Integrand size = 29, antiderivative size = 237 \[ \int \frac {\sec (e+f x) (a+b \sec (e+f x))}{(c+d \sec (e+f x))^4} \, dx=\frac {\left (2 a c^3-4 b c^2 d+3 a c d^2-b d^3\right ) \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{(c-d)^{7/2} (c+d)^{7/2} f}+\frac {(b c-a d) \tan (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sec (e+f x))^3}+\frac {\left (2 b c^2-5 a c d+3 b d^2\right ) \tan (e+f x)}{6 \left (c^2-d^2\right )^2 f (c+d \sec (e+f x))^2}+\frac {\left (2 b c^3-11 a c^2 d+13 b c d^2-4 a d^3\right ) \tan (e+f x)}{6 \left (c^2-d^2\right )^3 f (c+d \sec (e+f x))} \]
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Time = 0.82 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {4088, 12, 3916, 2738, 214} \[ \int \frac {\sec (e+f x) (a+b \sec (e+f x))}{(c+d \sec (e+f x))^4} \, dx=\frac {\left (2 a c^3+3 a c d^2-4 b c^2 d-b d^3\right ) \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{f (c-d)^{7/2} (c+d)^{7/2}}+\frac {\left (-5 a c d+2 b c^2+3 b d^2\right ) \tan (e+f x)}{6 f \left (c^2-d^2\right )^2 (c+d \sec (e+f x))^2}+\frac {(b c-a d) \tan (e+f x)}{3 f \left (c^2-d^2\right ) (c+d \sec (e+f x))^3}+\frac {\left (-11 a c^2 d-4 a d^3+2 b c^3+13 b c d^2\right ) \tan (e+f x)}{6 f \left (c^2-d^2\right )^3 (c+d \sec (e+f x))} \]
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Rule 12
Rule 214
Rule 2738
Rule 3916
Rule 4088
Rubi steps \begin{align*} \text {integral}& = \frac {(b c-a d) \tan (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sec (e+f x))^3}-\frac {\int \frac {\sec (e+f x) (-3 (a c-b d)-2 (b c-a d) \sec (e+f x))}{(c+d \sec (e+f x))^3} \, dx}{3 \left (c^2-d^2\right )} \\ & = \frac {(b c-a d) \tan (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sec (e+f x))^3}+\frac {\left (2 b c^2-5 a c d+3 b d^2\right ) \tan (e+f x)}{6 \left (c^2-d^2\right )^2 f (c+d \sec (e+f x))^2}+\frac {\int \frac {\sec (e+f x) \left (2 \left (3 a c^2-5 b c d+2 a d^2\right )+\left (2 b c^2-5 a c d+3 b d^2\right ) \sec (e+f x)\right )}{(c+d \sec (e+f x))^2} \, dx}{6 \left (c^2-d^2\right )^2} \\ & = \frac {(b c-a d) \tan (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sec (e+f x))^3}+\frac {\left (2 b c^2-5 a c d+3 b d^2\right ) \tan (e+f x)}{6 \left (c^2-d^2\right )^2 f (c+d \sec (e+f x))^2}+\frac {\left (2 b c^3-11 a c^2 d+13 b c d^2-4 a d^3\right ) \tan (e+f x)}{6 \left (c^2-d^2\right )^3 f (c+d \sec (e+f x))}-\frac {\int -\frac {3 \left (2 a c^3-4 b c^2 d+3 a c d^2-b d^3\right ) \sec (e+f x)}{c+d \sec (e+f x)} \, dx}{6 \left (c^2-d^2\right )^3} \\ & = \frac {(b c-a d) \tan (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sec (e+f x))^3}+\frac {\left (2 b c^2-5 a c d+3 b d^2\right ) \tan (e+f