Integrand size = 23, antiderivative size = 154 \[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^{3/2} \, dx=\frac {a e^{3/2} \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {a e^{3/2} \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {2 a e^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d \sqrt {e \sin (c+d x)}}-\frac {2 a e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d} \]
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Time = 0.23 (sec) , antiderivative size = 154, 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, 327, 335, 218, 212, 209, 2715, 2721, 2720} \[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^{3/2} \, dx=\frac {a e^{3/2} \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {a e^{3/2} \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {2 a e^2 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 d \sqrt {e \sin (c+d x)}}-\frac {2 a e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d} \]
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Rule 209
Rule 212
Rule 218
Rule 327
Rule 335
Rule 2644
Rule 2715
Rule 2720
Rule 2721
Rule 2917
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int (-a-a \cos (c+d x)) \sec (c+d x) (e \sin (c+d x))^{3/2} \, dx \\ & = a \int (e \sin (c+d x))^{3/2} \, dx+a \int \sec (c+d x) (e \sin (c+d x))^{3/2} \, dx \\ & = -\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {a \text {Subst}\left (\int \frac {x^{3/2}}{1-\frac {x^2}{e^2}} \, dx,x,e \sin (c+d x)\right )}{d e}+\frac {1}{3} \left (a e^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx \\ & = -\frac {2 a e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {(a e) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{e^2}\right )} \, dx,x,e \sin (c+d x)\right )}{d}+\frac {\left (a e^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 \sqrt {e \sin (c+d x)}} \\ & = \frac {2 a e^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d \sqrt {e \sin (c+d x)}}-\frac {2 a e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {(2 a e) \text {Subst}\left (\int \frac {1}{1-\frac {x^4}{e^2}} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d} \\ & = \frac {2 a e^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d \sqrt {e \sin (c+d x)}}-\frac {2 a e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d}+\frac {\left (a e^2\right ) \text {Subst}\left (\int \frac {1}{e-x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d}+\frac {\left (a e^2\right ) \text {Subst}\left (\int \frac {1}{e+x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{d} \\ & = \frac {a e^{3/2} \arctan \left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {a e^{3/2} \text {arctanh}\left (\frac {\sqrt {e \sin (c+d x)}}{\sqrt {e}}\right )}{d}+\frac {2 a e^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 d \sqrt {e \sin (c+d x)}}-\frac {2 a e \sqrt {e \sin (c+d x)}}{d}-\frac {2 a e \cos (c+d x) \sqrt {e \sin (c+d x)}}{3 d} \\ \end{align*}
Time = 0.62 (sec) , antiderivative size = 170, normalized size of antiderivative = 1.10 \[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^{3/2} \, dx=\frac {a (e \sin (c+d x))^{3/2} \left (12 \arctan \left (\sqrt {\sin (c+d x)}\right )+6 \text {arctanh}\left (\sqrt {\sin (c+d x)}\right )-8 \operatorname {EllipticF}\left (\frac {1}{4} (-2 c+\pi -2 d x),2\right )-3 \log \left (1-\sqrt {\sin (c+d x)}\right )+3 \log \left (1+\sqrt {\sin (c+d x)}\right )-24 \sqrt {\sin (c+d x)}-8 \cos (c+d x) \sec (2 (c+d x)) \sqrt {\sin (c+d x)}+16 \cos (c+d x) \sec (2 (c+d x)) \sin ^{\frac {5}{2}}(c+d x)\right )}{12 d \sin ^{\frac {3}{2}}(c+d x)} \]
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Time = 10.28 (sec) , antiderivative size = 154, normalized size of antiderivative = 1.00
method | result | size |
default | \(\frac {a \,e^{\frac {3}{2}} \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )+a \,e^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )-2 a e \sqrt {e \sin \left (d x +c \right )}-\frac {a \,e^{2} \left (\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (d x +c \right )^{3}+2 \sin \left (d x +c \right )\right )}{3 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(154\) |
parts | \(-\frac {a \,e^{2} \left (\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )-2 \sin \left (d x +c \right )^{3}+2 \sin \left (d x +c \right )\right )}{3 \cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}\, d}+\frac {a \left (e^{\frac {3}{2}} \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )+e^{\frac {3}{2}} \operatorname {arctanh}\left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\sqrt {e}}\right )-2 e \sqrt {e \sin \left (d x +c \right )}\right )}{d}\) | \(156\) |
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 0.37 (sec) , antiderivative size = 603, normalized size of antiderivative = 3.92 \[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^{3/2} \, dx=\left [\frac {8 \, \sqrt {2} a \sqrt {-i \, e} e {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 8 \, \sqrt {2} a \sqrt {i \, e} e {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) - 6 \, a \sqrt {-e} 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 ) + 3 \, a \sqrt {-e} 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 ) - 16 \, {\left (a e \cos \left (d x + c\right ) + 3 \, a e\right )} \sqrt {e \sin \left (d x + c\right )}}{24 \, d}, \frac {8 \, \sqrt {2} a \sqrt {-i \, e} e {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) + i \, \sin \left (d x + c\right )\right ) + 8 \, \sqrt {2} a \sqrt {i \, e} e {\rm weierstrassPInverse}\left (4, 0, \cos \left (d x + c\right ) - i \, \sin \left (d x + c\right )\right ) + 6 \, a e^{\frac {3}{2}} \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 ) + 3 \, a e^{\frac {3}{2}} \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 ) - 16 \, {\left (a e \cos \left (d x + c\right ) + 3 \, a e\right )} \sqrt {e \sin \left (d x + c\right )}}{24 \, d}\right ] \]
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Timed out. \[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^{3/2} \, dx=\text {Timed out} \]
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\[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}} \,d x } \]
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\[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^{3/2} \, dx=\int { {\left (a \sec \left (d x + c\right ) + a\right )} \left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}} \,d x } \]
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Timed out. \[ \int (a+a \sec (c+d x)) (e \sin (c+d x))^{3/2} \, dx=\int {\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}\,\left (a+\frac {a}{\cos \left (c+d\,x\right )}\right ) \,d x \]
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