Optimal. Leaf size=209 \[ \frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2}-\frac {3 x}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \text {ArcTan}(a x)}+\frac {x}{2 a^2 c^2 \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)}-\frac {\sqrt {1+a^2 x^2} \text {CosIntegral}(\text {ArcTan}(a x))}{8 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {9 \sqrt {1+a^2 x^2} \text {CosIntegral}(3 \text {ArcTan}(a x))}{8 a^3 c^2 \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.62, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps
used = 20, number of rules used = 11, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.458, Rules used = {5084, 5022,
5062, 5025, 5024, 3383, 5088, 5091, 5090, 4491, 3393} \begin {gather*} \frac {x}{2 a^2 c^2 \text {ArcTan}(a x) \sqrt {a^2 c x^2+c}}-\frac {3 x}{2 a^2 c \text {ArcTan}(a x) \left (a^2 c x^2+c\right )^{3/2}}-\frac {\sqrt {a^2 x^2+1} \text {CosIntegral}(\text {ArcTan}(a x))}{8 a^3 c^2 \sqrt {a^2 c x^2+c}}+\frac {9 \sqrt {a^2 x^2+1} \text {CosIntegral}(3 \text {ArcTan}(a x))}{8 a^3 c^2 \sqrt {a^2 c x^2+c}}-\frac {1}{2 a^3 c^2 \text {ArcTan}(a x)^2 \sqrt {a^2 c x^2+c}}+\frac {1}{2 a^3 c \text {ArcTan}(a x)^2 \left (a^2 c x^2+c\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 3383
Rule 3393
Rule 4491
Rule 5022
Rule 5024
Rule 5025
Rule 5062
Rule 5084
Rule 5088
Rule 5090
Rule 5091
Rubi steps
\begin {align*} \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx &=-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^3} \, dx}{a^2}+\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^3} \, dx}{a^2 c}\\ &=\frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}+\frac {3 \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)^2} \, dx}{2 a}-\frac {\int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2} \, dx}{2 a c}\\ &=\frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {3 x}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{2 a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-3 \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx+\frac {3 \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \tan ^{-1}(a x)} \, dx}{2 a^2}-\frac {\int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{2 a^2 c}\\ &=\frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {3 x}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{2 a^2 c^2 \sqrt {c+a^2 c x^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}{c^2 \sqrt {c+a^2 c x^2}}-\frac {\sqrt {1+a^2 x^2} \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \tan ^{-1}(a x)} \, dx}{2 a^2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (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}{2 a^2 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {3 x}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{2 a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos ^3(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x) \sin ^2(x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {3 x}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{2 a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {1+a^2 x^2} \text {Ci}\left (\tan ^{-1}(a x)\right )}{2 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {3 \cos (x)}{4 x}+\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{2 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \left (\frac {\cos (x)}{4 x}-\frac {\cos (3 x)}{4 x}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {3 x}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{2 a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {1+a^2 x^2} \text {Ci}\left (\tan ^{-1}(a x)\right )}{2 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 c^2 \sqrt {c+a^2 c x^2}}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{4 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (9 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\tan ^{-1}(a x)\right )}{8 a^3 c^2 \sqrt {c+a^2 c x^2}}\\ &=\frac {1}{2 a^3 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)^2}-\frac {1}{2 a^3 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)^2}-\frac {3 x}{2 a^2 c \left (c+a^2 c x^2\right )^{3/2} \tan ^{-1}(a x)}+\frac {x}{2 a^2 c^2 \sqrt {c+a^2 c x^2} \tan ^{-1}(a x)}-\frac {\sqrt {1+a^2 x^2} \text {Ci}\left (\tan ^{-1}(a x)\right )}{8 a^3 c^2 \sqrt {c+a^2 c x^2}}+\frac {9 \sqrt {1+a^2 x^2} \text {Ci}\left (3 \tan ^{-1}(a x)\right )}{8 a^3 c^2 \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.17, size = 119, normalized size = 0.57 \begin {gather*} \frac {4 a x \left (-a x+\left (-2+a^2 x^2\right ) \text {ArcTan}(a x)\right )-\left (1+a^2 x^2\right )^{3/2} \text {ArcTan}(a x)^2 \text {CosIntegral}(\text {ArcTan}(a x))+9 \left (1+a^2 x^2\right )^{3/2} \text {ArcTan}(a x)^2 \text {CosIntegral}(3 \text {ArcTan}(a x))}{8 a^3 c^2 \left (1+a^2 x^2\right ) \sqrt {c+a^2 c x^2} \text {ArcTan}(a x)^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 1.55, size = 844, normalized size = 4.04
method | result | size |
default | \(-\frac {\left (9 \arctan \left (a x \right )^{2} \expIntegral \left (1, 3 i \arctan \left (a x \right )\right ) a^{4} x^{4}-3 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+18 \arctan \left (a x \right )^{2} \expIntegral \left (1, 3 i \arctan \left (a x \right )\right ) a^{2} x^{2}-i \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}-9 i \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+3 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+9 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +9 \expIntegral \left (1, 3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+3 i \sqrt {a^{2} x^{2}+1}\, a x +3 i \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}-\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}\, \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{2} a^{3} c^{3}}-\frac {\left (9 \arctan \left (a x \right )^{2} \expIntegral \left (1, -3 i \arctan \left (a x \right )\right ) a^{4} x^{4}-3 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+18 \arctan \left (a x \right )^{2} \expIntegral \left (1, -3 i \arctan \left (a x \right )\right ) a^{2} x^{2}+i \sqrt {a^{2} x^{2}+1}\, a^{3} x^{3}+9 i \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+3 \sqrt {a^{2} x^{2}+1}\, a^{2} x^{2}+9 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x -3 i \sqrt {a^{2} x^{2}+1}\, a x +9 \expIntegral \left (1, -3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-3 i \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}-\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}\, \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right ) \arctan \left (a x \right )^{2} a^{3} c^{3}}+\frac {\left (\arctan \left (a x \right )^{2} \expIntegral \left (1, i \arctan \left (a x \right )\right ) a^{2} x^{2}+\arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +i \sqrt {a^{2} x^{2}+1}\, a x +\expIntegral \left (1, i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+i \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}-\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} a^{3} c^{3}}+\frac {\left (\arctan \left (a x \right )^{2} \expIntegral \left (1, -i \arctan \left (a x \right )\right ) a^{2} x^{2}+\arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +\expIntegral \left (1, -i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-i \sqrt {a^{2} x^{2}+1}\, a x -i \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}-\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} a^{3} c^{3}}\) | \(844\) |
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 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^{2}}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \end {gather*}
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 \begin {gather*} \text {could not integrate} \end {gather*}
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 \begin {gather*} \int \frac {x^2}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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