x)}{6 \left (c^2-d^2\right )^2 f (c+d \sec (e+f x))^2}+\frac {\left (2 b c^3-11 a c^2 d+13 b c d^2-4 a d^3\right ) \tan (e+f x)}{6 \left (c^2-d^2\right )^3 f (c+d \sec (e+f x))}+\frac {\left (2 a c^3-4 b c^2 d+3 a c d^2-b d^3\right ) \int \frac {\sec (e+f x)}{c+d \sec (e+f x)} \, dx}{2 \left (c^2-d^2\right )^3} \\ & = \frac {(b c-a d) \tan (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sec (e+f x))^3}+\frac {\left (2 b c^2-5 a c d+3 b d^2\right ) \tan (e+f x)}{6 \left (c^2-d^2\right )^2 f (c+d \sec (e+f x))^2}+\frac {\left (2 b c^3-11 a c^2 d+13 b c d^2-4 a d^3\right ) \tan (e+f x)}{6 \left (c^2-d^2\right )^3 f (c+d \sec (e+f x))}+\frac {\left (2 a c^3-4 b c^2 d+3 a c d^2-b d^3\right ) \int \frac {1}{1+\frac {c \cos (e+f x)}{d}} \, dx}{2 d \left (c^2-d^2\right )^3} \\ & = \frac {(b c-a d) \tan (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sec (e+f x))^3}+\frac {\left (2 b c^2-5 a c d+3 b d^2\right ) \tan (e+f x)}{6 \left (c^2-d^2\right )^2 f (c+d \sec (e+f x))^2}+\frac {\left (2 b c^3-11 a c^2 d+13 b c d^2-4 a d^3\right ) \tan (e+f x)}{6 \left (c^2-d^2\right )^3 f (c+d \sec (e+f x))}+\frac {\left (2 a c^3-4 b c^2 d+3 a c d^2-b d^3\right ) \text {Subst}\left (\int \frac {1}{1+\frac {c}{d}+\left (1-\frac {c}{d}\right ) x^2} \, dx,x,\tan \left (\frac {1}{2} (e+f x)\right )\right )}{d \left (c^2-d^2\right )^3 f} \\ & = \frac {\left (2 a c^3-4 b c^2 d+3 a c d^2-b d^3\right ) \text {arctanh}\left (\frac {\sqrt {c-d} \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c+d}}\right )}{(c-d)^{7/2} (c+d)^{7/2} f}+\frac {(b c-a d) \tan (e+f x)}{3 \left (c^2-d^2\right ) f (c+d \sec (e+f x))^3}+\frac {\left (2 b c^2-5 a c d+3 b d^2\right ) \tan (e+f x)}{6 \left (c^2-d^2\right )^2 f (c+d \sec (e+f x))^2}+\frac {\left (2 b c^3-11 a c^2 d+13 b c d^2-4 a d^3\right ) \tan (e+f x)}{6 \left (c^2-d^2\right )^3 f (c+d \sec (e+f x))} \\ \end{align*}
Time = 2.01 (sec) , antiderivative size = 405, normalized size of antiderivative = 1.71 \[ \int \frac {\sec (e+f x) (a+b \sec (e+f x))}{(c+d \sec (e+f x))^4} \, dx=\frac {(d+c \cos (e+f x)) \sec ^3(e+f x) (a+b \sec (e+f x)) \left (\frac {24 \left (-b d \left (4 c^2+d^2\right )+a \left (2 c^3+3 c d^2\right )\right ) \text {arctanh}\left (\frac {(-c+d) \tan \left (\frac {1}{2} (e+f x)\right )}{\sqrt {c^2-d^2}}\right ) (d+c \cos (e+f x))^3}{\sqrt {c^2-d^2}}-6 b c^5 \sin (e+f x)+18 a c^4 d \sin (e+f x)-18 b c^3 d^2 \sin (e+f x)+39 a c^2 d^3 \sin (e+f x)-51 b c d^4 \sin (e+f x)+18 a d^5 \sin (e+f x)-12 b c^4 d \sin (2 (e+f x))+54 a c^3 d^2 \sin (2 (e+f x))-54 b c^2 d^3 \sin (2 (e+f x))+6 a c d^4 \sin (2 (e+f x))+6 b d^5 \sin (2 (e+f x))-6 b c^5 \sin (3 (e+f x))+18 a c^4 d \sin (3 (e+f x))-10 b c^3 d^2 \sin (3 (e+f x))-5 a c^2 d^3 \sin (3 (e+f x))+b c d^4 \sin (3 (e+f x))+2 a d^5 \sin (3 (e+f x))\right )}{24 \left (-c^2+d^2\right )^3 f (b+a \cos (e+f x)) (c+d \sec (e+f x))^4} \]
