Integrand size = 23, antiderivative size = 78 \[ \int \cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=-\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {a \sqrt {a+b \sin ^2(e+f x)}}{f}+\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f} \]
[Out]
Time = 0.10 (sec) , antiderivative size = 78, 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 = {3273, 52, 65, 214} \[ \int \cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=-\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {a \sqrt {a+b \sin ^2(e+f x)}}{f}+\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f} \]
[In]
[Out]
Rule 52
Rule 65
Rule 214
Rule 3273
Rubi steps \begin{align*} \text {integral}& = \frac {\text {Subst}\left (\int \frac {(a+b x)^{3/2}}{x} \, dx,x,\sin ^2(e+f x)\right )}{2 f} \\ & = \frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}+\frac {a \text {Subst}\left (\int \frac {\sqrt {a+b x}}{x} \, dx,x,\sin ^2(e+f x)\right )}{2 f} \\ & = \frac {a \sqrt {a+b \sin ^2(e+f x)}}{f}+\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}+\frac {a^2 \text {Subst}\left (\int \frac {1}{x \sqrt {a+b x}} \, dx,x,\sin ^2(e+f x)\right )}{2 f} \\ & = \frac {a \sqrt {a+b \sin ^2(e+f x)}}{f}+\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f}+\frac {a^2 \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b \sin ^2(e+f x)}\right )}{b f} \\ & = -\frac {a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )}{f}+\frac {a \sqrt {a+b \sin ^2(e+f x)}}{f}+\frac {\left (a+b \sin ^2(e+f x)\right )^{3/2}}{3 f} \\ \end{align*}
Time = 0.12 (sec) , antiderivative size = 69, normalized size of antiderivative = 0.88 \[ \int \cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\frac {-3 a^{3/2} \text {arctanh}\left (\frac {\sqrt {a+b \sin ^2(e+f x)}}{\sqrt {a}}\right )+\sqrt {a+b \sin ^2(e+f x)} \left (4 a+b \sin ^2(e+f x)\right )}{3 f} \]
[In]
[Out]
Time = 0.93 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.10
method | result | size |
default | \(\frac {\frac {b \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{3}+\frac {4 a \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{3}-a^{\frac {3}{2}} \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {a +b \left (\sin ^{2}\left (f x +e \right )\right )}}{\sin \left (f x +e \right )}\right )}{f}\) | \(86\) |
[In]
[Out]
none
Time = 0.72 (sec) , antiderivative size = 175, normalized size of antiderivative = 2.24 \[ \int \cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\left [\frac {3 \, a^{\frac {3}{2}} \log \left (\frac {2 \, {\left (b \cos \left (f x + e\right )^{2} + 2 \, \sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {a} - 2 \, a - b\right )}}{\cos \left (f x + e\right )^{2} - 1}\right ) - 2 \, {\left (b \cos \left (f x + e\right )^{2} - 4 \, a - b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{6 \, f}, \frac {3 \, \sqrt {-a} a \arctan \left (\frac {\sqrt {-b \cos \left (f x + e\right )^{2} + a + b} \sqrt {-a}}{a}\right ) - {\left (b \cos \left (f x + e\right )^{2} - 4 \, a - b\right )} \sqrt {-b \cos \left (f x + e\right )^{2} + a + b}}{3 \, f}\right ] \]
[In]
[Out]
\[ \int \cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int \left (a + b \sin ^{2}{\left (e + f x \right )}\right )^{\frac {3}{2}} \cot {\left (e + f x \right )}\, dx \]
[In]
[Out]
none
Time = 0.20 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.78 \[ \int \cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=-\frac {3 \, a^{\frac {3}{2}} \operatorname {arsinh}\left (\frac {a}{\sqrt {a b} {\left | \sin \left (f x + e\right ) \right |}}\right ) - {\left (b \sin \left (f x + e\right )^{2} + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b \sin \left (f x + e\right )^{2} + a} a}{3 \, f} \]
[In]
[Out]
Timed out. \[ \int \cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\text {Timed out} \]
[In]
[Out]
Timed out. \[ \int \cot (e+f x) \left (a+b \sin ^2(e+f x)\right )^{3/2} \, dx=\int \mathrm {cot}\left (e+f\,x\right )\,{\left (b\,{\sin \left (e+f\,x\right )}^2+a\right )}^{3/2} \,d x \]
[In]
[Out]