Integrand size = 25, antiderivative size = 104 \[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{f}-\frac {a \sqrt {a+b \sec ^2(e+f x)}}{f}-\frac {\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 f}+\frac {\left (a+b \sec ^2(e+f x)\right )^{5/2}}{5 b f} \]
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Time = 0.15 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.240, Rules used = {4224, 457, 81, 52, 65, 214} \[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {\left (a+b \sec ^2(e+f x)\right )^{5/2}}{5 b f}-\frac {\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 f}-\frac {a \sqrt {a+b \sec ^2(e+f x)}}{f} \]
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Rule 52
Rule 65
Rule 81
Rule 214
Rule 457
Rule 4224
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {\left (-1+x^2\right ) \left (a+b x^2\right )^{3/2}}{x} \, dx,x,\sec (e+f x)\right )}{f} \\ & = \frac {\text {Subst}\left (\int \frac {(-1+x) (a+b x)^{3/2}}{x} \, dx,x,\sec ^2(e+f x)\right )}{2 f} \\ & = \frac {\left (a+b \sec ^2(e+f x)\right )^{5/2}}{5 b f}-\frac {\text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,\sec ^2(e+f x)\right )}{2 f} \\ & = -\frac {\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 f}+\frac {\left (a+b \sec ^2(e+f x)\right )^{5/2}}{5 b f}-\frac {a \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\sec ^2(e+f x)\right )}{2 f} \\ & = -\frac {a \sqrt {a+b \sec ^2(e+f x)}}{f}-\frac {\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 f}+\frac {\left (a+b \sec ^2(e+f x)\right )^{5/2}}{5 b f}-\frac {a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sec ^2(e+f x)\right )}{2 f} \\ & = -\frac {a \sqrt {a+b \sec ^2(e+f x)}}{f}-\frac {\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 f}+\frac {\left (a+b \sec ^2(e+f x)\right )^{5/2}}{5 b f}-\frac {a^2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sec ^2(e+f x)}\right )}{b f} \\ & = \frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )}{f}-\frac {a \sqrt {a+b \sec ^2(e+f x)}}{f}-\frac {\left (a+b \sec ^2(e+f x)\right )^{3/2}}{3 f}+\frac {\left (a+b \sec ^2(e+f x)\right )^{5/2}}{5 b f} \\ \end{align*}
Time = 1.72 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.95 \[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=\frac {15 a^{3/2} b \text {arctanh}\left (\frac {\sqrt {a+b \sec ^2(e+f x)}}{\sqrt {a}}\right )-15 a b \sqrt {a+b \sec ^2(e+f x)}-5 b \left (a+b \sec ^2(e+f x)\right )^{3/2}+3 \left (a+b \sec ^2(e+f x)\right )^{5/2}}{15 b f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(517\) vs. \(2(88)=176\).
Time = 14.22 (sec) , antiderivative size = 518, normalized size of antiderivative = 4.98
method | result | size |
default | \(\frac {\left (a +b \sec \left (f x +e \right )^{2}\right )^{\frac {3}{2}} \left (15 \cos \left (f x +e \right )^{3} a^{\frac {3}{2}} \ln \left (4 \cos \left (f x +e \right ) \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \sqrt {a}+4 \cos \left (f x +e \right ) a +4 \sqrt {a}\, \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\right ) b +3 \cos \left (f x +e \right )^{3} \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, a^{2}-20 \cos \left (f x +e \right )^{3} \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, a b +3 \cos \left (f x +e \right )^{2} \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, a^{2}-20 \cos \left (f x +e \right )^{2} \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, a b +6 \cos \left (f x +e \right ) \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, a b -5 \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, b^{2} \cos \left (f x +e \right )+6 \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, a b -5 \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, b^{2}+3 b^{2} \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \sec \left (f x +e \right )+3 b^{2} \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \sec \left (f x +e \right )^{2}\right )}{15 f b \sqrt {\frac {b +a \cos \left (f x +e \right )^{2}}{\left (1+\cos \left (f x +e \right )\right )^{2}}}\, \left (b +a \cos \left (f x +e \right )^{2}\right ) \left (1+\cos \left (f x +e \right )\right )}\) | \(518\) |
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Leaf count of result is larger than twice the leaf count of optimal. 198 vs. \(2 (88) = 176\).
Time = 1.97 (sec) , antiderivative size = 443, normalized size of antiderivative = 4.26 \[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=\left [\frac {15 \, a^{\frac {3}{2}} b \cos \left (f x + e\right )^{4} \log \left (128 \, a^{4} \cos \left (f x + e\right )^{8} + 256 \, a^{3} b \cos \left (f x + e\right )^{6} + 160 \, a^{2} b^{2} \cos \left (f x + e\right )^{4} + 32 \, a b^{3} \cos \left (f x + e\right )^{2} + b^{4} + 8 \, {\left (16 \, a^{3} \cos \left (f x + e\right )^{8} + 24 \, a^{2} b \cos \left (f x + e\right )^{6} + 10 \, a b^{2} \cos \left (f x + e\right )^{4} + b^{3} \cos \left (f x + e\right )^{2}\right )} \sqrt {a} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}\right ) + 8 \, {\left ({\left (3 \, a^{2} - 20 \, a b\right )} \cos \left (f x + e\right )^{4} + {\left (6 \, a b - 5 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 3 \, b^{2}\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{120 \, b f \cos \left (f x + e\right )^{4}}, -\frac {15 \, \sqrt {-a} a b \arctan \left (\frac {{\left (8 \, a^{2} \cos \left (f x + e\right )^{4} + 8 \, a b \cos \left (f x + e\right )^{2} + b^{2}\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{4 \, {\left (2 \, a^{3} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b \cos \left (f x + e\right )^{2} + a b^{2}\right )}}\right ) \cos \left (f x + e\right )^{4} - 4 \, {\left ({\left (3 \, a^{2} - 20 \, a b\right )} \cos \left (f x + e\right )^{4} + {\left (6 \, a b - 5 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + 3 \, b^{2}\right )} \sqrt {\frac {a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{60 \, b f \cos \left (f x + e\right )^{4}}\right ] \]
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\[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=\int \left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \tan ^{3}{\left (e + f x \right )}\, dx \]
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\[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=\int { {\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \tan \left (f x + e\right )^{3} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 1431 vs. \(2 (88) = 176\).
Time = 1.67 (sec) , antiderivative size = 1431, normalized size of antiderivative = 13.76 \[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=\text {Too large to display} \]
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Timed out. \[ \int \left (a+b \sec ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^3\,{\left (a+\frac {b}{{\cos \left (e+f\,x\right )}^2}\right )}^{3/2} \,d x \]
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