Optimal. Leaf size=238 \[ \frac {2 a \text {Int}\left (\frac {1}{x^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)},x\right )}{c^2}+\frac {\text {Int}\left (\frac {1}{x^3 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)^2},x\right )}{c^2}+\frac {2 a \sqrt {a^2 c x^2+c}}{c^3 x \tan ^{-1}(a x)}+\frac {9 a^2 \sqrt {a^2 x^2+1} \text {Ci}\left (\tan ^{-1}(a x)\right )}{4 c^2 \sqrt {a^2 c x^2+c}}+\frac {3 a^2 \sqrt {a^2 x^2+1} \text {Ci}\left (3 \tan ^{-1}(a x)\right )}{4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 a^3 x}{c^2 \sqrt {a^2 c x^2+c} \tan ^{-1}(a x)}-\frac {a^3 x}{c \left (a^2 c x^2+c\right )^{3/2} \tan ^{-1}(a x)} \]
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Rubi [A] time = 2.05, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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Rubi steps
\begin {align*} \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx &=-\left (a^2 \int \frac {1}{x \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx\right )+\frac {\int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{c}\\ &=a^4 \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx+\frac {\int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-2 \frac {a^2 \int \frac {1}{x \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{c}\\ &=-\frac {a^3 x}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+a^3 \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx-\left (2 a^5\right ) \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx+\frac {\int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-2 \left (\frac {a^2 \int \frac {1}{x \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-\frac {a^4 \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{c}\right )\\ &=-\frac {a^3 x}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {\int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-2 \left (\frac {a^3 x}{c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {a \sqrt {c+a^2 c x^2}}{c^3 x \tan ^{-1}(a x)}-\frac {a \int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)} \, dx}{c^2}-\frac {a^3 \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{c}\right )+\frac {\left (a^3 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\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 (2 a^5 \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}{c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {a^3 x}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {\int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}+\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos ^3(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (2 a^2 \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 )}{c^2 \sqrt {c+a^2 c x^2}}-2 \left (\frac {a^3 x}{c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {a \sqrt {c+a^2 c x^2}}{c^3 x \tan ^{-1}(a x)}-\frac {a \int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)} \, dx}{c^2}-\frac {\left (a^3 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{c^2 \sqrt {c+a^2 c x^2}}\right )\\ &=-\frac {a^3 x}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {\int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-2 \left (\frac {a^3 x}{c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {a \sqrt {c+a^2 c x^2}}{c^3 x \tan ^{-1}(a x)}-\frac {a \int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)} \, dx}{c^2}-\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}\right )+\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \left (\frac {3 \cos (x)}{4 x}+\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (2 a^2 \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 )}{c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {a^3 x}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {\int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-2 \left (\frac {a^3 x}{c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {a \sqrt {c+a^2 c x^2}}{c^3 x \tan ^{-1}(a x)}-\frac {a^2 \sqrt {1+a^2 x^2} \text {Ci}\left (\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {a \int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)} \, dx}{c^2}\right )+\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (a^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 \sqrt {1+a^2 x^2}\right ) \operatorname {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {a^3 x}{c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {a^2 \sqrt {1+a^2 x^2} \text {Ci}\left (\tan ^{-1}(a x)\right )}{4 c^2 \sqrt {c+a^2 c x^2}}+\frac {3 a^2 \sqrt {1+a^2 x^2} \text {Ci}\left (3 \tan ^{-1}(a x)\right )}{4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\int \frac {1}{x^3 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2} \, dx}{c^2}-2 \left (\frac {a^3 x}{c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {a \sqrt {c+a^2 c x^2}}{c^3 x \tan ^{-1}(a x)}-\frac {a^2 \sqrt {1+a^2 x^2} \text {Ci}\left (\tan ^{-1}(a x)\right )}{c^2 \sqrt {c+a^2 c x^2}}-\frac {a \int \frac {1}{x^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)} \, dx}{c^2}\right )\\ \end {align*}
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Mathematica [A] time = 7.95, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^3 \left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx \]
Verification is Not applicable to the result.
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fricas [A] time = 0.43, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c}}{{\left (a^{6} c^{3} x^{9} + 3 \, a^{4} c^{3} x^{7} + 3 \, a^{2} c^{3} x^{5} + c^{3} x^{3}\right )} \arctan \left (a x\right )^{2}}, x\right ) \]
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 [A] time = 1.97, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \left (a^{2} c \,x^{2}+c \right )^{\frac {5}{2}} \arctan \left (a x \right )^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} x^{3} \arctan \left (a x\right )^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [A] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{x^3\,{\mathrm {atan}\left (a\,x\right )}^2\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{x^{3} \left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{2}{\left (a x \right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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