Integrand size = 33, antiderivative size = 574 \[ \int \frac {(g \cos (e+f x))^{3/2} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx=\frac {g^{3/2} \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{4 a f}-\frac {b^2 g^{3/2} \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^3 f}+\frac {b^{3/2} \sqrt [4]{-a^2+b^2} g^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a^3 f}+\frac {g^{3/2} \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{4 a f}-\frac {b^2 g^{3/2} \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^3 f}+\frac {b^{3/2} \sqrt [4]{-a^2+b^2} g^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a^3 f}+\frac {b g \sqrt {g \cos (e+f x)} \csc (e+f x)}{a^2 f}-\frac {g \sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{2 a f}-\frac {b g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{a^2 f \sqrt {g \cos (e+f x)}}+\frac {b \left (a^2-b^2\right ) g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a^2 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) f \sqrt {g \cos (e+f x)}}+\frac {b \left (a^2-b^2\right ) g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a^2 \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) f \sqrt {g \cos (e+f x)}} \]
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Time = 1.00 (sec) , antiderivative size = 574, normalized size of antiderivative = 1.00, number of steps used = 30, number of rules used = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.545, Rules used = {2977, 2645, 327, 335, 218, 212, 209, 2647, 2721, 2720, 294, 2774, 2946, 2781, 2886, 2884, 214, 211} \[ \int \frac {(g \cos (e+f x))^{3/2} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx=-\frac {b^2 g^{3/2} \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^3 f}-\frac {b^2 g^{3/2} \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^3 f}+\frac {b g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (e+f x),2\right )}{a^2 f \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {g \cos (e+f x)}}+\frac {b g^2 \left (a^2-b^2\right ) \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (e+f x),2\right )}{a^2 f \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {g \cos (e+f x)}}-\frac {b g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{a^2 f \sqrt {g \cos (e+f x)}}+\frac {b g \csc (e+f x) \sqrt {g \cos (e+f x)}}{a^2 f}+\frac {b^{3/2} g^{3/2} \sqrt [4]{b^2-a^2} \arctan \left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{a^3 f}+\frac {b^{3/2} g^{3/2} \sqrt [4]{b^2-a^2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt {g} \sqrt [4]{b^2-a^2}}\right )}{a^3 f}+\frac {g^{3/2} \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{4 a f}+\frac {g^{3/2} \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{4 a f}-\frac {g \csc ^2(e+f x) \sqrt {g \cos (e+f x)}}{2 a f} \]
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Rule 209
Rule 211
Rule 212
Rule 214
Rule 218
Rule 294
Rule 327
Rule 335
Rule 2645
Rule 2647
Rule 2720
Rule 2721
Rule 2774
Rule 2781
Rule 2884
Rule 2886
Rule 2946
Rule 2977
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {b^2 (g \cos (e+f x))^{3/2} \csc (e+f x)}{a^3}-\frac {b (g \cos (e+f x))^{3/2} \csc ^2(e+f x)}{a^2}+\frac {(g \cos (e+f x))^{3/2} \csc ^3(e+f x)}{a}-\frac {b^3 (g \cos (e+f x))^{3/2}}{a^3 (a+b \sin (e+f x))}\right ) \, dx \\ & = \frac {\int (g \cos (e+f x))^{3/2} \csc ^3(e+f x) \, dx}{a}-\frac {b \int (g \cos (e+f x))^{3/2} \csc ^2(e+f x) \, dx}{a^2}+\frac {b^2 \int (g \cos (e+f x))^{3/2} \csc (e+f x) \, dx}{a^3}-\frac {b^3 \int \frac {(g \cos (e+f x))^{3/2}}{a+b \sin (e+f x)} \, dx}{a^3} \\ & = -\frac {2 b^2 g \sqrt {g \cos (e+f x)}}{a^3 f}+\frac {b g \sqrt {g \cos (e+f x)} \csc (e+f x)}{a^2 f}-\frac {\text {Subst}\left (\int \frac {x^{3/2}}{\left (1-\frac {x^2}{g^2}\right )^2} \, dx,x,g \cos (e+f x)\right )}{a f g}-\frac {b^2 \text {Subst}\left (\int \frac {x^{3/2}}{1-\frac {x^2}{g^2}} \, dx,x,g \cos (e+f x)\right )}{a^3 f g}+\frac {\left (b g^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)}} \, dx}{2 