Integrand size = 25, antiderivative size = 520 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3} \, dx=-\frac {3 \sqrt {b} \left (5 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{11/4} d \sqrt {e}}-\frac {3 \sqrt {b} \left (5 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{11/4} d \sqrt {e}}-\frac {7 a \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 \left (a^2-b^2\right )^2 d \sqrt {e \cos (c+d x)}}+\frac {3 a \left (5 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{8 \left (a^2-b^2\right )^2 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}}+\frac {3 a \left (5 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{8 \left (a^2-b^2\right )^2 \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \cos (c+d x)}}+\frac {b \sqrt {e \cos (c+d x)}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}+\frac {7 a b \sqrt {e \cos (c+d x)}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))} \]
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Time = 0.84 (sec) , antiderivative size = 520, normalized size of antiderivative = 1.00, number of steps used = 14, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {2773, 2943, 2946, 2721, 2720, 2781, 2886, 2884, 335, 218, 214, 211} \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3} \, dx=-\frac {3 \sqrt {b} \left (5 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d \sqrt {e} \left (b^2-a^2\right )^{11/4}}-\frac {3 \sqrt {b} \left (5 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{8 d \sqrt {e} \left (b^2-a^2\right )^{11/4}}+\frac {7 a b \sqrt {e \cos (c+d x)}}{4 d e \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}+\frac {b \sqrt {e \cos (c+d x)}}{2 d e \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {7 a \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 d \left (a^2-b^2\right )^2 \sqrt {e \cos (c+d x)}}+\frac {3 a \left (5 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{8 d \left (a^2-b^2\right )^2 \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {e \cos (c+d x)}}+\frac {3 a \left (5 a^2+2 b^2\right ) \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{8 d \left (a^2-b^2\right )^2 \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {e \cos (c+d x)}} \]
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Rule 211
Rule 214
Rule 218
Rule 335
Rule 2720
Rule 2721
Rule 2773
Rule 2781
Rule 2884
Rule 2886
Rule 2943
Rule 2946
Rubi steps \begin{align*} \text {integral}& = \frac {b \sqrt {e \cos (c+d x)}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}-\frac {\int \frac {-2 a+\frac {3}{2} b \sin (c+d x)}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^2} \, dx}{2 \left (a^2-b^2\right )} \\ & = \frac {b \sqrt {e \cos (c+d x)}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}+\frac {7 a b \sqrt {e \cos (c+d x)}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))}+\frac {\int \frac {\frac {1}{2} \left (4 a^2+3 b^2\right )-\frac {7}{4} a b \sin (c+d x)}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))} \, dx}{2 \left (a^2-b^2\right )^2} \\ & = \frac {b \sqrt {e \cos (c+d x)}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}+\frac {7 a b \sqrt {e \cos (c+d x)}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))}-\frac {(7 a) \int \frac {1}{\sqrt {e \cos (c+d x)}} \, dx}{8 \left (a^2-b^2\right )^2}+\frac {\left (3 \left (5 a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))} \, dx}{8 \left (a^2-b^2\right )^2} \\ & = \frac {b \sqrt {e \cos (c+d x)}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}+\frac {7 a b \sqrt {e \cos (c+d x)}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))}-\frac {\left (3 a \left (5 a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}-b \cos (c+d x)\right )} \, dx}{16 \left (-a^2+b^2\right )^{5/2}}-\frac {\left (3 a \left (5 a^2+2 b^2\right )\right ) \int \frac {1}{\sqrt {e \cos (c+d x)} \left (\sqrt {-a^2+b^2}+b \cos (c+d x)\right )} \, dx}{16 \left (-a^2+b^2\right )^{5/2}}+\frac {\left (3 b \left (5 a^2+2 b^2\right ) e\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \left (\left (a^2-b^2\right ) e^2+b^2 x^2\right )} \, dx,x,e \cos (c+d x)\right )}{8 \left (a^2-b^2\right )^2 d}-\frac {\left (7 a \sqrt {\cos (c+d x)}\right ) \int \frac {1}{\sqrt {\cos (c+d x)}} \, dx}{8 \left (a^2-b^2\right )^2 \sqrt {e \cos (c+d x)}} \\ & = -\frac {7 a \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 \left (a^2-b^2\right )^2 d \sqrt {e \cos (c+d x)}}+\frac {b \sqrt {e \cos (c+d x)}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}+\frac {7 a b \sqrt {e \cos (c+d x)}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))}+\frac {\left (3 b \left (5 a^2+2 b^2\right ) e\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{4 \left (a^2-b^2\right )^2 d}-\frac {\left (3 a \left (5 a^2+2 b^2\right ) \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}{16 \left (-a^2+b^2\right )^{5/2} \sqrt {e \cos (c+d x)}}-\frac {\left (3 a \left (5 a^2+2 b^2\right ) \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}{16 \left (-a^2+b^2\right )^{5/2} \sqrt {e \cos (c+d x)}} \\ & = -\frac {7 a \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 \left (a^2-b^2\right )^2 d \sqrt {e \cos (c+d x)}}+\frac {3 a \left (5 a^2+2 b^2\right ) \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 )}{8 \left (-a^2+b^2\right )^{5/2} \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {3 a \left (5 a^2+2 b^2\right ) \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 )}{8 \left (-a^2+b^2\right )^{5/2} \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {b \sqrt {e \cos (c+d x)}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}+\frac {7 a b \sqrt {e \cos (c+d x)}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))}-\frac {\left (3 b \left (5 a^2+2 b^2\right )\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 )}{8 \left (-a^2+b^2\right )^{5/2} d}-\frac {\left (3 b \left (5 a^2+2 b^2\right )\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 )}{8 \left (-a^2+b^2\right )^{5/2} d} \\ & = -\frac {3 \sqrt {b} \left (5 a^2+2 b^2\right ) \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{11/4} d \sqrt {e}}-\frac {3 \sqrt {b} \left (5 a^2+2 b^2\right ) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{8 \left (-a^2+b^2\right )^{11/4} d \sqrt {e}}-\frac {7 a \sqrt {\cos (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} (c+d x),2\right )}{4 \left (a^2-b^2\right )^2 d \sqrt {e \cos (c+d x)}}+\frac {3 a \left (5 a^2+2 b^2\right ) \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 )}{8 \left (-a^2+b^2\right )^{5/2} \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}-\frac {3 a \left (5 a^2+2 b^2\right ) \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 )}{8 \left (-a^2+b^2\right )^{5/2} \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \cos (c+d x)}}+\frac {b \sqrt {e \cos (c+d x)}}{2 \left (a^2-b^2\right ) d e (a+b \sin (c+d x))^2}+\frac {7 a b \sqrt {e \cos (c+d x)}}{4 \left (a^2-b^2\right )^2 d e (a+b \sin (c+d x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 20.93 (sec) , antiderivative size = 1226, normalized size of antiderivative = 2.36 \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3} \, dx=\frac {\cos (c+d x) \left (\frac {b}{2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}+\frac {7 a b}{4 \left (a^2-b^2\right )^2 (a+b \sin (c+d x))}\right )}{d \sqrt {e \cos (c+d x)}}+\frac {\sqrt {\cos (c+d x)} \left (-\frac {2 \left (8 a^2+6 b^2\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (\frac {5 a \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \sqrt {\cos (c+d x)}}{\sqrt {1-\cos ^2(c+d x)} \left (5 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right )-2 \left (2 b^2 \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right )+\left (-a^2+b^2\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right )\right ) \cos ^2(c+d x)\right ) \left (a^2+b^2 \left (-1+\cos ^2(c+d x)\right )\right )}-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \sqrt {b} \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 )}{\left (-a^2+b^2\right )^{3/4}}\right ) \sin (c+d x)}{\sqrt {1-\cos ^2(c+d x)} (a+b \sin (c+d x))}+\frac {14 a b \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \left (\frac {5 b \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{2},1,\frac {5}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right ) \sqrt {\cos (c+d x)} \sqrt {1-\cos ^2(c+d x)}}{\left (-5 \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{2},1,\frac {5}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right )+2 \left (2 b^2 \operatorname {AppellF1}\left (\frac {5}{4},-\frac {1}{2},2,\frac {9}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right )+\left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\cos ^2(c+d x),\frac {b^2 \cos ^2(c+d x)}{-a^2+b^2}\right )\right ) \cos ^2(c+d x)\right ) \left (a^2+b^2 \left (-1+\cos ^2(c+d x)\right )\right )}+\frac {a \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 )}{4 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4}}\right ) \sin ^2(c+d x)}{\left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}\right )}{8 (a-b)^2 (a+b)^2 d \sqrt {e \cos (c+d x)}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 10.54 (sec) , antiderivative size = 2295, normalized size of antiderivative = 4.41
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Timed out. \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3} \, dx=\int { \frac {1}{\sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
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\[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3} \, dx=\int { \frac {1}{\sqrt {e \cos \left (d x + c\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{3}} \,d x } \]
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Timed out. \[ \int \frac {1}{\sqrt {e \cos (c+d x)} (a+b \sin (c+d x))^3} \, dx=\int \frac {1}{\sqrt {e\,\cos \left (c+d\,x\right )}\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^3} \,d x \]
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