Optimal. Leaf size=308 \[ -\frac {x^3 \tan ^{-1}(a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}}+\frac {i \sqrt {a^2 x^2+1} \text {Li}_2\left (-\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {i \sqrt {a^2 x^2+1} \text {Li}_2\left (\frac {i \sqrt {i a x+1}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {4}{3 a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {1}{9 a^5 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {x \tan ^{-1}(a x)}{a^4 c^2 \sqrt {a^2 c x^2+c}} \]
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Rubi [A] time = 0.37, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.318, Rules used = {4964, 4934, 4890, 4886, 4944, 266, 43} \[ \frac {i \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {i \sqrt {a^2 x^2+1} \text {PolyLog}\left (2,\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {4}{3 a^5 c^2 \sqrt {a^2 c x^2+c}}-\frac {x \tan ^{-1}(a x)}{a^4 c^2 \sqrt {a^2 c x^2+c}}-\frac {2 i \sqrt {a^2 x^2+1} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {a^2 c x^2+c}}+\frac {1}{9 a^5 c \left (a^2 c x^2+c\right )^{3/2}}-\frac {x^3 \tan ^{-1}(a x)}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rule 4886
Rule 4890
Rule 4934
Rule 4944
Rule 4964
Rubi steps
\begin {align*} \int \frac {x^4 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=-\frac {\int \frac {x^2 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{a^2}+\frac {\int \frac {x^2 \tan ^{-1}(a x)}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^2 c}\\ &=-\frac {1}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)}{a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\int \frac {x^3}{\left (c+a^2 c x^2\right )^{5/2}} \, dx}{3 a}+\frac {\int \frac {\tan ^{-1}(a x)}{\sqrt {c+a^2 c x^2}} \, dx}{a^4 c^2}\\ &=-\frac {1}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)}{a^4 c^2 \sqrt {c+a^2 c x^2}}+\frac {\operatorname {Subst}\left (\int \frac {x}{\left (c+a^2 c x\right )^{5/2}} \, dx,x,x^2\right )}{6 a}+\frac {\sqrt {1+a^2 x^2} \int \frac {\tan ^{-1}(a x)}{\sqrt {1+a^2 x^2}} \, dx}{a^4 c^2 \sqrt {c+a^2 c x^2}}\\ &=-\frac {1}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {\operatorname {Subst}\left (\int \left (-\frac {1}{a^2 \left (c+a^2 c x\right )^{5/2}}+\frac {1}{a^2 c \left (c+a^2 c x\right )^{3/2}}\right ) \, dx,x,x^2\right )}{6 a}\\ &=\frac {1}{9 a^5 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {4}{3 a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {x^3 \tan ^{-1}(a x)}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac {x \tan ^{-1}(a x)}{a^4 c^2 \sqrt {c+a^2 c x^2}}-\frac {2 i \sqrt {1+a^2 x^2} \tan ^{-1}(a x) \tan ^{-1}\left (\frac {\sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}+\frac {i \sqrt {1+a^2 x^2} \text {Li}_2\left (-\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}-\frac {i \sqrt {1+a^2 x^2} \text {Li}_2\left (\frac {i \sqrt {1+i a x}}{\sqrt {1-i a x}}\right )}{a^5 c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.40, size = 177, normalized size = 0.57 \[ \frac {\sqrt {c \left (a^2 x^2+1\right )} \left (-\frac {45}{\sqrt {a^2 x^2+1}}-\frac {45 a x \tan ^{-1}(a x)}{\sqrt {a^2 x^2+1}}+36 i \left (\text {Li}_2\left (-i e^{i \tan ^{-1}(a x)}\right )-\text {Li}_2\left (i e^{i \tan ^{-1}(a x)}\right )\right )+36 \tan ^{-1}(a x) \left (\log \left (1-i e^{i \tan ^{-1}(a x)}\right )-\log \left (1+i e^{i \tan ^{-1}(a x)}\right )\right )+3 \tan ^{-1}(a x) \sin \left (3 \tan ^{-1}(a x)\right )+\cos \left (3 \tan ^{-1}(a x)\right )\right )}{36 a^5 c^3 \sqrt {a^2 x^2+1}} \]
Warning: Unable to verify antiderivative.
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fricas [F] time = 0.57, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {a^{2} c x^{2} + c} x^{4} \arctan \left (a x\right )}{a^{6} c^{3} x^{6} + 3 \, a^{4} c^{3} x^{4} + 3 \, a^{2} c^{3} x^{2} + c^{3}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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maple [A] time = 3.24, size = 389, normalized size = 1.26 \[ -\frac {\left (i+3 \arctan \left (a x \right )\right ) \left (a^{3} x^{3}-3 i x^{2} a^{2}-3 a x +i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{72 \left (a^{2} x^{2}+1\right )^{2} c^{3} a^{5}}-\frac {5 \left (i+\arctan \left (a x \right )\right ) \left (a x -i\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{8 a^{5} c^{3} \left (a^{2} x^{2}+1\right )}-\frac {5 \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a x +i\right ) \left (\arctan \left (a x \right )-i\right )}{8 a^{5} c^{3} \left (a^{2} x^{2}+1\right )}-\frac {\left (-i+3 \arctan \left (a x \right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}\, \left (a^{3} x^{3}+3 i x^{2} a^{2}-3 a x -i\right )}{72 \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) c^{3} a^{5}}+\frac {i \left (i \arctan \left (a x \right ) \ln \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-i \arctan \left (a x \right ) \ln \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )+\dilog \left (1+\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )-\dilog \left (1-\frac {i \left (i a x +1\right )}{\sqrt {a^{2} x^{2}+1}}\right )\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{\sqrt {a^{2} x^{2}+1}\, a^{5} c^{3}} \]
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 \[ \int \frac {x^{4} \arctan \left (a x\right )}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {x^4\,\mathrm {atan}\left (a\,x\right )}{{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {x^{4} \operatorname {atan}{\left (a x \right )}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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