Integrand size = 25, antiderivative size = 146 \[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\frac {2 a \left (a^2+6 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sec ^2(e+f x)^{3/4}}{3 f (d \sec (e+f x))^{3/2}}-\frac {2 (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{3 f (d \sec (e+f x))^{3/2}}-\frac {2 b \sec ^2(e+f x) \left (2 \left (a^2-2 b^2\right )+a b \tan (e+f x)\right )}{3 f (d \sec (e+f x))^{3/2}} \]
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Time = 0.14 (sec) , antiderivative size = 146, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {3593, 753, 794, 237} \[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\frac {2 a \left (a^2+6 b^2\right ) \sec ^2(e+f x)^{3/4} \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right )}{3 f (d \sec (e+f x))^{3/2}}-\frac {2 b \sec ^2(e+f x) \left (2 \left (a^2-2 b^2\right )+a b \tan (e+f x)\right )}{3 f (d \sec (e+f x))^{3/2}}-\frac {2 (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{3 f (d \sec (e+f x))^{3/2}} \]
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Rule 237
Rule 753
Rule 794
Rule 3593
Rubi steps \begin{align*} \text {integral}& = \frac {\sec ^2(e+f x)^{3/4} \text {Subst}\left (\int \frac {(a+x)^3}{\left (1+\frac {x^2}{b^2}\right )^{7/4}} \, dx,x,b \tan (e+f x)\right )}{b f (d \sec (e+f x))^{3/2}} \\ & = -\frac {2 (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{3 f (d \sec (e+f x))^{3/2}}+\frac {\left (2 b \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {(a+x) \left (\frac {1}{2} \left (4+\frac {a^2}{b^2}\right )-\frac {3 a x}{2 b^2}\right )}{\left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{3 f (d \sec (e+f x))^{3/2}} \\ & = -\frac {2 (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{3 f (d \sec (e+f x))^{3/2}}-\frac {2 b \sec ^2(e+f x) \left (2 \left (a^2-2 b^2\right )+a b \tan (e+f x)\right )}{3 f (d \sec (e+f x))^{3/2}}+\frac {\left (a \left (6+\frac {a^2}{b^2}\right ) b \sec ^2(e+f x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {x^2}{b^2}\right )^{3/4}} \, dx,x,b \tan (e+f x)\right )}{3 f (d \sec (e+f x))^{3/2}} \\ & = \frac {2 a \left (a^2+6 b^2\right ) \operatorname {EllipticF}\left (\frac {1}{2} \arctan (\tan (e+f x)),2\right ) \sec ^2(e+f x)^{3/4}}{3 f (d \sec (e+f x))^{3/2}}-\frac {2 (b-a \tan (e+f x)) (a+b \tan (e+f x))^2}{3 f (d \sec (e+f x))^{3/2}}-\frac {2 b \sec ^2(e+f x) \left (2 \left (a^2-2 b^2\right )+a b \tan (e+f x)\right )}{3 f (d \sec (e+f x))^{3/2}} \\ \end{align*}
Time = 3.80 (sec) , antiderivative size = 117, normalized size of antiderivative = 0.80 \[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\frac {\sec ^2(e+f x) \left (-3 a^2 b+7 b^3+\left (-3 a^2 b+b^3\right ) \cos (2 (e+f x))+2 a \left (a^2+6 b^2\right ) \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )+a^3 \sin (2 (e+f x))-3 a b^2 \sin (2 (e+f x))\right )}{3 f (d \sec (e+f x))^{3/2}} \]
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Result contains complex when optimal does not.
Time = 23.96 (sec) , antiderivative size = 323, normalized size of antiderivative = 2.21
method | result | size |
default | \(-\frac {2 \left (i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a^{3}+6 i \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, a \,b^{2}+i \sec \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) a^{3}+6 i \sec \left (f x +e \right ) \sqrt {\frac {1}{\cos \left (f x +e \right )+1}}\, \sqrt {\frac {\cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, F\left (i \left (\csc \left (f x +e \right )-\cot \left (f x +e \right )\right ), i\right ) a \,b^{2}+3 \cos \left (f x +e \right ) a^{2} b -\cos \left (f x +e \right ) b^{3}-a^{3} \sin \left (f x +e \right )+3 a \,b^{2} \sin \left (f x +e \right )-3 \sec \left (f x +e \right ) b^{3}\right )}{3 d f \sqrt {d \sec \left (f x +e \right )}}\) | \(323\) |
parts | \(\text {Expression too large to display}\) | \(1146\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.10 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.01 \[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\frac {\sqrt {2} {\left (-i \, a^{3} - 6 i \, a b^{2}\right )} \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right ) + \sqrt {2} {\left (i \, a^{3} + 6 i \, a b^{2}\right )} \sqrt {d} {\rm weierstrassPInverse}\left (-4, 0, \cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right ) + 2 \, {\left (3 \, b^{3} - {\left (3 \, a^{2} b - b^{3}\right )} \cos \left (f x + e\right )^{2} + {\left (a^{3} - 3 \, a b^{2}\right )} \cos \left (f x + e\right ) \sin \left (f x + e\right )\right )} \sqrt {\frac {d}{\cos \left (f x + e\right )}}}{3 \, d^{2} f} \]
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\[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\int \frac {\left (a + b \tan {\left (e + f x \right )}\right )^{3}}{\left (d \sec {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\int { \frac {{\left (b \tan \left (f x + e\right ) + a\right )}^{3}}{\left (d \sec \left (f x + e\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(a+b \tan (e+f x))^3}{(d \sec (e+f x))^{3/2}} \, dx=\int \frac {{\left (a+b\,\mathrm {tan}\left (e+f\,x\right )\right )}^3}{{\left (\frac {d}{\cos \left (e+f\,x\right )}\right )}^{3/2}} \,d x \]
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