Integrand size = 25, antiderivative size = 82 \[ \int \sec ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {a \text {arctanh}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 \sqrt {a+b} f}+\frac {\sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{2 f} \]
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Time = 0.17 (sec) , antiderivative size = 82, 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 = {3269, 386, 385, 212} \[ \int \sec ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {a \text {arctanh}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 f \sqrt {a+b}}+\frac {\tan (e+f x) \sec (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{2 f} \]
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Rule 212
Rule 385
Rule 386
Rule 3269
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\sqrt {a+b x^2}}{\left (1-x^2\right )^2} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{2 f}+\frac {a \text {Subst}\left (\int \frac {1}{\left (1-x^2\right ) \sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{2 f} \\ & = \frac {\sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{2 f}+\frac {a \text {Subst}\left (\int \frac {1}{1-(a+b) x^2} \, dx,x,\frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 f} \\ & = \frac {a \text {arctanh}\left (\frac {\sqrt {a+b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{2 \sqrt {a+b} f}+\frac {\sec (e+f x) \sqrt {a+b \sin ^2(e+f x)} \tan (e+f x)}{2 f} \\ \end{align*}
Time = 2.23 (sec) , antiderivative size = 164, normalized size of antiderivative = 2.00 \[ \int \sec ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\frac {\sin (e+f x) \left (\sqrt {2} a \text {arctanh}\left (\frac {\sqrt {\frac {(a+b) \sin ^2(e+f x)}{a}}}{\sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}\right ) \sqrt {\frac {2 a+b-b \cos (2 (e+f x))}{a}}+(2 a+b-b \cos (2 (e+f x))) \sec ^2(e+f x) \sqrt {\frac {(a+b) \sin ^2(e+f x)}{a}}\right )}{4 f \sqrt {\frac {(a+b) \sin ^2(e+f x)}{a}} \sqrt {a+b \sin ^2(e+f x)}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(289\) vs. \(2(70)=140\).
Time = 1.34 (sec) , antiderivative size = 290, normalized size of antiderivative = 3.54
method | result | size |
default | \(\frac {2 b \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, \sqrt {a +b}\, \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+a \left (\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) a +\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right ) b -\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) a -\ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right ) b \right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 {\left (a +b -b \left (\cos ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}} \sqrt {a +b}\, \sin \left (f x +e \right )}{4 \left (a +b \right )^{\frac {3}{2}} \cos \left (f x +e \right )^{2} f}\) | \(290\) |
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Leaf count of result is larger than twice the leaf count of optimal. 155 vs. \(2 (70) = 140\).
Time = 0.33 (sec) , antiderivative size = 337, normalized size of antiderivative = 4.11 \[ \int \sec ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\left [\frac {\sqrt {a + b} a \cos \left (f x + e\right )^{2} \log \left (\frac {{\left (a^{2} + 8 \, a b + 8 \, b^{2}\right )} \cos \left (f x + e\right )^{4} - 8 \, {\left (a^{2} + 3 \, a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} - 4 \, {\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - 2 \, a - 2 \, b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a + b} \sin \left (f x + e\right ) + 8 \, a^{2} + 16 \, a b + 8 \, b^{2}}{\cos \left (f x + e\right )^{4}}\right ) + 4 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (a + b\right )} \sin \left (f x + e\right )}{8 \, {\left (a + b\right )} f \cos \left (f x + e\right )^{2}}, -\frac {a \sqrt {-a - b} \arctan \left (\frac {{\left ({\left (a + 2 \, b\right )} \cos \left (f x + e\right )^{2} - 2 \, a - 2 \, b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a - b}}{2 \, {\left ({\left (a b + b^{2}\right )} \cos \left (f x + e\right )^{2} - a^{2} - 2 \, a b - b^{2}\right )} \sin \left (f x + e\right )}\right ) \cos \left (f x + e\right )^{2} - 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} {\left (a + b\right )} \sin \left (f x + e\right )}{4 \, {\left (a + b\right )} f \cos \left (f x + e\right )^{2}}\right ] \]
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\[ \int \sec ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int \sqrt {a + b \sin ^{2}{\left (e + f x \right )}} \sec ^{3}{\left (e + f x \right )}\, dx \]
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\[ \int \sec ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int { \sqrt {b \sin \left (f x + e\right )^{2} + a} \sec \left (f x + e\right )^{3} \,d x } \]
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\[ \int \sec ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int { \sqrt {b \sin \left (f x + e\right )^{2} + a} \sec \left (f x + e\right )^{3} \,d x } \]
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Timed out. \[ \int \sec ^3(e+f x) \sqrt {a+b \sin ^2(e+f x)} \, dx=\int \frac {\sqrt {b\,{\sin \left (e+f\,x\right )}^2+a}}{{\cos \left (e+f\,x\right )}^3} \,d x \]
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