Integrand size = 23, antiderivative size = 104 \[ \int \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{8 \sqrt {b} f}+\frac {3 a \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 f}+\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 f} \]
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Time = 0.08 (sec) , antiderivative size = 104, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.174, Rules used = {3269, 201, 223, 212} \[ \int \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{8 \sqrt {b} f}+\frac {3 a \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 f}+\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 f} \]
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Rule 201
Rule 212
Rule 223
Rule 3269
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \left (a+b x^2\right )^{3/2} \, dx,x,\sin (e+f x)\right )}{f} \\ & = \frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 f}+\frac {(3 a) \text {Subst}\left (\int \sqrt {a+b x^2} \, dx,x,\sin (e+f x)\right )}{4 f} \\ & = \frac {3 a \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 f}+\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 f}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sin (e+f x)\right )}{8 f} \\ & = \frac {3 a \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 f}+\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 f}+\frac {\left (3 a^2\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{8 f} \\ & = \frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a+b \sin ^2(e+f x)}}\right )}{8 \sqrt {b} f}+\frac {3 a \sin (e+f x) \sqrt {a+b \sin ^2(e+f x)}}{8 f}+\frac {\sin (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2}}{4 f} \\ \end{align*}
Time = 0.51 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.89 \[ \int \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {\sqrt {a+b \sin ^2(e+f x)} \left (5 a \sin (e+f x)+2 b \sin ^3(e+f x)+\frac {3 a^{3/2} \text {arcsinh}\left (\frac {\sqrt {b} \sin (e+f x)}{\sqrt {a}}\right )}{\sqrt {b} \sqrt {1+\frac {b \sin ^2(e+f x)}{a}}}\right )}{8 f} \]
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Time = 0.35 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.87
method | result | size |
derivativedivides | \(\frac {\sin \left (f x +e \right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}}}{4 f}+\frac {3 a \sin \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{8 f}+\frac {3 a^{2} \ln \left (\sqrt {b}\, \sin \left (f x +e \right )+\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\right )}{8 f \sqrt {b}}\) | \(90\) |
default | \(\frac {\sin \left (f x +e \right ) {\left (a +b \left (\sin ^{2}\left (f x +e \right )\right )\right )}^{\frac {3}{2}}}{4 f}+\frac {3 a \sin \left (f x +e \right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{8 f}+\frac {3 a^{2} \ln \left (\sqrt {b}\, \sin \left (f x +e \right )+\sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}\right )}{8 f \sqrt {b}}\) | \(90\) |
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Leaf count of result is larger than twice the leaf count of optimal. 197 vs. \(2 (88) = 176\).
Time = 0.51 (sec) , antiderivative size = 503, normalized size of antiderivative = 4.84 \[ \int \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\left [\frac {3 \, a^{2} \sqrt {b} \log \left (128 \, b^{4} \cos \left (f x + e\right )^{8} - 256 \, {\left (a b^{3} + 2 \, b^{4}\right )} \cos \left (f x + e\right )^{6} + 32 \, {\left (5 \, a^{2} b^{2} + 24 \, a b^{3} + 24 \, b^{4}\right )} \cos \left (f x + e\right )^{4} + a^{4} + 32 \, a^{3} b + 160 \, a^{2} b^{2} + 256 \, a b^{3} + 128 \, b^{4} - 32 \, {\left (a^{3} b + 10 \, a^{2} b^{2} + 24 \, a b^{3} + 16 \, b^{4}\right )} \cos \left (f x + e\right )^{2} - 8 \, {\left (16 \, b^{3} \cos \left (f x + e\right )^{6} - 24 \, {\left (a b^{2} + 2 \, b^{3}\right )} \cos \left (f x + e\right )^{4} - a^{3} - 10 \, a^{2} b - 24 \, a b^{2} - 16 \, b^{3} + 2 \, {\left (5 \, a^{2} b + 24 \, a b^{2} + 24 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {b} \sin \left (f x + e\right )\right ) - 8 \, {\left (2 \, b^{2} \cos \left (f x + e\right )^{2} - 5 \, a b - 2 \, b^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{64 \, b f}, -\frac {3 \, a^{2} \sqrt {-b} \arctan \left (\frac {{\left (8 \, b^{2} \cos \left (f x + e\right )^{4} - 8 \, {\left (a b + 2 \, b^{2}\right )} \cos \left (f x + e\right )^{2} + a^{2} + 8 \, a b + 8 \, b^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-b}}{4 \, {\left (2 \, b^{3} \cos \left (f x + e\right )^{4} + a^{2} b + 3 \, a b^{2} + 2 \, b^{3} - {\left (3 \, a b^{2} + 4 \, b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )}\right ) + 4 \, {\left (2 \, b^{2} \cos \left (f x + e\right )^{2} - 5 \, a b - 2 \, b^{2}\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sin \left (f x + e\right )}{32 \, b f}\right ] \]
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Timed out. \[ \int \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
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none
Time = 0.33 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.70 \[ \int \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {\frac {3 \, a^{2} \operatorname {arsinh}\left (\frac {b \sin \left (f x + e\right )}{\sqrt {a b}}\right )}{\sqrt {b}} + 2 \, {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} \sin \left (f x + e\right ) + 3 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} a \sin \left (f x + e\right )}{8 \, f} \]
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Timed out. \[ \int \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
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Time = 14.24 (sec) , antiderivative size = 60, normalized size of antiderivative = 0.58 \[ \int \cos (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {\sin \left (e+f\,x\right )\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2}\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},\frac {1}{2};\ \frac {3}{2};\ -\frac {b\,{\sin \left (e+f\,x\right )}^2}{a}\right )}{f\,{\left (\frac {b\,{\sin \left (e+f\,x\right )}^2}{a}+1\right )}^{3/2}} \]
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