Integrand size = 25, antiderivative size = 348 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {(5 a-3 b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^3 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {8 (a-b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^4 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {8 (a-b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 (a+b)^4 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {(5 a-3 b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 (a+b)^3 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (2 a-b) \tan (e+f x)}{3 (a+b)^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
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Time = 0.52 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3275, 481, 541, 538, 437, 435, 432, 430} \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=-\frac {(5 a-3 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{3 f (a+b)^3 \sqrt {a+b \sin ^2(e+f x)}}+\frac {8 (a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 f (a+b)^4 \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}-\frac {2 (2 a-b) \tan (e+f x)}{3 f (a+b)^2 \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {8 b (a-b) \sin (e+f x) \cos (e+f x)}{3 f (a+b)^4 \sqrt {a+b \sin ^2(e+f x)}}+\frac {b (5 a-3 b) \sin (e+f x) \cos (e+f x)}{3 f (a+b)^3 \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\tan (e+f x) \sec ^2(e+f x)}{3 f (a+b) \left (a+b \sin ^2(e+f x)\right )^{3/2}} \]
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Rule 430
Rule 432
Rule 435
Rule 437
Rule 481
Rule 538
Rule 541
Rule 3275
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {x^4}{\left (1-x^2\right )^{5/2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {a+(3 a-2 b) x^2}{\left (1-x^2\right )^{3/2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{3 (a+b) f} \\ & = -\frac {2 (2 a-b) \tan (e+f x)}{3 (a+b)^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-3 a (a-b)+6 (2 a-b) b x^2}{\sqrt {1-x^2} \left (a+b x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{3 (a+b)^2 f} \\ & = \frac {(5 a-3 b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^3 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {2 (2 a-b) \tan (e+f x)}{3 (a+b)^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {3 a^2 (3 a-5 b)-3 a (5 a-3 b) b x^2}{\sqrt {1-x^2} \left (a+b x^2\right )^{3/2}} \, dx,x,\sin (e+f x)\right )}{9 a (a+b)^3 f} \\ & = \frac {(5 a-3 b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^3 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {8 (a-b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^4 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (2 a-b) \tan (e+f x)}{3 (a+b)^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-3 a^2 (a-3 b) (3 a-b)-24 a^2 (a-b) b x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{9 a^2 (a+b)^4 f} \\ & = \frac {(5 a-3 b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^3 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {8 (a-b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^4 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (2 a-b) \tan (e+f x)}{3 (a+b)^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\left (8 (a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 (a+b)^4 f}-\frac {\left ((5 a-3 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 (a+b)^3 f} \\ & = \frac {(5 a-3 b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^3 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {8 (a-b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^4 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (2 a-b) \tan (e+f x)}{3 (a+b)^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\left (8 (a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {b x^2}{a}}}{\sqrt {1-x^2}} \, dx,x,\sin (e+f x)\right )}{3 (a+b)^4 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {\left ((5 a-3 b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-x^2} \sqrt {1+\frac {b x^2}{a}}} \, dx,x,\sin (e+f x)\right )}{3 (a+b)^3 f \sqrt {a+b \sin ^2(e+f x)}} \\ & = \frac {(5 a-3 b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^3 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {8 (a-b) b \cos (e+f x) \sin (e+f x)}{3 (a+b)^4 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {8 (a-b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 (a+b)^4 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}-\frac {(5 a-3 b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 (a+b)^3 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (2 a-b) \tan (e+f x)}{3 (a+b)^2 f \left (a+b \sin ^2(e+f x)\right )^{3/2}}+\frac {\sec ^2(e+f x) \tan (e+f x)}{3 (a+b) f \left (a+b \sin ^2(e+f x)\right )^{3/2}} \\ \end{align*}
Time = 4.17 (sec) , antiderivative size = 235, normalized size of antiderivative = 0.68 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\frac {2 a b \left (\frac {2 a+b-b \cos (2 (e+f x))}{a}\right )^{3/2} \left (8 a (a-b) E\left (e+f x\left |-\frac {b}{a}\right .\right )+\left (-5 a^2-2 a b+3 b^2\right ) \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )\right )+\sqrt {2} b \left (2 a b (a+b) \sin (2 (e+f x))+4 (a-b) b (2 a+b-b \cos (2 (e+f x))) \sin (2 (e+f x))-4 (a-b) (2 a+b-b \cos (2 (e+f x)))^2 \tan (e+f x)+(a+b) (2 a+b-b \cos (2 (e+f x)))^2 \sec ^2(e+f x) \tan (e+f x)\right )}{6 b (a+b)^4 f (2 a+b-b \cos (2 (e+f x)))^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(666\) vs. \(2(318)=636\).
Time = 5.88 (sec) , antiderivative size = 667, normalized size of antiderivative = 1.92
method | result | size |
default | \(-\frac {-8 \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, b^{2} \left (a -b \right ) \left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, b \left (13 a^{2}+2 a b -11 b^{2}\right ) \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, b \left (5 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}+2 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b -3 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{2}-8 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2}+8 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a b \right ) \left (\cos ^{4}\left (f x +e \right )\right )-2 \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (2 a^{3}+3 a^{2} b -b^{3}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \left (5 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+7 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{2} b -F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}-3 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b^{3}-8 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a^{3}+8 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a \,b^{2}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (a^{3}+3 a^{2} b +3 a \,b^{2}+b^{3}\right ) \sin \left (f x +e \right )}{3 {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} \left (1+\sin \left (f x +e \right )\right ) \sqrt {-\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right ) \left (\sin \left (f x +e \right )-1\right ) \left (1+\sin \left (f x +e \right )\right )}\, \left (\sin \left (f x +e \right )-1\right ) \left (a +b \right )^{4} \cos \left (f x +e \right ) f}\) | \(667\) |
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Result contains complex when optimal does not.
Time = 0.33 (sec) , antiderivative size = 1730, normalized size of antiderivative = 4.97 \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Too large to display} \]
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\[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {\tan ^{4}{\left (e + f x \right )}}{\left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx \]
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Timed out. \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int { \frac {\tan \left (f x + e\right )^{4}}{{\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\tan ^4(e+f x)}{\left (a+b \sin ^2(e+f x)\right )^{5/2}} \, dx=\int \frac {{\mathrm {tan}\left (e+f\,x\right )}^4}{{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{5/2}} \,d x \]
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