Optimal. Leaf size=118 \[ \frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left (\tan ^{-1}(a x)\right )}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left (3 \tan ^{-1}(a x)\right )}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{x^3}{a c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.402728, antiderivative size = 118, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208, Rules used = {4942, 4971, 4970, 4406, 3302} \[ \frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left (\tan ^{-1}(a x)\right )}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{3 \sqrt{a^2 x^2+1} \text{CosIntegral}\left (3 \tan ^{-1}(a x)\right )}{4 a^4 c^2 \sqrt{a^2 c x^2+c}}-\frac{x^3}{a c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4942
Rule 4971
Rule 4970
Rule 4406
Rule 3302
Rubi steps
\begin{align*} \int \frac{x^3}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx &=-\frac{x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{3 \int \frac{x^2}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{a}\\ &=-\frac{x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \int \frac{x^2}{\left (1+a^2 x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{a c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x) \sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \left (\frac{\cos (x)}{4 x}-\frac{\cos (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{\left (3 \sqrt{1+a^2 x^2}\right ) \operatorname{Subst}\left (\int \frac{\cos (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^4 c^2 \sqrt{c+a^2 c x^2}}\\ &=-\frac{x^3}{a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac{3 \sqrt{1+a^2 x^2} \text{Ci}\left (\tan ^{-1}(a x)\right )}{4 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{3 \sqrt{1+a^2 x^2} \text{Ci}\left (3 \tan ^{-1}(a x)\right )}{4 a^4 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [A] time = 0.204778, size = 82, normalized size = 0.69 \[ \frac{3 c \sqrt{a^2 x^2+1} \left (\text{CosIntegral}\left (\tan ^{-1}(a x)\right )-\text{CosIntegral}\left (3 \tan ^{-1}(a x)\right )\right )-\frac{4 a^3 c x^3}{\left (a^2 x^2+1\right ) \tan ^{-1}(a x)}}{4 a^4 c^3 \sqrt{a^2 c x^2+c}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [C] time = 1.072, size = 582, normalized size = 4.9 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{\sqrt{a^{2} c x^{2} + c} x^{3}}{{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac{5}{2}} \operatorname{atan}^{2}{\left (a x \right )}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}} \arctan \left (a x\right )^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]