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Time = 1.28 (sec) , antiderivative size = 376, normalized size of antiderivative = 1.59
method | result | size |
derivativedivides | \(\frac {-\frac {2 \left (-\frac {\left (6 a \,c^{2} d +3 a c \,d^{2}+2 a \,d^{3}-2 b \,c^{3}-2 b \,c^{2} d -6 b c \,d^{2}-b \,d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{2 \left (c -d \right ) \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}+\frac {2 \left (9 a \,c^{2} d +a \,d^{3}-3 b \,c^{3}-7 b c \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 \left (c^{2}+2 c d +d^{2}\right ) \left (c^{2}-2 c d +d^{2}\right )}-\frac {\left (6 a \,c^{2} d -3 a c \,d^{2}+2 a \,d^{3}-2 b \,c^{3}+2 b \,c^{2} d -6 b c \,d^{2}+b \,d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (c^{3}-3 c^{2} d +3 c \,d^{2}-d^{3}\right )}\right )}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )^{3}}+\frac {\left (2 a \,c^{3}+3 a c \,d^{2}-4 b \,c^{2} d -b \,d^{3}\right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}}{f}\) | \(376\) |
default | \(\frac {-\frac {2 \left (-\frac {\left (6 a \,c^{2} d +3 a c \,d^{2}+2 a \,d^{3}-2 b \,c^{3}-2 b \,c^{2} d -6 b c \,d^{2}-b \,d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{5}}{2 \left (c -d \right ) \left (c^{3}+3 c^{2} d +3 c \,d^{2}+d^{3}\right )}+\frac {2 \left (9 a \,c^{2} d +a \,d^{3}-3 b \,c^{3}-7 b c \,d^{2}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{3}}{3 \left (c^{2}+2 c d +d^{2}\right ) \left (c^{2}-2 c d +d^{2}\right )}-\frac {\left (6 a \,c^{2} d -3 a c \,d^{2}+2 a \,d^{3}-2 b \,c^{3}+2 b \,c^{2} d -6 b c \,d^{2}+b \,d^{3}\right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{2 \left (c +d \right ) \left (c^{3}-3 c^{2} d +3 c \,d^{2}-d^{3}\right )}\right )}{\left (\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} c -\tan \left (\frac {f x}{2}+\frac {e}{2}\right )^{2} d -c -d \right )^{3}}+\frac {\left (2 a \,c^{3}+3 a c \,d^{2}-4 b \,c^{2} d -b \,d^{3}\right ) \operatorname {arctanh}\left (\frac {\left (c -d \right ) \tan \left (\frac {f x}{2}+\frac {e}{2}\right )}{\sqrt {\left (c +d \right ) \left (c -d \right )}}\right )}{\left (c^{6}-3 c^{4} d^{2}+3 c^{2} d^{4}-d^{6}\right ) \sqrt {\left (c +d \right ) \left (c -d \right )}}}{f}\) | \(376\) |
risch | \(\text {Expression too large to display}\) | \(1386\) |
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Leaf count of result is larger than twice the leaf count of optimal. 590 vs. \(2 (222) = 444\).
