Integrand size = 23, antiderivative size = 155 \[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=-\frac {a \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{3/2}}+\frac {a \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{3/2}}-\frac {2 a}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}} \]
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Time = 0.23 (sec) , antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.478, Rules used = {3957, 2917, 2644, 331, 335, 304, 209, 212, 2716, 2721, 2719} \[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=-\frac {a \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{3/2}}+\frac {a \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{3/2}}-\frac {2 a E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}}-\frac {2 a}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}} \]
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Rule 209
Rule 212
Rule 304
Rule 331
Rule 335
Rule 2644
Rule 2716
Rule 2719
Rule 2721
Rule 2917
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {(-a-a \cos (c+d x)) \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx \\ & = a \int \frac {1}{(e \sin (c+d x))^{3/2}} \, dx+a \int \frac {\sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx \\ & = -\frac {2 a \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}-\frac {a \int \sqrt {e \sin (c+d x)} \, dx}{e^2}+\frac {a \text {Subst}\left (\int \frac {1}{x^{3/2} \left (1-\frac {x^2}{e^2}\right )} \, dx,x,e \sin (c+d x)\right )}{d e} \\ & = -\frac {2 a}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}+\frac {a \text {Subst}\left (\int \frac {\sqrt {x}}{1-\frac {x^2}{e^2}} \, dx,x,e \sin (c+d x)\right )}{d e^3}-\frac {\left (a \sqrt {e \sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{e^2 \sqrt {\sin (c+d x)}} \\ & = -\frac {2 a}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}}+\frac {(2 a) \text {Subst}\left (\int \frac {x^2}{1-\frac {x^4}{e^2}} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d e^3} \\ & = -\frac {2 a}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}}+\frac {a \text {Subst}\left (\int \frac {1}{e-x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d e}-\frac {a \text {Subst}\left (\int \frac {1}{e+x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d e} \\ & = -\frac {a \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{3/2}}+\frac {a \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d e^{3/2}}-\frac {2 a}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a \cos (c+d x)}{d e \sqrt {e \sin (c+d x)}}-\frac {2 a E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {e \sin (c+d x)}}{d e^2 \sqrt {\sin (c+d x)}} \\ \end{align*}
Time = 0.57 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.92 \[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=-\frac {a (1+\cos (c+d x)) \sec \left (\frac {1}{2} (c+d x)\right ) \left (\arctan \left (\sqrt {\sin (c+d x)}\right ) \sec \left (\frac {1}{2} (c+d x)\right )-\text {arctanh}\left (\sqrt {\sin (c+d x)}\right ) \sec \left (\frac {1}{2} (c+d x)\right )-2 E\left (\left .\frac {1}{4} (-2 c+\pi -2 d x)\right |2\right ) \sec \left (\frac {1}{2} (c+d x)\right )+2 \csc \left (\frac {1}{2} (c+d x)\right ) \sqrt {\sin (c+d x)}\right ) \sin ^{\frac {3}{2}}(c+d x)}{2 d (e \sin (c+d x))^{3/2}} \]
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Time = 10.07 (sec) , antiderivative size = 201, normalized size of antiderivative = 1.30
method | result | size |
default | \(\frac {-\frac {a \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{e^{\frac {3}{2}}}-\frac {2 a}{e \sqrt {e \sin \left (d x +c \right )}}+\frac {a \,\operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{e^{\frac {3}{2}}}+\frac {a \left (2 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \cos \left (d x +c \right )^{2}\right )}{e \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(201\) |
parts | \(\frac {a \left (2 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticE}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \cos \left (d x +c \right )^{2}\right )}{e \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {a \left (-\frac {\arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{e^{\frac {3}{2}}}-\frac {2}{e \sqrt {e \sin \left (d x +c \right )}}+\frac {\operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )}{e^{\frac {3}{2}}}\right )}{d}\) | \(203\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.20 (sec) , antiderivative size = 670, normalized size of antiderivative = 4.32 \[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=\left [\frac {-8 i \, \sqrt {2} a \sqrt {-i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 8 i \, \sqrt {2} a \sqrt {i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, a \sqrt {-e} \arctan \left (\frac {{\left (\cos \left (d x + c\right )^{2} - 6 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {e \sin \left (d x + c\right )} \sqrt {-e}}{4 \, {\left (e \cos \left (d x + c\right )^{2} - e \sin \left (d x + c\right ) - e\right )}}\right ) \sin \left (d x + c\right ) - a \sqrt {-e} \log \left (\frac {e \cos \left (d x + c\right )^{4} - 72 \, e \cos \left (d x + c\right )^{2} + 8 \, {\left (7 \, \cos \left (d x + c\right )^{2} - {\left (\cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) - 8\right )} \sqrt {e \sin \left (d x + c\right )} \sqrt {-e} + 28 \, {\left (e \cos \left (d x + c\right )^{2} - 2 \, e\right )} \sin \left (d x + c\right ) + 72 \, e}{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8}\right ) \sin \left (d x + c\right ) - 16 \, {\left (a \cos \left (d x + c\right ) + a\right )} \sqrt {e \sin \left (d x + c\right )}}{8 \, d e^{2} \sin \left (d x + c\right )}, \frac {-8 i \, \sqrt {2} a \sqrt {-i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right )\right ) + 8 i \, \sqrt {2} a \sqrt {i \, e} \sin \left (d x + c\right ) {\rm weierstrassZeta}\left (4, 0, {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right )\right ) - 2 \, a \sqrt {e} \arctan \left (\frac {{\left (\cos \left (d x + c\right )^{2} + 6 \, \sin \left (d x + c\right ) - 2\right )} \sqrt {e \sin \left (d x + c\right )} \sqrt {e}}{4 \, {\left (e \cos \left (d x + c\right )^{2} + e \sin \left (d x + c\right ) - e\right )}}\right ) \sin \left (d x + c\right ) + a \sqrt {e} \log \left (\frac {e \cos \left (d x + c\right )^{4} - 72 \, e \cos \left (d x + c\right )^{2} - 8 \, {\left (7 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 8\right )} \sin \left (d x + c\right ) - 8\right )} \sqrt {e \sin \left (d x + c\right )} \sqrt {e} - 28 \, {\left (e \cos \left (d x + c\right )^{2} - 2 \, e\right )} \sin \left (d x + c\right ) + 72 \, e}{\cos \left (d x + c\right )^{4} - 8 \, \cos \left (d x + c\right )^{2} + 4 \, {\left (\cos \left (d x + c\right )^{2} - 2\right )} \sin \left (d x + c\right ) + 8}\right ) \sin \left (d x + c\right ) - 16 \, {\left (a \cos \left (d x + c\right ) + a\right )} \sqrt {e \sin \left (d x + c\right )}}{8 \, d e^{2} \sin \left (d x + c\right )}\right ] \]
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\[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=a \left (\int \frac {1}{\left (e \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx + \int \frac {\sec {\left (c + d x \right )}}{\left (e \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx\right ) \]
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Timed out. \[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=\int { \frac {a \sec \left (d x + c\right ) + a}{\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {a+a \sec (c+d x)}{(e \sin (c+d x))^{3/2}} \, dx=\int \frac {a+\frac {a}{\cos \left (c+d\,x\right )}}{{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}} \,d x \]
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