a^2}-\frac {\left (b^2 g^2\right ) \int \frac {b+a \sin (e+f x)}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx}{a^3} \\ & = \frac {b g \sqrt {g \cos (e+f x)} \csc (e+f x)}{a^2 f}-\frac {g \sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{2 a f}+\frac {g \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{g^2}\right )} \, dx,x,g \cos (e+f x)\right )}{4 a f}-\frac {\left (b^2 g\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (1-\frac {x^2}{g^2}\right )} \, dx,x,g \cos (e+f x)\right )}{a^3 f}-\frac {\left (b g^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)}} \, dx}{a^2}+\frac {\left (b \left (a^2-b^2\right ) g^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} (a+b \sin (e+f x))} \, dx}{a^3}+\frac {\left (b g^2 \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{2 a^2 \sqrt {g \cos (e+f x)}} \\ & = \frac {b g \sqrt {g \cos (e+f x)} \csc (e+f x)}{a^2 f}-\frac {g \sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{2 a f}+\frac {b g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{a^2 f \sqrt {g \cos (e+f x)}}+\frac {g \text {Subst}\left (\int \frac {1}{1-\frac {x^4}{g^2}} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{2 a f}-\frac {\left (2 b^2 g\right ) \text {Subst}\left (\int \frac {1}{1-\frac {x^4}{g^2}} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a^3 f}+\frac {\left (b \sqrt {-a^2+b^2} g^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 a^2}+\frac {\left (b \sqrt {-a^2+b^2} g^2\right ) \int \frac {1}{\sqrt {g \cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 a^2}+\frac {\left (b^2 \left (a^2-b^2\right ) g^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (\left (a^2-b^2\right ) g^2+b^2 x^2\right )} \, dx,x,g \cos (e+f x)\right )}{a^3 f}-\frac {\left (b g^2 \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)}} \, dx}{a^2 \sqrt {g \cos (e+f x)}} \\ & = \frac {b g \sqrt {g \cos (e+f x)} \csc (e+f x)}{a^2 f}-\frac {g \sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{2 a f}-\frac {b g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{a^2 f \sqrt {g \cos (e+f x)}}+\frac {g^2 \text {Subst}\left (\int \frac {1}{g-x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{4 a f}+\frac {g^2 \text {Subst}\left (\int \frac {1}{g+x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{4 a f}-\frac {\left (b^2 g^2\right ) \text {Subst}\left (\int \frac {1}{g-x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a^3 f}-\frac {\left (b^2 g^2\right ) \text {Subst}\left (\int \frac {1}{g+x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a^3 f}+\frac {\left (2 b^2 \left (a^2-b^2\right ) g^3\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-b^2\right ) g^2+b^2 x^4} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a^3 f}+\frac {\left (b \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {-a^2+b^2}-b \cos (e+f x)\right )} \, dx}{2 a^2 \sqrt {g \cos (e+f x)}}+\frac {\left (b \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)}\right ) \int \frac {1}{\sqrt {\cos (e+f x)} \left (\sqrt {-a^2+b^2}+b \cos (e+f x)\right )} \, dx}{2 a^2 \sqrt {g \cos (e+f x)}} \\ & = \frac {g^{3/2} \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{4 a f}-\frac {b^2 g^{3/2} \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^3 f}+\frac {g^{3/2} \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{4 a f}-\frac {b^2 g^{3/2} \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^3 f}+\frac {b g \sqrt {g \cos (e+f x)} \csc (e+f x)}{a^2 f}-\frac {g \sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{2 a f}-\frac {b g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{a^2 f \sqrt {g \cos (e+f x)}}-\frac {b \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a^2 \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {b \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a^2 \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {\left (b^2 \sqrt {-a^2+b^2} g^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g-b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a^3 