Integrand size = 23, antiderivative size = 123 \[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}+\frac {10 a \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {d \sec (e+f x)}}{21 d^4 f}+\frac {2 a \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}+\frac {10 a \sin (e+f x)}{21 d^3 f \sqrt {d \sec (e+f x)}} \]
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Time = 0.11 (sec) , antiderivative size = 123, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3567, 3854, 3856, 2720} \[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=\frac {10 a \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {d \sec (e+f x)}}{21 d^4 f}+\frac {10 a \sin (e+f x)}{21 d^3 f \sqrt {d \sec (e+f x)}}+\frac {2 a \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}-\frac {2 b}{7 f (d \sec (e+f x))^{7/2}} \]
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Rule 2720
Rule 3567
Rule 3854
Rule 3856
Rubi steps \begin{align*} \text {integral}& = -\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}+a \int \frac {1}{(d \sec (e+f x))^{7/2}} \, dx \\ & = -\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}+\frac {2 a \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}+\frac {(5 a) \int \frac {1}{(d \sec (e+f x))^{3/2}} \, dx}{7 d^2} \\ & = -\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}+\frac {2 a \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}+\frac {10 a \sin (e+f x)}{21 d^3 f \sqrt {d \sec (e+f x)}}+\frac {(5 a) \int \sqrt {d \sec (e+f x)} \, dx}{21 d^4} \\ & = -\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}+\frac {2 a \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}+\frac {10 a \sin (e+f x)}{21 d^3 f \sqrt {d \sec (e+f x)}}+\frac {\left (5 a \sqrt {\cos (e+f x)} \sqrt {d \sec (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{21 d^4} \\ & = -\frac {2 b}{7 f (d \sec (e+f x))^{7/2}}+\frac {10 a \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right ) \sqrt {d \sec (e+f x)}}{21 d^4 f}+\frac {2 a \sin (e+f x)}{7 d f (d \sec (e+f x))^{5/2}}+\frac {10 a \sin (e+f x)}{21 d^3 f \sqrt {d \sec (e+f x)}} \\ \end{align*}
Time = 1.11 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.76 \[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=\frac {\sqrt {d \sec (e+f x)} \left (-9 b-12 b \cos (2 (e+f x))-3 b \cos (4 (e+f x))+40 a \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )+26 a \sin (2 (e+f x))+3 a \sin (4 (e+f x))\right )}{84 d^4 f} \]
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Result contains complex when optimal does not.
Time = 6.18 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.45
method | result | size |
default | \(-\frac {2 a \left (5 i \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}+5 i \sec \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}-3 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-5 \sin \left (f x +e \right )\right )}{21 f \sqrt {d \sec \left (f x +e \right )}\, d^{3}}-\frac {2 b}{7 f \left (d \sec \left (f x +e \right )\right )^{\frac {7}{2}}}\) | \(178\) |
parts | \(-\frac {2 a \left (5 i \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}+5 i \sec \left (f x +e \right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}-3 \sin \left (f x +e \right ) \left (\cos ^{2}\left (f x +e \right )\right )-5 \sin \left (f x +e \right )\right )}{21 f \sqrt {d \sec \left (f x +e \right )}\, d^{3}}-\frac {2 b}{7 f \left (d \sec \left (f x +e \right )\right )^{\frac {7}{2}}}\) | \(178\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.09 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.96 \[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=\frac {-5 i \, \sqrt {2} a \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + 5 i \, \sqrt {2} a \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) - 2 \, {\left (3 \, b \cos \left (f x + e\right )^{4} - {\left (3 \, a \cos \left (f x + e\right )^{3} + 5 \, a \cos \left (f x + e\right )\right )} \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{21 \, d^{4} f} \]
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\[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=\int \frac {a + b \tan {\left (e + f x \right )}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {7}{2}}}\, dx \]
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\[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=\int { \frac {b \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}} \,d x } \]
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\[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=\int { \frac {b \tan \left (f x + e\right ) + a}{\left (d \sec \left (f x + e\right )\right )^{\frac {7}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+b \tan (e+f x)}{(d \sec (e+f x))^{7/2}} \, dx=\int \frac {a+b\,\mathrm {tan}\left (e+f\,x\right )}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{7/2}} \,d x \]
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