Integrand size = 25, antiderivative size = 486 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=\frac {b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{\left (-a^2+b^2\right )^{9/4} d e^{7/2}}-\frac {b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{\left (-a^2+b^2\right )^{9/4} d e^{7/2}}-\frac {2 a \left (3 a^2-8 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 \left (a^2-b^2\right )^2 d e^4 \sqrt {\cos (c+d x)}}+\frac {a b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{\left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {a b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} (c+d x),2\right )}{\left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}-\frac {2 (b-a \sin (c+d x))}{5 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2}}+\frac {2 \left (5 b^3+a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \cos (c+d x)}} \]
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Time = 0.86 (sec) , antiderivative size = 486, 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 = {2775, 2945, 2946, 2721, 2719, 2780, 2886, 2884, 335, 304, 211, 214} \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=\frac {b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d e^{7/2} \left (b^2-a^2\right )^{9/4}}-\frac {b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{d e^{7/2} \left (b^2-a^2\right )^{9/4}}-\frac {2 a \left (3 a^2-8 b^2\right ) E\left (\left .\frac {1}{2} (c+d x)\right |2\right ) \sqrt {e \cos (c+d x)}}{5 d e^4 \left (a^2-b^2\right )^2 \sqrt {\cos (c+d x)}}-\frac {2 (b-a \sin (c+d x))}{5 d e \left (a^2-b^2\right ) (e \cos (c+d x))^{5/2}}+\frac {2 \left (a \left (3 a^2-8 b^2\right ) \sin (c+d x)+5 b^3\right )}{5 d e^3 \left (a^2-b^2\right )^2 \sqrt {e \cos (c+d x)}}+\frac {a b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{d e^3 \left (a^2-b^2\right )^2 \left (b-\sqrt {b^2-a^2}\right ) \sqrt {e \cos (c+d x)}}+\frac {a b^3 \sqrt {\cos (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} (c+d x),2\right )}{d e^3 \left (a^2-b^2\right )^2 \left (\sqrt {b^2-a^2}+b\right ) \sqrt {e \cos (c+d x)}} \]
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Rule 211
Rule 214
Rule 304
Rule 335
Rule 2719
Rule 2721
Rule 2775
Rule 2780
Rule 2884
Rule 2886
Rule 2945
Rule 2946
Rubi steps \begin{align*} \text {integral}& = -\frac {2 (b-a \sin (c+d x))}{5 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2}}-\frac {2 \int \frac {-\frac {3 a^2}{2}+\frac {5 b^2}{2}-\frac {3}{2} a b \sin (c+d x)}{(e \cos (c+d x))^{3/2} (a+b \sin (c+d x))} \, dx}{5 \left (a^2-b^2\right ) e^2} \\ & = -\frac {2 (b-a \sin (c+d x))}{5 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2}}+\frac {2 \left (5 b^3+a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \cos (c+d x)}}+\frac {4 \int \frac {\sqrt {e \cos (c+d x)} \left (\frac {1}{4} \left (-3 a^4+8 a^2 b^2+5 b^4\right )-\frac {1}{4} a b \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{a+b \sin (c+d x)} \, dx}{5 \left (a^2-b^2\right )^2 e^4} \\ & = -\frac {2 (b-a \sin (c+d x))}{5 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2}}+\frac {2 \left (5 b^3+a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \cos (c+d x)}}+\frac {b^4 \int \frac {\sqrt {e \cos (c+d x)}}{a+b \sin (c+d x)} \, dx}{\left (a^2-b^2\right )^2 e^4}-\frac {\left (a \left (3 a^2-8 b^2\right )\right ) \int \sqrt {e \cos (c+d x)} \, dx}{5 \left (a^2-b^2\right )^2 e^4} \\ & = -\frac {2 (b-a \sin (c+d x))}{5 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2}}+\frac {2 \left (5 b^3+a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \cos (c+d x)}}-\frac {\left (a b^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 \left (a^2-b^2\right )^2 e^3}+\frac {\left (a b^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 \left (a^2-b^2\right )^2 e^3}+\frac {b^5 \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 )}{\left (a^2-b^2\right )^2 d e^3}-\frac {\left (a \left (3 a^2-8 b^2\right ) \sqrt {e \cos (c+d x)}\right ) \int \sqrt {\cos (c+d x)} \, dx}{5 \left (a^2-b^2\right )^2 e^4 \sqrt {\cos (c+d x)}} \\ & = -\frac {2 a \left (3 a^2-8 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 \left (a^2-b^2\right )^2 d e^4 \sqrt {\cos (c+d x)}}-\frac {2 (b-a \sin (c+d x))}{5 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2}}+\frac {2 \left (5 b^3+a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \cos (c+d x)}}+\frac {\left (2 b^5\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 )}{\left (a^2-b^2\right )^2 d e^3}-\frac {\left (a b^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 \left (a^2-b^2\right )^2 e^3 \sqrt {e \cos (c+d x)}}+\frac {\left (a b^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 \left (a^2-b^2\right )^2 e^3 \sqrt {e \cos (c+d x)}} \\ & = -\frac {2 a \left (3 a^2-8 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 \left (a^2-b^2\right )^2 d e^4 \sqrt {\cos (c+d x)}}+\frac {a b^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 )}{\left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {a b^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 )}{\left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}-\frac {2 (b-a \sin (c+d x))}{5 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2}}+\frac {2 \left (5 b^3+a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \cos (c+d x)}}-\frac {b^4 \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{\left (a^2-b^2\right )^2 d e^3}+\frac {b^4 \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \cos (c+d x)}\right )}{\left (a^2-b^2\right )^2 d e^3} \\ & = \frac {b^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{\left (-a^2+b^2\right )^{9/4} d e^{7/2}}-\frac {b^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \cos (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{\left (-a^2+b^2\right )^{9/4} d e^{7/2}}-\frac {2 a \left (3 a^2-8 b^2\right ) \sqrt {e \cos (c+d x)} E\left (\left .\frac {1}{2} (c+d x)\right |2\right )}{5 \left (a^2-b^2\right )^2 d e^4 \sqrt {\cos (c+d x)}}+\frac {a b^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 )}{\left (a^2-b^2\right )^2 \left (b-\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}+\frac {a b^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 )}{\left (a^2-b^2\right )^2 \left (b+\sqrt {-a^2+b^2}\right ) d e^3 \sqrt {e \cos (c+d x)}}-\frac {2 (b-a \sin (c+d x))}{5 \left (a^2-b^2\right ) d e (e \cos (c+d x))^{5/2}}+\frac {2 \left (5 b^3+a \left (3 a^2-8 b^2\right ) \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^2 d e^3 \sqrt {e \cos (c+d x)}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 6.55 (sec) , antiderivative size = 881, normalized size of antiderivative = 1.81 \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=-\frac {\cos ^{\frac {7}{2}}(c+d x) \left (-\frac {2 \left (3 a^4-8 a^2 b^2-5 b^4\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \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)}{\sqrt {1-\cos ^2(c+d x)} (a+b \sin (c+d x))}-\frac {\left (3 a^3 b-8 a b^3\right ) \left (a+b \sqrt {1-\cos ^2(c+d x)}\right ) \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 ) \sin ^2(c+d x)}{12 b^{3/2} \left (-a^2+b^2\right ) \left (1-\cos ^2(c+d x)\right ) (a+b \sin (c+d x))}\right )}{5 (a-b)^2 (a+b)^2 d (e \cos (c+d x))^{7/2}}+\frac {\cos ^4(c+d x) \left (\frac {2 \sec ^3(c+d x) (-b+a \sin (c+d x))}{5 \left (a^2-b^2\right )}+\frac {2 \sec (c+d x) \left (5 b^3+3 a^3 \sin (c+d x)-8 a b^2 \sin (c+d x)\right )}{5 \left (a^2-b^2\right )^2}\right )}{d (e \cos (c+d x))^{7/2}} \]
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Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 9.45 (sec) , antiderivative size = 1723, normalized size of antiderivative = 3.55
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=\text {Timed out} \]
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\[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}} \,d x } \]
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\[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=\int { \frac {1}{\left (e \cos \left (d x + c\right )\right )^{\frac {7}{2}} {\left (b \sin \left (d x + c\right ) + a\right )}} \,d x } \]
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Timed out. \[ \int \frac {1}{(e \cos (c+d x))^{7/2} (a+b \sin (c+d x))} \, dx=\int \frac {1}{{\left (e\,\cos \left (c+d\,x\right )\right )}^{7/2}\,\left (a+b\,\sin \left (c+d\,x\right )\right )} \,d x \]
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