f}+\frac {\left (b^2 \sqrt {-a^2+b^2} g^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} g+b x^2} \, dx,x,\sqrt {g \cos (e+f x)}\right )}{a^3 f} \\ & = \frac {g^{3/2} \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{4 a f}-\frac {b^2 g^{3/2} \arctan \left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^3 f}+\frac {b^{3/2} \sqrt [4]{-a^2+b^2} g^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a^3 f}+\frac {g^{3/2} \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{4 a f}-\frac {b^2 g^{3/2} \text {arctanh}\left (\frac {\sqrt {g \cos (e+f x)}}{\sqrt {g}}\right )}{a^3 f}+\frac {b^{3/2} \sqrt [4]{-a^2+b^2} g^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {g \cos (e+f x)}}{\sqrt [4]{-a^2+b^2} \sqrt {g}}\right )}{a^3 f}+\frac {b g \sqrt {g \cos (e+f x)} \csc (e+f x)}{a^2 f}-\frac {g \sqrt {g \cos (e+f x)} \csc ^2(e+f x)}{2 a f}-\frac {b g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticF}\left (\frac {1}{2} (e+f x),2\right )}{a^2 f \sqrt {g \cos (e+f x)}}-\frac {b \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a^2 \left (b-\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}}+\frac {b \sqrt {-a^2+b^2} g^2 \sqrt {\cos (e+f x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (e+f x),2\right )}{a^2 \left (b+\sqrt {-a^2+b^2}\right ) f \sqrt {g \cos (e+f x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 26.15 (sec) , antiderivative size = 2129, normalized size of antiderivative = 3.71 \[ \int \frac {(g \cos (e+f x))^{3/2} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx=\text {Result too large to show} \]
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Time = 0.75 (sec) , antiderivative size = 289, normalized size of antiderivative = 0.50
method | result | size |
default | \(-\frac {g^{2} \left (\frac {\sqrt {2 g \left (\cos ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-g}}{2 g \cos \left (\frac {f x}{2}+\frac {e}{2}\right )^{2}}+\frac {\ln \left (\frac {-2 g +2 \sqrt {-g}\, \sqrt {2 g \left (\cos ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )-g}}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )}\right )}{\sqrt {-g}}-\frac {\ln \left (\frac {-4 g \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+2 \sqrt {g}\, \sqrt {-2 g \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+g}-2 g}{\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1}\right )}{2 \sqrt {g}}-\frac {\ln \left (\frac {4 g \cos \left (\frac {f x}{2}+\frac {e}{2}\right )+2 \sqrt {g}\, \sqrt {-2 g \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+g}-2 g}{-1+\cos \left (\frac {f x}{2}+\frac {e}{2}\right )}\right )}{2 \sqrt {g}}-\frac {\sqrt {-2 g \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+g}}{4 g \left (-1+\cos \left (\frac {f x}{2}+\frac {e}{2}\right )\right )}+\frac {\sqrt {-2 g \left (\sin ^{2}\left (\frac {f x}{2}+\frac {e}{2}\right )\right )+g}}{4 g \left (\cos \left (\frac {f x}{2}+\frac {e}{2}\right )+1\right )}\right )}{4 a f}\) | \(289\) |
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\[ \int \frac {(g \cos (e+f x))^{3/2} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \csc \left (f x + e\right )^{3}}{b \sin \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(g \cos (e+f x))^{3/2} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx=\text {Exception raised: RuntimeError} \]
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\[ \int \frac {(g \cos (e+f x))^{3/2} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx=\int { \frac {\left (g \cos \left (f x + e\right )\right )^{\frac {3}{2}} \csc \left (f x + e\right )^{3}}{b \sin \left (f x + e\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(g \cos (e+f x))^{3/2} \csc ^3(e+f x)}{a+b \sin (e+f x)} \, dx=\int \frac {{\left (g\,\cos \left (e+f\,x\right )\right )}^{3/2}}{{\sin \left (e+f\,x\right )}^3\,\left (a+b\,\sin \left (e+f\,x\right )\right )} \,d x \]
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