Integrand size = 25, antiderivative size = 148 \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=-\frac {\sqrt {a+b} (2 a+5 b) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a+b}}\right )}{2 f}+\frac {(2 a+5 b) \sqrt {a+b \sin ^2(e+f x)}}{2 f}+\frac {(2 a+5 b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{6 (a+b) f}+\frac {\sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{2 (a+b) f} \]
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Time = 0.15 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {3273, 79, 52, 65, 214} \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=-\frac {\sqrt {a+b} (2 a+5 b) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a+b}}\right )}{2 f}+\frac {(2 a+5 b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{6 f (a+b)}+\frac {(2 a+5 b) \sqrt {a+b \sin ^2(e+f x)}}{2 f}+\frac {\sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{2 f (a+b)} \]
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Rule 52
Rule 65
Rule 79
Rule 214
Rule 3273
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {x (a+b x)^{3/2}}{(1-x)^2} \, dx,x,\sin ^2(e+f x)\right )}{2 f} \\ & = \frac {\sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{2 (a+b) f}-\frac {(2 a+5 b) \text {Subst}\left (\int \frac {(a+b x)^{3/2}}{1-x} \, dx,x,\sin ^2(e+f x)\right )}{4 (a+b) f} \\ & = \frac {(2 a+5 b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{6 (a+b) f}+\frac {\sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{2 (a+b) f}-\frac {(2 a+5 b) \text {Subst}\left (\int \frac {\sqrt {a+b x}}{1-x} \, dx,x,\sin ^2(e+f x)\right )}{4 f} \\ & = \frac {(2 a+5 b) \sqrt {a+b \sin ^2(e+f x)}}{2 f}+\frac {(2 a+5 b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{6 (a+b) f}+\frac {\sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{2 (a+b) f}-\frac {((a+b) (2 a+5 b)) \text {Subst}\left (\int \frac {1}{(1-x) \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{4 f} \\ & = \frac {(2 a+5 b) \sqrt {a+b \sin ^2(e+f x)}}{2 f}+\frac {(2 a+5 b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{6 (a+b) f}+\frac {\sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{2 (a+b) f}-\frac {((a+b) (2 a+5 b)) \text {Subst}\left (\int \frac {1}{1+\frac {a}{b}-\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^2(e+f x)}\right )}{2 b f} \\ & = -\frac {\sqrt {a+b} (2 a+5 b) \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a+b}}\right )}{2 f}+\frac {(2 a+5 b) \sqrt {a+b \sin ^2(e+f x)}}{2 f}+\frac {(2 a+5 b) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{6 (a+b) f}+\frac {\sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}}{2 (a+b) f} \\ \end{align*}
Time = 0.54 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.78 \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=\frac {3 \sec ^2(e+f x) \left (a+b \sin ^2(e+f x)\right )^{5/2}+(2 a+5 b) \left (-3 (a+b)^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a+b}}\right )+\sqrt {a+b \sin ^2(e+f x)} \left (4 a+3 b+b \sin ^2(e+f x)\right )\right )}{6 (a+b) f} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(425\) vs. \(2(128)=256\).
Time = 1.78 (sec) , antiderivative size = 426, normalized size of antiderivative = 2.88
method | result | size |
default | \(\frac {-\frac {\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}\, \left (b \left (\cos ^{2}\left (f x +e \right )\right )+2 a -b \right )}{3}+2 b \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}+2 a \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}-\frac {\left (\frac {1}{2} a^{2}+2 a b +\frac {3}{2} b^{2}\right ) \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right )}{\sqrt {a +b}}-\frac {\left (\frac {1}{2} a^{2}+2 a b +\frac {3}{2} b^{2}\right ) \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right )}{\sqrt {a +b}}+\left (\frac {1}{4} a^{2}+\frac {1}{2} a b +\frac {1}{4} b^{2}\right ) \left (-\frac {\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}}{\left (a +b \right ) \left (\sin \left (f x +e \right )-1\right )}+\frac {b \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}+2 b \sin \left (f x +e \right )+2 a}{\sin \left (f x +e \right )-1}\right )}{\left (a +b \right )^{\frac {3}{2}}}\right )+\left (-\frac {1}{4} a^{2}-\frac {1}{2} a b -\frac {1}{4} b^{2}\right ) \left (-\frac {\sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}}{\left (a +b \right ) \left (1+\sin \left (f x +e \right )\right )}-\frac {b \ln \left (\frac {2 \sqrt {a +b}\, \sqrt {a +b -b \left (\cos ^{2}\left (f x +e \right )\right )}-2 b \sin \left (f x +e \right )+2 a}{1+\sin \left (f x +e \right )}\right )}{\left (a +b \right )^{\frac {3}{2}}}\right )}{f}\) | \(426\) |
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Time = 0.53 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.79 \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=\left [\frac {3 \, {\left (2 \, a + 5 \, b\right )} \sqrt {a + b} \cos \left (f x + e\right )^{2} \log \left (\frac {b \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a + b} - 2 \, a - 2 \, b}{\cos \left (f x + e\right )^{2}}\right ) - 2 \, {\left (2 \, b \cos \left (f x + e\right )^{4} - 2 \, {\left (4 \, a + 7 \, b\right )} \cos \left (f x + e\right )^{2} - 3 \, a - 3 \, b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{12 \, f \cos \left (f x + e\right )^{2}}, \frac {3 \, {\left (2 \, a + 5 \, b\right )} \sqrt {-a - b} \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a - b}}{a + b}\right ) \cos \left (f x + e\right )^{2} - {\left (2 \, b \cos \left (f x + e\right )^{4} - 2 \, {\left (4 \, a + 7 \, b\right )} \cos \left (f x + e\right )^{2} - 3 \, a - 3 \, b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{6 \, f \cos \left (f x + e\right )^{2}}\right ] \]
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Timed out. \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=\text {Timed out} \]
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Time = 0.29 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.14 \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=\frac {4 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} b^{2} + 12 \, {\left (a b^{2} + 2 \, b^{3}\right )} \sqrt {b \sin \left (f x + e\right )^{2} + a} + \frac {3 \, {\left (2 \, a^{2} b^{2} + 7 \, a b^{3} + 5 \, b^{4}\right )} \log \left (\frac {\sqrt {b \sin \left (f x + e\right )^{2} + a} - \sqrt {a + b}}{\sqrt {b \sin \left (f x + e\right )^{2} + a} + \sqrt {a + b}}\right )}{\sqrt {a + b}} - \frac {6 \, {\left (a b^{3} + b^{4}\right )} \sqrt {b \sin \left (f x + e\right )^{2} + a}}{b \sin \left (f x + e\right )^{2} - b}}{12 \, b^{2} f} \]
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Exception generated. \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \left (a+b \sin ^2(e+f x)\right )^{3/2} \tan ^3(e+f x) \, dx=\int {\mathrm {tan}\left (e+f\,x\right )}^3\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]
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