Integrand size = 21, antiderivative size = 145 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=-\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {3 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{8 a c^2 \sqrt {c+a^2 c x^2}}-\frac {9 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{8 a c^2 \sqrt {c+a^2 c x^2}} \]
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Time = 0.40 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.429, Rules used = {5022, 5088, 5091, 5090, 4491, 3383, 5025, 5024, 3393} \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=-\frac {3 \sqrt {a^2 x^2+1} \operatorname {CosIntegral}(\arctan (a x))}{8 a c^2 \sqrt {a^2 c x^2+c}}-\frac {9 \sqrt {a^2 x^2+1} \operatorname {CosIntegral}(3 \arctan (a x))}{8 a c^2 \sqrt {a^2 c x^2+c}}+\frac {3 x}{2 c \arctan (a x) \left (a^2 c x^2+c\right )^{3/2}}-\frac {1}{2 a c \arctan (a x)^2 \left (a^2 c x^2+c\right )^{3/2}} \]
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Rule 3383
Rule 3393
Rule 4491
Rule 5022
Rule 5024
Rule 5025
Rule 5088
Rule 5090
Rule 5091
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}-\frac {1}{2} (3 a) \int \frac {x}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^2} \, dx \\ & = -\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {3}{2} \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx+\left (3 a^2\right ) \int \frac {x^2}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)} \, dx \\ & = -\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \int \frac {1}{\left (1+a^2 x^2\right )^{5/2} \arctan (a x)} \, dx}{2 c^2 \sqrt {c+a^2 c x^2}}+\frac {\left (3 a^2 \sqrt {1+a^2 x^2}\right ) \int \frac {x^2}{\left (1+a^2 x^2\right )^{5/2} \arctan (a x)} \, dx}{c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos ^3(x)}{x} \, dx,x,\arctan (a x)\right )}{2 a 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,\arctan (a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\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,\arctan (a x)\right )}{2 a 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,\arctan (a x)\right )}{a c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {\left (3 \sqrt {1+a^2 x^2}\right ) \text {Subst}\left (\int \frac {\cos (3 x)}{x} \, dx,x,\arctan (a x)\right )}{8 a 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,\arctan (a x)\right )}{4 a 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,\arctan (a x)\right )}{4 a 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,\arctan (a x)\right )}{8 a c^2 \sqrt {c+a^2 c x^2}} \\ & = -\frac {1}{2 a c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)^2}+\frac {3 x}{2 c \left (c+a^2 c x^2\right )^{3/2} \arctan (a x)}-\frac {3 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(\arctan (a x))}{8 a c^2 \sqrt {c+a^2 c x^2}}-\frac {9 \sqrt {1+a^2 x^2} \operatorname {CosIntegral}(3 \arctan (a x))}{8 a c^2 \sqrt {c+a^2 c x^2}} \\ \end{align*}
Time = 0.19 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.70 \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\frac {-4+12 a x \arctan (a x)-3 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2 \operatorname {CosIntegral}(\arctan (a x))-9 \left (1+a^2 x^2\right )^{3/2} \arctan (a x)^2 \operatorname {CosIntegral}(3 \arctan (a x))}{8 c^2 \left (a+a^3 x^2\right ) \sqrt {c+a^2 c x^2} \arctan (a x)^2} \]
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Result contains complex when optimal does not.
Time = 18.91 (sec) , antiderivative size = 346, normalized size of antiderivative = 2.39
method | result | size |
default | \(\frac {\left (9 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{4} x^{4}+3 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{4} x^{4}+3 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{4} x^{4}+9 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{4} x^{4}+18 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) a^{2} x^{2}+6 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) a^{2} x^{2}+6 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) a^{2} x^{2}+18 \arctan \left (a x \right )^{2} \operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) a^{2} x^{2}+24 \arctan \left (a x \right ) \sqrt {a^{2} x^{2}+1}\, a x +9 \,\operatorname {Ei}_{1}\left (3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+3 \,\operatorname {Ei}_{1}\left (-i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+3 \,\operatorname {Ei}_{1}\left (i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}+9 \,\operatorname {Ei}_{1}\left (-3 i \arctan \left (a x \right )\right ) \arctan \left (a x \right )^{2}-8 \sqrt {a^{2} x^{2}+1}\right ) \sqrt {c \left (a x -i\right ) \left (a x +i\right )}}{16 \sqrt {a^{2} x^{2}+1}\, \arctan \left (a x \right )^{2} a \,c^{3} \left (a^{4} x^{4}+2 a^{2} x^{2}+1\right )}\) | \(346\) |
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {1}{\left (c \left (a^{2} x^{2} + 1\right )\right )^{\frac {5}{2}} \operatorname {atan}^{3}{\left (a x \right )}}\, dx \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \]
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\[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int { \frac {1}{{\left (a^{2} c x^{2} + c\right )}^{\frac {5}{2}} \arctan \left (a x\right )^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\left (c+a^2 c x^2\right )^{5/2} \arctan (a x)^3} \, dx=\int \frac {1}{{\mathrm {atan}\left (a\,x\right )}^3\,{\left (c\,a^2\,x^2+c\right )}^{5/2}} \,d x \]
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