Integrand size = 25, antiderivative size = 384 \[ \int \frac {(e \cos (c+d x))^{5/2}}{a+b \sin (c+d x)} \, dx=\frac {\left (-a^2+b^2\right )^{3/4} e^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{5/2} d}-\frac {\left (-a^2+b^2\right )^{3/4} e^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{5/2} d}+\frac {2 e (e \cos (c+d x))^{3/2}}{3 b d}+\frac {2 a e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt {\cos (c+d x)}}-\frac {a \left (a^2-b^2\right ) e^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{b^3 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {a \left (a^2-b^2\right ) e^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{b^3 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}} \]
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Time = 0.57 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.440, Rules used = {2774, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \[ \int \frac {(e \cos (c+d x))^{5/2}}{a+b \sin (c+d x)} \, dx=\frac {e^{5/2} \left (b^2-a^2\right )^{3/4} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{b^{5/2} d}-\frac {e^{5/2} \left (b^2-a^2\right )^{3/4} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{b^{5/2} d}-\frac {a e^3 \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{b^3 d \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}-\frac {a e^3 \left (a^2-b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{b^3 d \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}}+\frac {2 a e^2 E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{b^2 d \sqrt {\cos (c+d x)}}+\frac {2 e (e \cos (c+d x))^{3/2}}{3 b d} \]
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Rule 211
Rule 214
Rule 304
Rule 335
Rule 2719
Rule 2721
Rule 2774
Rule 2780
Rule 2884
Rule 2886
Rule 2946
Rubi steps \begin{align*} \text {integral}& = \frac {2 e (e \cos (c+d x))^{3/2}}{3 b d}+\frac {e^2 \int \frac {\sqrt {e \cos (c+d x)} (b+a \sin (c+d x))}{a+b \sin (c+d x)} \, dx}{b} \\ & = \frac {2 e (e \cos (c+d x))^{3/2}}{3 b d}+\frac {\left (a e^2\right ) \int \sqrt {e \cos (c+d x)} \, dx}{b^2}+\frac {\left (\left (-a^2+b^2\right ) e^2\right ) \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{b^2} \\ & = \frac {2 e (e \cos (c+d x))^{3/2}}{3 b d}+\frac {\left (a \left (a^2-b^2\right ) e^3\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{2 b^3}-\frac {\left (a \left (a^2-b^2\right ) e^3\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{2 b^3}-\frac {\left (\left (a^2-b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\left (a^2-b^2\right ) e^2+b^2 x^2} \, dx,x,e \cos (c+d x)\right )}{b d}+\frac {\left (a e^2 \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{b^2 \sqrt {\cos (c+d x)}} \\ & = \frac {2 e (e \cos (c+d x))^{3/2}}{3 b d}+\frac {2 a e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt {\cos (c+d x)}}-\frac {\left (2 \left (a^2-b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {x^2}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{b d}+\frac {\left (a \left (a^2-b^2\right ) e^3 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{2 b^3 \sqrt {e \cos (c+d x)}}-\frac {\left (a \left (a^2-b^2\right ) e^3 \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{2 b^3 \sqrt {e \cos (c+d x)}} \\ & = \frac {2 e (e \cos (c+d x))^{3/2}}{3 b d}+\frac {2 a e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt {\cos (c+d x)}}-\frac {a \left (a^2-b^2\right ) e^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{b^3 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {a \left (a^2-b^2\right ) e^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{b^3 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {\left (\left (a^2-b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{b^2 d}-\frac {\left (\left (a^2-b^2\right ) e^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{b^2 d} \\ & = \frac {\left (-a^2+b^2\right )^{3/4} e^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{5/2} d}-\frac {\left (-a^2+b^2\right )^{3/4} e^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{5/2} d}+\frac {2 e (e \cos (c+d x))^{3/2}}{3 b d}+\frac {2 a e^2 \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{b^2 d \sqrt {\cos (c+d x)}}-\frac {a \left (a^2-b^2\right ) e^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{b^3 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {a \left (a^2-b^2\right ) e^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{b^3 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 15.13 (sec) , antiderivative size = 709, normalized size of antiderivative = 1.85 \[ \int \frac {(e \cos (c+d x))^{5/2}}{a+b \sin (c+d x)} \, dx=\frac {(e \cos (c+d x))^{5/2} \left (2 \cos ^{\frac {3}{2}}(c+d x)-\frac {a \left (8 b^{5/2} \operatorname {AppellF1}\left (\frac {3}{4},-\frac {1}{2},1,\frac {7}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)+3 \sqrt {2} a \left (a^2-b^2\right )^{3/4} \left (2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )-\log \left (\sqrt {a^2-b^2}-\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\cos (c+d x)}+b \cos (c+d x)\right )\right )\right ) \left (a+b \sqrt {\sin ^2(c+d x)}\right )}{4 b^{3/2} \left (-a^2+b^2\right ) (a+b \sin (c+d x))}-\frac {6 b \left (\frac {a \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \cos ^{\frac {3}{2}}(c+d x)}{3 \left (a^2-b^2\right )}+\frac {\left (\frac {1}{8}+\frac {i}{8}\right ) \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\cos (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-\log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )+\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\cos (c+d x)}+i b \cos (c+d x)\right )\right )}{\sqrt {b} \sqrt [4]{-a^2+b^2}}\right ) \sin (c+d x) \left (a+b \sqrt {\sin ^2(c+d x)}\right )}{\sqrt {\sin ^2(c+d x)} (a+b \sin (c+d x))}\right )}{3 b d \cos ^{\frac {5}{2}}(c+d x)} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 5.28 (sec) , antiderivative size = 1025, normalized size of antiderivative = 2.67
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Timed out. \[ \int \frac {(e \cos (c+d x))^{5/2}}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e \cos (c+d x))^{5/2}}{a+b \sin (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \cos (c+d x))^{5/2}}{a+b \sin (c+d x)} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}{b \sin \left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {(e \cos (c+d x))^{5/2}}{a+b \sin (c+d x)} \, dx=\int { \frac {\left (e \cos \left (d x + c\right )\right )^{\frac {5}{2}}}{b \sin \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e \cos (c+d x))^{5/2}}{a+b \sin (c+d x)} \, dx=\int \frac {{\left (e\,\cos \left (c+d\,x\right )\right )}^{5/2}}{a+b\,\sin \left (c+d\,x\right )} \,d x \]
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