Integrand size = 27, antiderivative size = 802 \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3} \, dx=-\frac {\tan (e+f x)}{2 a (c-d)^3 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} (c-4 d) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} \sqrt {a} (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {3 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{4 \sqrt {a} c (c-d)^2 (c+d)^{5/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(3 c-d) d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{\sqrt {a} c^2 (c-d)^3 (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {2 d^{5/2} \left (6 c^2-4 c d+d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{\sqrt {a} c^3 (c-d)^4 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {d^3 \tan (e+f x)}{2 a c (c-d)^2 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^2}-\frac {(3 c-d) d^3 \tan (e+f x)}{a c^2 (c-d)^3 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac {3 d^3 \tan (e+f x)}{4 a c \left (c^2-d^2\right )^2 f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \]
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Time = 0.94 (sec) , antiderivative size = 802, normalized size of antiderivative = 1.00, number of steps used = 19, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {4025, 186, 65, 212, 44, 214} \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3} \, dx=-\frac {(3 c-d) \tan (e+f x) d^3}{a c^2 (c-d)^3 (c+d) f \sqrt {\sec (e+f x) a+a} (c+d \sec (e+f x))}-\frac {3 \tan (e+f x) d^3}{4 a c \left (c^2-d^2\right )^2 f \sqrt {\sec (e+f x) a+a} (c+d \sec (e+f x))}-\frac {\tan (e+f x) d^3}{2 a c (c-d)^2 (c+d) f \sqrt {\sec (e+f x) a+a} (c+d \sec (e+f x))^2}-\frac {2 \left (6 c^2-4 d c+d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x) d^{5/2}}{\sqrt {a} c^3 (c-d)^4 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}-\frac {(3 c-d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x) d^{5/2}}{\sqrt {a} c^2 (c-d)^3 (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}-\frac {3 \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x) d^{5/2}}{4 \sqrt {a} c (c-d)^2 (c+d)^{5/2} f \sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}-\frac {\text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} \sqrt {a} (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}-\frac {\sqrt {2} (c-4 d) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {\sec (e+f x) a+a}}-\frac {\tan (e+f x)}{2 a (c-d)^3 f (\sec (e+f x)+1) \sqrt {\sec (e+f x) a+a}} \]
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Rule 44
Rule 65
Rule 186
Rule 212
Rule 214
Rule 4025
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{x \sqrt {a-a x} (a+a x)^2 (c+d x)^3} \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {\left (a^2 \tan (e+f x)\right ) \text {Subst}\left (\int \left (\frac {1}{a^2 c^3 x \sqrt {a-a x}}-\frac {1}{a^2 (c-d)^3 (1+x)^2 \sqrt {a-a x}}+\frac {-c+4 d}{a^2 (c-d)^4 (1+x) \sqrt {a-a x}}-\frac {d^3}{a^2 c (c-d)^2 \sqrt {a-a x} (c+d x)^3}-\frac {(3 c-d) d^3}{a^2 c^2 (c-d)^3 \sqrt {a-a x} (c+d x)^2}-\frac {d^3 \left (6 c^2-4 c d+d^2\right )}{a^2 c^3 (c-d)^4 \sqrt {a-a x} (c+d x)}\right ) \, dx,x,\sec (e+f x)\right )}{f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {\tan (e+f x) \text {Subst}\left (\int \frac {1}{x \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {((c-4 d) \tan (e+f x)) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{(c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {1}{(1+x)^2 \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{(c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (d^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (c+d x)^3} \, dx,x,\sec (e+f x)\right )}{c (c-d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left ((3 c-d) d^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{c^2 (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (d^3 \left (6 c^2-4 c d+d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{c^3 (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {\tan (e+f x)}{2 a (c-d)^3 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}-\frac {d^3 \tan (e+f x)}{2 a c (c-d)^2 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^2}-\frac {(3 c-d) d^3 \tan (e+f x)}{a c^2 (c-d)^3 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}+\frac {(2 \tan (e+f x)) \text {Subst}\left (\int \frac {1}{1-\frac {x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{a c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(2 (c-4 d) \tan (e+f x)) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{a (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\tan (e+f x) \text {Subst}\left (\int \frac {1}{(1+x) \sqrt {a-a x}} \, dx,x,\sec (e+f x)\right )}{4 (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (3 d^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (c+d x)^2} \, dx,x,\sec (e+f x)\right )}{4 c (c-d)^2 (c+d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left ((3 c-d) d^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{2 c^2 (c-d)^3 (c+d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left (2 d^3 \left (6 c^2-4 c d+d^2\right ) \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+d-\frac {d x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{a c^3 (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {\tan (e+f x)}{2 a (c-d)^3 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} (c-4 d) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {2 d^{5/2} \left (6 c^2-4 c d+d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{\sqrt {a} c^3 (c-d)^4 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {d^3 \tan (e+f x)}{2 a c (c-d)^2 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^2}-\frac {(3 c-d) d^3 \tan (e+f x)}{a c^2 (c-d)^3 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac {3 d^3 \tan (e+f x)}{4 a c \left (c^2-d^2\right )^2 f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac {\tan (e+f x) \text {Subst}\left (\int \frac {1}{2-\frac {x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{2 a (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}+\frac {\left (3 d^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-a x} (c+d x)} \, dx,x,\sec (e+f x)\right )}{8 c (c-d)^2 (c+d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\left ((3 c-d) d^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+d-\frac {d x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{a c^2 (c-d)^3 (c+d) f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {\tan (e+f x)}{2 a (c-d)^3 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} (c-4 d) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} \sqrt {a} (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(3 c-d) d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{\sqrt {a} c^2 (c-d)^3 (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {2 d^{5/2} \left (6 c^2-4 c d+d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{\sqrt {a} c^3 (c-d)^4 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {d^3 \tan (e+f x)}{2 a c (c-d)^2 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^2}-\frac {(3 c-d) d^3 \tan (e+f x)}{a c^2 (c-d)^3 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac {3 d^3 \tan (e+f x)}{4 a c \left (c^2-d^2\right )^2 f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac {\left (3 d^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{c+d-\frac {d x^2}{a}} \, dx,x,\sqrt {a-a \sec (e+f x)}\right )}{4 a c (c-d)^2 (c+d)^2 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}} \\ & = -\frac {\tan (e+f x)}{2 a (c-d)^3 f (1+\sec (e+f x)) \sqrt {a+a \sec (e+f x)}}+\frac {2 \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} c^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\sqrt {2} (c-4 d) \text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{\sqrt {a} (c-d)^4 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {\text {arctanh}\left (\frac {\sqrt {a-a \sec (e+f x)}}{\sqrt {2} \sqrt {a}}\right ) \tan (e+f x)}{2 \sqrt {2} \sqrt {a} (c-d)^3 f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {3 d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{4 \sqrt {a} c (c-d)^2 (c+d)^{5/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {(3 c-d) d^{5/2} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{\sqrt {a} c^2 (c-d)^3 (c+d)^{3/2} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {2 d^{5/2} \left (6 c^2-4 c d+d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a-a \sec (e+f x)}}{\sqrt {a} \sqrt {c+d}}\right ) \tan (e+f x)}{\sqrt {a} c^3 (c-d)^4 \sqrt {c+d} f \sqrt {a-a \sec (e+f x)} \sqrt {a+a \sec (e+f x)}}-\frac {d^3 \tan (e+f x)}{2 a c (c-d)^2 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))^2}-\frac {(3 c-d) d^3 \tan (e+f x)}{a c^2 (c-d)^3 (c+d) f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))}-\frac {3 d^3 \tan (e+f x)}{4 a c \left (c^2-d^2\right )^2 f \sqrt {a+a \sec (e+f x)} (c+d \sec (e+f x))} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(2632\) vs. \(2(802)=1604\).
Time = 18.76 (sec) , antiderivative size = 2632, normalized size of antiderivative = 3.28 \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3} \, dx=\text {Result too large to show} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(117746\) vs. \(2(694)=1388\).
Time = 21.10 (sec) , antiderivative size = 117747, normalized size of antiderivative = 146.82
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Timed out. \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3} \, dx=\int \frac {1}{\left (a \left (\sec {\left (e + f x \right )} + 1\right )\right )^{\frac {3}{2}} \left (c + d \sec {\left (e + f x \right )}\right )^{3}}\, dx \]
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Timed out. \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {1}{(a+a \sec (e+f x))^{3/2} (c+d \sec (e+f x))^3} \, dx=\text {Hanged} \]
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