Time = 0.36 (sec) , antiderivative size = 1238, normalized size of antiderivative = 5.22 \[ \int \frac {\sec (e+f x) (a+b \sec (e+f x))}{(c+d \sec (e+f x))^4} \, dx=\text {Too large to display} \]
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\[ \int \frac {\sec (e+f x) (a+b \sec (e+f x))}{(c+d \sec (e+f x))^4} \, dx=\int \frac {\left (a + b \sec {\left (e + f x \right )}\right ) \sec {\left (e + f x \right )}}{\left (c + d \sec {\left (e + f x \right )}\right )^{4}}\, dx \]
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Exception generated. \[ \int \frac {\sec (e+f x) (a+b \sec (e+f x))}{(c+d \sec (e+f x))^4} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 693 vs. \(2 (222) = 444\).
Time = 0.40 (sec) , antiderivative size = 693, normalized size of antiderivative = 2.92 \[ \int \frac {\sec (e+f x) (a+b \sec (e+f x))}{(c+d \sec (e+f x))^4} \, dx=-\frac {\frac {3 \, {\left (2 \, a c^{3} - 4 \, b c^{2} d + 3 \, a c d^{2} - b d^{3}\right )} {\left (\pi \left \lfloor \frac {f x + e}{2 \, \pi } + \frac {1}{2} \right \rfloor \mathrm {sgn}\left (2 \, c - 2 \, d\right ) + \arctan \left (\frac {c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c^{2} + d^{2}}}\right )\right )}}{{\left (c^{6} - 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} - d^{6}\right )} \sqrt {-c^{2} + d^{2}}} + \frac {6 \, b c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 18 \, a c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 6 \, b c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 27 \, a c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 12 \, b c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 6 \, a c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 27 \, b c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, a c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 12 \, b c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 6 \, a d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} + 3 \, b d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 12 \, b c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 36 \, a c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 16 \, b c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 32 \, a c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 28 \, b c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 4 \, a d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 6 \, b c^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 18 \, a c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 6 \, b c^{4} d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 27 \, a c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, b c^{3} d^{2} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 27 \, b c^{2} d^{3} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, a c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 12 \, b c d^{4} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 6 \, a d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 3 \, b d^{5} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{{\left (c^{6} - 3 \, c^{4} d^{2} + 3 \, c^{2} d^{4} - d^{6}\right )} {\left (c \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - d \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}^{3}}}{3 \, f} \]
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Time = 18.53 (sec) , antiderivative size = 439, normalized size of antiderivative = 1.85 \[ \int \frac {\sec (e+f x) (a+b \sec (e+f x))}{(c+d \sec (e+f x))^4} \, dx=\frac {\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^5\,\left (2\,b\,c^3-2\,a\,d^3+b\,d^3-3\,a\,c\,d^2-6\,a\,c^2\,d+6\,b\,c\,d^2+2\,b\,c^2\,d\right )}{{\left (c+d\right )}^3\,\left (c-d\right )}+\frac {4\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^3\,\left (-3\,b\,c^3+9\,a\,c^2\,d-7\,b\,c\,d^2+a\,d^3\right )}{3\,{\left (c+d\right )}^2\,\left (c^2-2\,c\,d+d^2\right )}-\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,a\,d^3-2\,b\,c^3+b\,d^3-3\,a\,c\,d^2+6\,a\,c^2\,d-6\,b\,c\,d^2+2\,b\,c^2\,d\right )}{\left (c+d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}}{f\,\left ({\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (-3\,c^3-3\,c^2\,d+3\,c\,d^2+3\,d^3\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (-3\,c^3+3\,c^2\,d+3\,c\,d^2-3\,d^3\right )+3\,c\,d^2+3\,c^2\,d+c^3+d^3-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )\right )}+\frac {\mathrm {atanh}\left (\frac {\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\,\left (2\,c-2\,d\right )\,\left (c^3-3\,c^2\,d+3\,c\,d^2-d^3\right )}{2\,\sqrt {c+d}\,{\left (c-d\right )}^{7/2}}\right )\,\left (2\,a\,c^3-4\,b\,c^2\,d+3\,a\,c\,d^2-b\,d^3\right )}{f\,{\left (c+d\right )}^{7/2}\,{\left (c-d\right )}^{7/2}} \]
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