Integrand size = 25, antiderivative size = 236 \[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=-\frac {2 (a-b) \sqrt {\cos ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right ) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {a (2 a-b) \sqrt {\cos ^2(e+f x)} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right ) \sec (e+f x) \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}{3 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {2 (a-b) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{3 f}+\frac {(a+b) \sec ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{3 f} \]
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Time = 0.31 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3271, 424, 541, 538, 437, 435, 432, 430} \[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {a (2 a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {\frac {b \sin ^2(e+f x)}{a}+1} \operatorname {EllipticF}\left (\arcsin (\sin (e+f x)),-\frac {b}{a}\right )}{3 f \sqrt {a+b \sin ^2(e+f x)}}-\frac {2 (a-b) \sqrt {\cos ^2(e+f x)} \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} E\left (\arcsin (\sin (e+f x))\left |-\frac {b}{a}\right .\right )}{3 f \sqrt {\frac {b \sin ^2(e+f x)}{a}+1}}+\frac {2 (a-b) \tan (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f}+\frac {(a+b) \tan (e+f x) \sec ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)}}{3 f} \]
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Rule 424
Rule 430
Rule 432
Rule 435
Rule 437
Rule 538
Rule 541
Rule 3271
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {\left (a+b x^2\right )^{3/2}}{\left (1-x^2\right )^{5/2}} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {(a+b) \sec ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{3 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-a (2 a-b)-(a-2 b) b x^2}{\left (1-x^2\right )^{3/2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 f} \\ & = \frac {2 (a-b) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{3 f}+\frac {(a+b) \sec ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{3 f}-\frac {\left (\sqrt {\cos ^2(e+f x)} \sec (e+f x)\right ) \text {Subst}\left (\int \frac {-a b (a+b)+2 b \left (a^2-b^2\right ) x^2}{\sqrt {1-x^2} \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{3 (a+b) f} \\ & = \frac {2 (a-b) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{3 f}+\frac {(a+b) \sec ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{3 f}+\frac {\left (a (2 a-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 f}-\frac {\left (2 \left (a^2-b^2\right ) \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) f} \\ & = \frac {2 (a-b) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{3 f}+\frac {(a+b) \sec ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{3 f}-\frac {\left (2 \left (a^2-b^2\right ) \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) f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {\left (a (2 a-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 f \sqrt {a+b \sin ^2(e+f x)}} \\ & = -\frac {2 (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 f \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}+\frac {a (2 a-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 f \sqrt {a+b \sin ^2(e+f x)}}+\frac {2 (a-b) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{3 f}+\frac {(a+b) \sec ^2(e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{3 f} \\ \end{align*}
Time = 2.37 (sec) , antiderivative size = 190, normalized size of antiderivative = 0.81 \[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {-4 a (a-b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} E\left (e+f x\left |-\frac {b}{a}\right .\right )+2 a (2 a-b) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}} \operatorname {EllipticF}\left (e+f x,-\frac {b}{a}\right )+\frac {\left (8 a^2+3 a b+b^2+\left (4 a^2-6 a b-2 b^2\right ) \cos (2 (e+f x))+b (-a+b) \cos (4 (e+f x))\right ) \sec ^2(e+f x) \tan (e+f x)}{\sqrt {2}}}{6 f \sqrt {2 a+b-b \cos (2 (e+f x))}} \]
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Time = 3.73 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.59
method | result | size |
default | \(\frac {2 \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, b \left (a -b \right ) \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (2 a^{2}-a b -3 b^{2}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-\sqrt {-\frac {b \left (\cos ^{2}\left (f x +e \right )\right )}{a}+\frac {a +b}{a}}\, \sqrt {\frac {\cos \left (2 f x +2 e \right )}{2}+\frac {1}{2}}\, \sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, a \left (2 F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a -F\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b -2 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) a +2 E\left (\sin \left (f x +e \right ), \sqrt {-\frac {b}{a}}\right ) b \right ) \left (\cos ^{2}\left (f x +e \right )\right )-\sqrt {-b \left (\cos ^{4}\left (f x +e \right )\right )+\left (a +b \right ) \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (a^{2}+2 a b +b^{2}\right ) \sin \left (f x +e \right )}{3 \sqrt {-\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right ) \left (\sin \left (f x +e \right )-1\right ) \left (1+\sin \left (f x +e \right )\right )}\, \left (\sin \left (f x +e \right )-1\right ) \left (1+\sin \left (f x +e \right )\right ) \cos \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\, f}\) | \(375\) |
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Result contains complex when optimal does not.
Time = 0.19 (sec) , antiderivative size = 784, normalized size of antiderivative = 3.32 \[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {{\left (2 \, {\left (-i \, a b + i \, b^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} \cos \left (f x + e\right )^{3} - {\left (2 i \, a^{2} - i \, a b - i \, b^{2}\right )} \sqrt {-b} \cos \left (f x + e\right )^{3}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left (i \, a b - i \, b^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} \cos \left (f x + e\right )^{3} - {\left (-2 i \, a^{2} + i \, a b + i \, b^{2}\right )} \sqrt {-b} \cos \left (f x + e\right )^{3}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} E(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left (i \, a b - 2 i \, b^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} \cos \left (f x + e\right )^{3} - {\left (-2 i \, a^{2} - i \, a b\right )} \sqrt {-b} \cos \left (f x + e\right )^{3}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) + i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left (-i \, a b + 2 i \, b^{2}\right )} \sqrt {-b} \sqrt {\frac {a^{2} + a b}{b^{2}}} \cos \left (f x + e\right )^{3} - {\left (2 i \, a^{2} + i \, a b\right )} \sqrt {-b} \cos \left (f x + e\right )^{3}\right )} \sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} F(\arcsin \left (\sqrt {\frac {2 \, b \sqrt {\frac {a^{2} + a b}{b^{2}}} + 2 \, a + b}{b}} {\left (\cos \left (f x + e\right ) - i \, \sin \left (f x + e\right )\right )}\right )\,|\,\frac {8 \, a^{2} + 8 \, a b + b^{2} - 4 \, {\left (2 \, a b + b^{2}\right )} \sqrt {\frac {a^{2} + a b}{b^{2}}}}{b^{2}}) + {\left (2 \, {\left (a b - b^{2}\right )} \cos \left (f x + e\right )^{2} + a b + b^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{3 \, b f \cos \left (f x + e\right )^{3}} \]
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Timed out. \[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
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\[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int { {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sec \left (f x + e\right )^{4} \,d x } \]
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Timed out. \[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
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Timed out. \[ \int \sec ^4(e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int \frac {{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}}{{\cos \left (e+f\,x\right )}^4} \,d x \]
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