\right )}{2 b}\right )}{2 \sqrt {2} b^2 \sqrt {-d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.26, size = 133, normalized size = 0.88 \begin {gather*} \frac {\sqrt {-d x^2 \left (2+d x^2\right )} \left (\frac {b}{a+b \text {ArcCos}\left (1+d x^2\right )}-\frac {\cos \left (\frac {1}{2} \text {ArcCos}\left (1+d x^2\right )\right ) \left (\text {CosIntegral}\left (\frac {a+b \text {ArcCos}\left (1+d x^2\right )}{2 b}\right ) \sin \left (\frac {a}{2 b}\right )-\cos \left (\frac {a}{2 b}\right ) \text {Si}\left (\frac {a+b \text {ArcCos}\left (1+d x^2\right )}{2 b}\right )\right )}{2+d x^2}\right )}{2 b^2 d x} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.05, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (a +b \arccos \left (d \,x^{2}+1\right )\right )^{2}}\, dx\]
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]
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]
________________________________________________________________________________________
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 )^{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 = -0.01 \begin {gather*} \int \frac {1}{{\left (a+b\,\mathrm {acos}\left (d\,x^2+1\right )\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________