Optimal. Leaf size=175 \[ -\frac {\sqrt {a^2 x^2+1} \text {Si}\left (\tan ^{-1}(a x)\right )}{8 a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {9 \sqrt {a^2 x^2+1} \text {Si}\left (3 \tan ^{-1}(a x)\right )}{8 a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {1}{a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}-\frac {x}{2 a c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {3}{2 a^2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]
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Rubi [A] time = 0.93, antiderivative size = 175, normalized size of antiderivative = 1.00, number of steps used = 20, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4968, 4964, 4902, 4971, 4970, 3299, 4406} \[ -\frac {\sqrt {a^2 x^2+1} \text {Si}\left (\tan ^{-1}(a x)\right )}{8 a^2 c^2 \sqrt {a^2 c x^2+c}}-\frac {9 \sqrt {a^2 x^2+1} \text {Si}\left (3 \tan ^{-1}(a x)\right )}{8 a^2 c^2 \sqrt {a^2 c x^2+c}}+\frac {1}{a^2 c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}-\frac {x}{2 a c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {3}{2 a^2 c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]
Antiderivative was successfully verified.
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Rule 3299
Rule 4406
Rule 4902
Rule 4964
Rule 4968
Rule 4970
Rule 4971
Rubi steps
\begin {align*} \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx &=-\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx}{2 a}-a \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx\\ &=-\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}-\frac {3}{2} \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx}{a}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{a c}\\ &=-\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {3}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {1}{a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-3 \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx+\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{c}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{2 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {3}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {1}{a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}+\frac {\sqrt {1+a^2 x^2} \int \frac {x}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {x}{\left (1+a^2 x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{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 ^2(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^2 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {3}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {1}{a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}+\frac {\sqrt {1+a^2 x^2} \operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\sin (x)}{4 x}+\frac {\sin (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^2 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 ^2(x) \sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {3}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {1}{a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}+\frac {\sqrt {1+a^2 x^2} \text {Si}\left (\tan ^{-1}(a x)\right )}{a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {\sin (x)}{4 x}+\frac {\sin (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^2 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {3}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {1}{a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}+\frac {5 \sqrt {1+a^2 x^2} \text {Si}\left (\tan ^{-1}(a x)\right )}{8 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {3 \sqrt {1+a^2 x^2} \text {Si}\left (3 \tan ^{-1}(a x)\right )}{8 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\sin (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^2 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {x}{2 a c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {3}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {1}{a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {1+a^2 x^2} \text {Si}\left (\tan ^{-1}(a x)\right )}{8 a^2 c^2 \sqrt {c+a^2 c x^2}}-\frac {9 \sqrt {1+a^2 x^2} \text {Si}\left (3 \tan ^{-1}(a x)\right )}{8 a^2 c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.30, size = 118, normalized size = 0.67 \[ \frac {-\left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x)^2 \text {Si}\left (\tan ^{-1}(a x)\right )-9 \left (a^2 x^2+1\right )^{3/2} \tan ^{-1}(a x)^2 \text {Si}\left (3 \tan ^{-1}(a x)\right )+8 a^2 x^2 \tan ^{-1}(a x)-4 a x-4 \tan ^{-1}(a x)}{8 a^2 c^2 \left (a^2 x^2+1\right ) \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} x}{{\left (a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}\right )} \arctan \left (a x\right )^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [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.
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maple [C] time = 1.30, size = 867, normalized size = 4.95 \[ -\frac {i \left (\Ei \left (1, -i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2} x^{2} a^{2}+\sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x a +\Ei \left (1, -i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-i \sqrt {a^{2} x^{2}+1}\, x a -\sqrt {a^{2} x^{2}+1}-i \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )\right ) \sqrt {a^{2} x^{2}+1}\, \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{16 \arctan \left (a x \right )^{2} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) c^{3} a^{2}}-\frac {i \left (9 \Ei \left (1, -3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2} x^{4} a^{4}-3 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x^{3} a^{3}+18 \Ei \left (1, -3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2} x^{2} a^{2}+i \sqrt {a^{2} x^{2}+1}\, x^{3} a^{3}+3 \sqrt {a^{2} x^{2}+1}\, x^{2} a^{2}+9 i \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x^{2} a^{2}+9 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x a -3 i \sqrt {a^{2} x^{2}+1}\, x a +9 \Ei \left (1, -3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-3 i \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )-\sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{16 \sqrt {a^{2} x^{2}+1}\, a^{2} c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{2}}+\frac {i \left (9 \Ei \left (1, 3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2} x^{4} a^{4}-3 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x^{3} a^{3}+18 \Ei \left (1, 3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2} x^{2} a^{2}-i \sqrt {a^{2} x^{2}+1}\, x^{3} a^{3}+3 \sqrt {a^{2} x^{2}+1}\, x^{2} a^{2}-9 i \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x^{2} a^{2}+9 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x a +9 \Ei \left (1, 3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+3 i \sqrt {a^{2} x^{2}+1}\, x a -\sqrt {a^{2} x^{2}+1}+3 i \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{16 \sqrt {a^{2} x^{2}+1}\, a^{2} c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{2}}+\frac {i \left (\Ei \left (1, i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2} x^{2} a^{2}+\sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right ) x a +i \sqrt {a^{2} x^{2}+1}\, x a +\Ei \left (1, i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+i \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )-\sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{16 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctan \left (a x \right )^{2} c^{3} a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {x}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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