Optimal. Leaf size=163 \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4 a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {a^2 x^2+1} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{12 a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\tan ^{-1}(a x)}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.25, antiderivative size = 163, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {4930, 4905, 4904, 3312, 3304, 3352} \[ \frac {\sqrt {\frac {\pi }{2}} \sqrt {a^2 x^2+1} \text {FresnelC}\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4 a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {a^2 x^2+1} \text {FresnelC}\left (\sqrt {\frac {6}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{12 a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {\sqrt {\tan ^{-1}(a x)}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3304
Rule 3312
Rule 3352
Rule 4904
Rule 4905
Rule 4930
Rubi steps
\begin {align*} \int \frac {x \sqrt {\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\frac {\sqrt {\tan ^{-1}(a x)}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{6 a}\\ &=-\frac {\sqrt {\tan ^{-1}(a x)}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {1+a^2 x^2} \int \frac {1}{\left (1+a^2 x^2\right )^{5/2} \sqrt {\tan ^{-1}(a x)}} \, dx}{6 a c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {\tan ^{-1}(a x)}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \frac {\cos ^3(x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{6 a^2 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {\tan ^{-1}(a x)}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \left (\frac {3 \cos (x)}{4 \sqrt {x}}+\frac {\cos (3 x)}{4 \sqrt {x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{6 a^2 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {\tan ^{-1}(a x)}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \frac {\cos (3 x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{24 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \frac {\cos (x)}{\sqrt {x}} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {\tan ^{-1}(a x)}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \cos \left (3 x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{12 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \cos \left (x^2\right ) \, dx,x,\sqrt {\tan ^{-1}(a x)}\right )}{4 a^2 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {\sqrt {\tan ^{-1}(a x)}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}+\frac {\sqrt {\frac {\pi }{2}} \sqrt {1+a^2 x^2} C\left (\sqrt {\frac {2}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{4 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\sqrt {\frac {\pi }{6}} \sqrt {1+a^2 x^2} C\left (\sqrt {\frac {6}{\pi }} \sqrt {\tan ^{-1}(a x)}\right )}{12 a^2 c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.44, size = 167, normalized size = 1.02 \[ \frac {-48 \tan ^{-1}(a x)-i \left (a^2 x^2+1\right )^{3/2} \left (9 \sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-i \tan ^{-1}(a x)\right )-9 \sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},i \tan ^{-1}(a x)\right )+\sqrt {3} \left (\sqrt {-i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},-3 i \tan ^{-1}(a x)\right )-\sqrt {i \tan ^{-1}(a x)} \Gamma \left (\frac {1}{2},3 i \tan ^{-1}(a x)\right )\right )\right )}{144 a^2 c \left (a^2 c x^2+c\right )^{3/2} \sqrt {\tan ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \mathit {sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 3.57, size = 0, normalized size = 0.00 \[ \int \frac {x \sqrt {\arctan \left (a x \right )}}{\left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x\,\sqrt {\mathrm {atan}\left (a\,x\right )}}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x \sqrt {\operatorname {atan}{\left (a x \right )}}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________