Integrand size = 25, antiderivative size = 474 \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \cos (c+d x)} \, dx=\frac {\left (-a^2+b^2\right )^{5/4} e^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{7/2} d}+\frac {\left (-a^2+b^2\right )^{5/4} e^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{7/2} d}-\frac {2 a \left (3 a^2-4 b^2\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 b^4 d \sqrt {e \sin (c+d x)}}+\frac {a \left (a^2-b^2\right )^2 e^4 \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{b^4 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \sin (c+d x)}}+\frac {a \left (a^2-b^2\right )^2 e^4 \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{b^4 \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 e^3 \left (3 \left (a^2-b^2\right )-a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{3 b^3 d}-\frac {2 e (e \sin (c+d x))^{5/2}}{5 b d} \]
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Time = 1.77 (sec) , antiderivative size = 474, 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 = {2774, 2944, 2946, 2721, 2720, 2781, 2886, 2884, 335, 218, 214, 211} \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \cos (c+d x)} \, dx=\frac {e^{7/2} \left (b^2-a^2\right )^{5/4} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{b^{7/2} d}+\frac {e^{7/2} \left (b^2-a^2\right )^{5/4} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{b^{7/2} d}-\frac {2 a e^4 \left (3 a^2-4 b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{3 b^4 d \sqrt {e \sin (c+d x)}}+\frac {a e^4 \left (a^2-b^2\right )^2 \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b^4 d \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {e \sin (c+d x)}}+\frac {a e^4 \left (a^2-b^2\right )^2 \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b^4 d \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {e \sin (c+d x)}}+\frac {2 e^3 \sqrt {e \sin (c+d x)} \left (3 \left (a^2-b^2\right )-a b \cos (c+d x)\right )}{3 b^3 d}-\frac {2 e (e \sin (c+d x))^{5/2}}{5 b d} \]
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Rule 211
Rule 214
Rule 218
Rule 335
Rule 2720
Rule 2721
Rule 2774
Rule 2781
Rule 2884
Rule 2886
Rule 2944
Rule 2946
Rubi steps \begin{align*} \text {integral}& = -\frac {2 e (e \sin (c+d x))^{5/2}}{5 b d}-\frac {e^2 \int \frac {(-b-a \cos (c+d x)) (e \sin (c+d x))^{3/2}}{a+b \cos (c+d x)} \, dx}{b} \\ & = \frac {2 e^3 \left (3 \left (a^2-b^2\right )-a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{3 b^3 d}-\frac {2 e (e \sin (c+d x))^{5/2}}{5 b d}-\frac {\left (2 e^4\right ) \int \frac {\frac {1}{2} b \left (2 a^2-3 b^2\right )+\frac {1}{2} a \left (3 a^2-4 b^2\right ) \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{3 b^3} \\ & = \frac {2 e^3 \left (3 \left (a^2-b^2\right )-a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{3 b^3 d}-\frac {2 e (e \sin (c+d x))^{5/2}}{5 b d}-\frac {\left (a \left (3 a^2-4 b^2\right ) e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{3 b^4}+\frac {\left (\left (a^2-b^2\right )^2 e^4\right ) \int \frac {1}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{b^4} \\ & = \frac {2 e^3 \left (3 \left (a^2-b^2\right )-a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{3 b^3 d}-\frac {2 e (e \sin (c+d x))^{5/2}}{5 b d}-\frac {\left (a \left (-a^2+b^2\right )^{3/2} e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{2 b^4}-\frac {\left (a \left (-a^2+b^2\right )^{3/2} e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)} \left (\sqrt {-a^2+b^2}+b \sin (c+d x)\right )} \, dx}{2 b^4}-\frac {\left (\left (a^2-b^2\right )^2 e^5\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 \sin (c+d x)\right )}{b^3 d}-\frac {\left (a \left (3 a^2-4 b^2\right ) e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 b^4 \sqrt {e \sin (c+d x)}} \\ & = -\frac {2 a \left (3 a^2-4 b^2\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 b^4 d \sqrt {e \sin (c+d x)}}+\frac {2 e^3 \left (3 \left (a^2-b^2\right )-a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{3 b^3 d}-\frac {2 e (e \sin (c+d x))^{5/2}}{5 b d}-\frac {\left (2 \left (a^2-b^2\right )^2 e^5\right ) \text {Subst}\left (\int \frac {1}{\left (a^2-b^2\right ) e^2+b^2 x^4} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{b^3 d}-\frac {\left (a \left (-a^2+b^2\right )^{3/2} e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {-a^2+b^2}-b \sin (c+d x)\right )} \, dx}{2 b^4 \sqrt {e \sin (c+d x)}}-\frac {\left (a \left (-a^2+b^2\right )^{3/2} e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)} \left (\sqrt {-a^2+b^2}+b \sin (c+d x)\right )} \, dx}{2 b^4 \sqrt {e \sin (c+d x)}} \\ & = -\frac {2 a \left (3 a^2-4 b^2\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 b^4 d \sqrt {e \sin (c+d x)}}+\frac {a \left (-a^2+b^2\right )^{3/2} e^4 \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{b^4 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {a \left (-a^2+b^2\right )^{3/2} e^4 \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{b^4 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 e^3 \left (3 \left (a^2-b^2\right )-a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{3 b^3 d}-\frac {2 e (e \sin (c+d x))^{5/2}}{5 b d}+\frac {\left (\left (-a^2+b^2\right )^{3/2} e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e-b x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{b^3 d}+\frac {\left (\left (-a^2+b^2\right )^{3/2} e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-a^2+b^2} e+b x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{b^3 d} \\ & = \frac {\left (-a^2+b^2\right )^{5/4} e^{7/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{7/2} d}+\frac {\left (-a^2+b^2\right )^{5/4} e^{7/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{7/2} d}-\frac {2 a \left (3 a^2-4 b^2\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{3 b^4 d \sqrt {e \sin (c+d x)}}+\frac {a \left (-a^2+b^2\right )^{3/2} e^4 \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{b^4 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {a \left (-a^2+b^2\right )^{3/2} e^4 \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{b^4 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 e^3 \left (3 \left (a^2-b^2\right )-a b \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{3 b^3 d}-\frac {2 e (e \sin (c+d x))^{5/2}}{5 b d} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 15.54 (sec) , antiderivative size = 1955, normalized size of antiderivative = 4.12 \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \cos (c+d x)} \, dx=\frac {\left (-\frac {2 a \cos (c+d x)}{3 b^2}+\frac {\cos (2 (c+d x))}{5 b}\right ) \csc ^3(c+d x) (e \sin (c+d x))^{7/2}}{d}+\frac {(e \sin (c+d x))^{7/2} \left (\frac {28 a b \cos ^2(c+d x) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right ) \left (\frac {a \left (-2 \arctan \left (1-\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{a^2-b^2}}\right )+2 \arctan \left (1+\frac {\sqrt {2} \sqrt {b} \sqrt {\sin (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 {\sin (c+d x)}+b \sin (c+d x)\right )+\log \left (\sqrt {a^2-b^2}+\sqrt {2} \sqrt {b} \sqrt [4]{a^2-b^2} \sqrt {\sin (c+d x)}+b \sin (c+d x)\right )\right )}{4 \sqrt {2} \sqrt {b} \left (a^2-b^2\right )^{3/4}}+\frac {5 b \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},-\frac {1}{2},1,\frac {5}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sqrt {\sin (c+d x)} \sqrt {1-\sin ^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},\sin ^2(c+d x),\frac {b^2 \sin ^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},\sin ^2(c+d x),\frac {b^2 \sin ^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},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )\right ) \sin ^2(c+d x)\right ) \left (a^2+b^2 \left (-1+\sin ^2(c+d x)\right )\right )}\right )}{(a+b \cos (c+d x)) \left (1-\sin ^2(c+d x)\right )}+\frac {2 \left (-10 a^2+27 b^2\right ) \cos (c+d x) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right ) \left (-\frac {\left (\frac {1}{8}-\frac {i}{8}\right ) \sqrt {b} \left (2 \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\sin (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 {\sin (c+d x)}+i b \sin (c+d x)\right )-\log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )\right )}{\left (-a^2+b^2\right )^{3/4}}+\frac {5 a \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sqrt {\sin (c+d x)}}{\sqrt {1-\sin ^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},\sin ^2(c+d x),\frac {b^2 \sin ^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},\sin ^2(c+d x),\frac {b^2 \sin ^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},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )\right ) \sin ^2(c+d x)\right ) \left (a^2+b^2 \left (-1+\sin ^2(c+d x)\right )\right )}\right )}{(a+b \cos (c+d x)) \sqrt {1-\sin ^2(c+d x)}}+\frac {\left (30 a^2-33 b^2\right ) \cos (c+d x) \cos (2 (c+d x)) \left (a+b \sqrt {1-\sin ^2(c+d x)}\right ) \left (\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (-2 a^2+b^2\right ) \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )}{b^{3/2} \left (-a^2+b^2\right )^{3/4}}-\frac {\left (\frac {1}{2}-\frac {i}{2}\right ) \left (-2 a^2+b^2\right ) \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )}{b^{3/2} \left (-a^2+b^2\right )^{3/4}}+\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (-2 a^2+b^2\right ) \log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )}{b^{3/2} \left (-a^2+b^2\right )^{3/4}}-\frac {\left (\frac {1}{4}-\frac {i}{4}\right ) \left (-2 a^2+b^2\right ) \log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )}{b^{3/2} \left (-a^2+b^2\right )^{3/4}}+\frac {4 \sqrt {\sin (c+d x)}}{b}-\frac {4 a \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sin ^{\frac {5}{2}}(c+d x)}{5 \left (a^2-b^2\right )}+\frac {10 a \left (a^2-b^2\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sqrt {\sin (c+d x)}}{\sqrt {1-\sin ^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},\sin ^2(c+d x),\frac {b^2 \sin ^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},\sin ^2(c+d x),\frac {b^2 \sin ^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},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right )\right ) \sin ^2(c+d x)\right ) \left (a^2+b^2 \left (-1+\sin ^2(c+d x)\right )\right )}\right )}{(a+b \cos (c+d x)) \left (1-2 \sin ^2(c+d x)\right ) \sqrt {1-\sin ^2(c+d x)}}\right )}{60 b^2 d \sin ^{\frac {7}{2}}(c+d x)} \]
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Time = 5.01 (sec) , antiderivative size = 773, normalized size of antiderivative = 1.63
method | result | size |
default | \(\frac {-2 e b \left (-\frac {\sqrt {e \sin \left (d x +c \right )}\, e^{2} \left (b^{2} \left (\cos ^{2}\left (d x +c \right )\right )+5 a^{2}-6 b^{2}\right )}{5 b^{4}}+\frac {e^{4} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{b^{2}}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {e \sin \left (d x +c \right )+\left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{b^{2}}\right )^{\frac {1}{4}} \sqrt {e \sin \left (d x +c \right )}\, \sqrt {2}+\sqrt {\frac {e^{2} \left (a^{2}-b^{2}\right )}{b^{2}}}}{e \sin \left (d x +c \right )-\left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{b^{2}}\right )^{\frac {1}{4}} \sqrt {e \sin \left (d x +c \right )}\, \sqrt {2}+\sqrt {\frac {e^{2} \left (a^{2}-b^{2}\right )}{b^{2}}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \sin \left (d x +c \right )}}{\left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{b^{2}}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {e \sin \left (d x +c \right )}}{\left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{b^{2}}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b^{4} \left (a^{2} e^{2}-b^{2} e^{2}\right )}\right )+\frac {\sqrt {\left (\cos ^{2}\left (d x +c \right )\right ) e \sin \left (d x +c \right )}\, e^{4} a \left (\frac {3 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) a^{2}-4 \sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right ) b^{2}-2 b^{2} \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )}{3 b^{4} \sqrt {\left (\cos ^{2}\left (d x +c \right )\right ) e \sin \left (d x +c \right )}}+\frac {\left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left (-\frac {\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \Pi \left (\sqrt {1-\sin \left (d x +c \right )}, \frac {1}{1-\frac {\sqrt {-a^{2}+b^{2}}}{b}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {-a^{2}+b^{2}}\, b \sqrt {\left (\cos ^{2}\left (d x +c \right )\right ) e \sin \left (d x +c \right )}\, \left (1-\frac {\sqrt {-a^{2}+b^{2}}}{b}\right )}+\frac {\sqrt {1-\sin \left (d x +c \right )}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \left (\sqrt {\sin }\left (d x +c \right )\right ) \Pi \left (\sqrt {1-\sin \left (d x +c \right )}, \frac {1}{1+\frac {\sqrt {-a^{2}+b^{2}}}{b}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {-a^{2}+b^{2}}\, b \sqrt {\left (\cos ^{2}\left (d x +c \right )\right ) e \sin \left (d x +c \right )}\, \left (1+\frac {\sqrt {-a^{2}+b^{2}}}{b}\right )}\right )}{b^{4}}\right )}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(773\) |
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Timed out. \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \cos (c+d x)} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \cos (c+d x)} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \cos (c+d x)} \, dx=\int \frac {{\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}}{a+b\,\cos \left (c+d\,x\right )} \,d x \]
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