Integrand size = 21, antiderivative size = 101 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=-\frac {1}{2 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}+\frac {x}{2 c \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{2 a c \sqrt {c+a^2 c x^2}} \]
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Time = 0.16 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {5022, 5062, 5025, 5024, 3383} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=-\frac {\sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{2 a c \sqrt {a^2 c x^2+c}}+\frac {x}{2 c \arctan (a x) \sqrt {a^2 c x^2+c}}-\frac {1}{2 a c \arctan (a x)^2 \sqrt {a^2 c x^2+c}} \]
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Rule 3383
Rule 5022
Rule 5024
Rule 5025
Rule 5062
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}-\frac {1}{2} a \int \frac {x}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2} \, dx \\ & = -\frac {1}{2 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}+\frac {x}{2 c \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {1}{2} \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)} \, dx \\ & = -\frac {1}{2 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}+\frac {x}{2 c \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \int \frac {1}{\left (1+a^2 x^2\right )^{3/2} \arctan (a x)} \, dx}{2 c \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{2 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}+\frac {x}{2 c \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \text {Subst}\left (\int \frac {\cos (x)}{x} \, dx,x,\arctan (a x)\right )}{2 a c \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{2 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2}+\frac {x}{2 c \sqrt {c+a^2 c x^2} \arctan (a x)}-\frac {\sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{2 a c \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.08 (sec) , antiderivative size = 65, normalized size of antiderivative = 0.64 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\frac {-1+a x \arctan (a x)-\sqrt {1+a^2 x^2} \arctan (a x)^2 \operatorname {CosIntegral}(\arctan (a x))}{2 a c \sqrt {c+a^2 c x^2} \arctan (a x)^2} \]
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Result contains complex when optimal does not.
Time = 9.04 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.50
method | result | size |
default | \(\frac {\left (\arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{2} x^{2}+\arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{2} x^{2}+2 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +\operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+\operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-2 \sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{4 \left (a^{2} x^{2}+1\right )^{\frac {3}{2}} \arctan \left (a x \right )^{2} a \,c^{2}}\) | \(151\) |
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int \frac {1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {3}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {3}{2}} \arctan \left (a x\right )^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^3} \, dx=\int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{3/2}} \,d x \]
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