Optimal. Leaf size=78 \[ -\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{a \text {ArcSin}(a x)}-\frac {5 c^2 \text {Si}(\text {ArcSin}(a x))}{8 a}-\frac {15 c^2 \text {Si}(3 \text {ArcSin}(a x))}{16 a}-\frac {5 c^2 \text {Si}(5 \text {ArcSin}(a x))}{16 a} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.11, antiderivative size = 78, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {4751, 4809,
4491, 3380} \begin {gather*} -\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{a \text {ArcSin}(a x)}-\frac {5 c^2 \text {Si}(\text {ArcSin}(a x))}{8 a}-\frac {15 c^2 \text {Si}(3 \text {ArcSin}(a x))}{16 a}-\frac {5 c^2 \text {Si}(5 \text {ArcSin}(a x))}{16 a} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 3380
Rule 4491
Rule 4751
Rule 4809
Rubi steps
\begin {align*} \int \frac {\left (c-a^2 c x^2\right )^2}{\sin ^{-1}(a x)^2} \, dx &=-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{a \sin ^{-1}(a x)}-\left (5 a c^2\right ) \int \frac {x \left (1-a^2 x^2\right )^{3/2}}{\sin ^{-1}(a x)} \, dx\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{a \sin ^{-1}(a x)}-\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\cos ^4(x) \sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{a \sin ^{-1}(a x)}-\frac {\left (5 c^2\right ) \text {Subst}\left (\int \left (\frac {\sin (x)}{8 x}+\frac {3 \sin (3 x)}{16 x}+\frac {\sin (5 x)}{16 x}\right ) \, dx,x,\sin ^{-1}(a x)\right )}{a}\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{a \sin ^{-1}(a x)}-\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\sin (5 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{16 a}-\frac {\left (5 c^2\right ) \text {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{8 a}-\frac {\left (15 c^2\right ) \text {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\sin ^{-1}(a x)\right )}{16 a}\\ &=-\frac {c^2 \left (1-a^2 x^2\right )^{5/2}}{a \sin ^{-1}(a x)}-\frac {5 c^2 \text {Si}\left (\sin ^{-1}(a x)\right )}{8 a}-\frac {15 c^2 \text {Si}\left (3 \sin ^{-1}(a x)\right )}{16 a}-\frac {5 c^2 \text {Si}\left (5 \sin ^{-1}(a x)\right )}{16 a}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.19, size = 70, normalized size = 0.90 \begin {gather*} -\frac {c^2 \left (16 \left (1-a^2 x^2\right )^{5/2}+10 \text {ArcSin}(a x) \text {Si}(\text {ArcSin}(a x))+15 \text {ArcSin}(a x) \text {Si}(3 \text {ArcSin}(a x))+5 \text {ArcSin}(a x) \text {Si}(5 \text {ArcSin}(a x))\right )}{16 a \text {ArcSin}(a x)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.10, size = 83, normalized size = 1.06
method | result | size |
derivativedivides | \(-\frac {c^{2} \left (10 \sinIntegral \left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+15 \sinIntegral \left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+5 \sinIntegral \left (5 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+10 \sqrt {-a^{2} x^{2}+1}+5 \cos \left (3 \arcsin \left (a x \right )\right )+\cos \left (5 \arcsin \left (a x \right )\right )\right )}{16 a \arcsin \left (a x \right )}\) | \(83\) |
default | \(-\frac {c^{2} \left (10 \sinIntegral \left (\arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+15 \sinIntegral \left (3 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+5 \sinIntegral \left (5 \arcsin \left (a x \right )\right ) \arcsin \left (a x \right )+10 \sqrt {-a^{2} x^{2}+1}+5 \cos \left (3 \arcsin \left (a x \right )\right )+\cos \left (5 \arcsin \left (a x \right )\right )\right )}{16 a \arcsin \left (a x \right )}\) | \(83\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[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*} c^{2} \left (\int \left (- \frac {2 a^{2} x^{2}}{\operatorname {asin}^{2}{\left (a x \right )}}\right )\, dx + \int \frac {a^{4} x^{4}}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx + \int \frac {1}{\operatorname {asin}^{2}{\left (a x \right )}}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.48, size = 81, normalized size = 1.04 \begin {gather*} -\frac {{\left (a^{2} x^{2} - 1\right )}^{2} \sqrt {-a^{2} x^{2} + 1} c^{2}}{a \arcsin \left (a x\right )} - \frac {5 \, c^{2} \operatorname {Si}\left (5 \, \arcsin \left (a x\right )\right )}{16 \, a} - \frac {15 \, c^{2} \operatorname {Si}\left (3 \, \arcsin \left (a x\right )\right )}{16 \, a} - \frac {5 \, c^{2} \operatorname {Si}\left (\arcsin \left (a x\right )\right )}{8 \, a} \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 {{\left (c-a^2\,c\,x^2\right )}^2}{{\mathrm {asin}\left (a\,x\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________