Integrand size = 25, antiderivative size = 516 \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \sec (c+d x)} \, dx=-\frac {b \left (a^2-b^2\right )^{5/4} e^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{9/2} d}-\frac {b \left (a^2-b^2\right )^{5/4} e^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{9/2} d}+\frac {2 \left (5 a^4-28 a^2 b^2+21 b^4\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a^5 d \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (a^2-b^2\right )^2 e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^5 \left (a^2-b^2-a \sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (a^2-b^2\right )^2 e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^5 \left (a^2-b^2+a \sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 e^3 \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{21 a^4 d}+\frac {2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d} \]
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Time = 2.05 (sec) , antiderivative size = 516, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.480, Rules used = {3957, 2944, 2946, 2721, 2720, 2781, 2886, 2884, 335, 218, 214, 211} \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \sec (c+d x)} \, dx=\frac {2 e (e \sin (c+d x))^{5/2} (7 b-5 a \cos (c+d x))}{35 a^2 d}-\frac {b e^{7/2} \left (a^2-b^2\right )^{5/4} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{a^{9/2} d}-\frac {b e^{7/2} \left (a^2-b^2\right )^{5/4} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{a^2-b^2}}\right )}{a^{9/2} d}+\frac {b^2 e^4 \left (a^2-b^2\right )^2 \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a^5 d \left (-a \sqrt {a^2-b^2}+a^2-b^2\right ) \sqrt {e \sin (c+d x)}}+\frac {b^2 e^4 \left (a^2-b^2\right )^2 \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{a^5 d \left (a \sqrt {a^2-b^2}+a^2-b^2\right ) \sqrt {e \sin (c+d x)}}+\frac {2 e^3 \sqrt {e \sin (c+d x)} \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right )}{21 a^4 d}+\frac {2 e^4 \left (5 a^4-28 a^2 b^2+21 b^4\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{21 a^5 d \sqrt {e \sin (c+d x)}} \]
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Rule 211
Rule 214
Rule 218
Rule 335
Rule 2720
Rule 2721
Rule 2781
Rule 2884
Rule 2886
Rule 2944
Rule 2946
Rule 3957
Rubi steps \begin{align*} \text {integral}& = -\int \frac {\cos (c+d x) (e \sin (c+d x))^{7/2}}{-b-a \cos (c+d x)} \, dx \\ & = \frac {2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d}-\frac {\left (2 e^2\right ) \int \frac {\left (-a b+\frac {1}{2} \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) (e \sin (c+d x))^{3/2}}{-b-a \cos (c+d x)} \, dx}{7 a^2} \\ & = \frac {2 e^3 \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{21 a^4 d}+\frac {2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d}-\frac {\left (4 e^4\right ) \int \frac {-\frac {1}{2} a b \left (8 a^2-7 b^2\right )+\frac {1}{4} \left (5 a^4-28 a^2 b^2+21 b^4\right ) \cos (c+d x)}{(-b-a \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{21 a^4} \\ & = \frac {2 e^3 \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{21 a^4 d}+\frac {2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d}+\frac {\left (b \left (a^2-b^2\right )^2 e^4\right ) \int \frac {1}{(-b-a \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{a^5}+\frac {\left (\left (5 a^4-28 a^2 b^2+21 b^4\right ) e^4\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{21 a^5} \\ & = \frac {2 e^3 \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{21 a^4 d}+\frac {2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d}+\frac {\left (b^2 \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}-a \sin (c+d x)\right )} \, dx}{2 a^5}+\frac {\left (b^2 \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}+a \sin (c+d x)\right )} \, dx}{2 a^5}+\frac {\left (b \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+a^2 x^2\right )} \, dx,x,e \sin (c+d x)\right )}{a^4 d}+\frac {\left (\left (5 a^4-28 a^2 b^2+21 b^4\right ) e^4 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{21 a^5 \sqrt {e \sin (c+d x)}} \\ & = \frac {2 \left (5 a^4-28 a^2 b^2+21 b^4\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a^5 d \sqrt {e \sin (c+d x)}}+\frac {2 e^3 \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{21 a^4 d}+\frac {2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d}+\frac {\left (2 b \left (a^2-b^2\right )^2 e^5\right ) \text {Subst}\left (\int \frac {1}{\left (-a^2+b^2\right ) e^2+a^2 x^4} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^4 d}+\frac {\left (b^2 \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}-a \sin (c+d x)\right )} \, dx}{2 a^5 \sqrt {e \sin (c+d x)}}+\frac {\left (b^2 \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}+a \sin (c+d x)\right )} \, dx}{2 a^5 \sqrt {e \sin (c+d x)}} \\ & = \frac {2 \left (5 a^4-28 a^2 b^2+21 b^4\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a^5 d \sqrt {e \sin (c+d x)}}-\frac {b^2 \left (a^2-b^2\right )^{3/2} e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^5 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (a^2-b^2\right )^{3/2} e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^5 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 e^3 \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{21 a^4 d}+\frac {2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d}-\frac {\left (b \left (a^2-b^2\right )^{3/2} e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e-a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^4 d}-\frac {\left (b \left (a^2-b^2\right )^{3/2} e^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a^2-b^2} e+a x^2} \, dx,x,\sqrt {e \sin (c+d x)}\right )}{a^4 d} \\ & = -\frac {b \left (a^2-b^2\right )^{5/4} e^{7/2} \arctan \left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{9/2} d}-\frac {b \left (a^2-b^2\right )^{5/4} e^{7/2} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {e \sin (c+d x)}}{\sqrt [4]{a^2-b^2} \sqrt {e}}\right )}{a^{9/2} d}+\frac {2 \left (5 a^4-28 a^2 b^2+21 b^4\right ) e^4 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{21 a^5 d \sqrt {e \sin (c+d x)}}-\frac {b^2 \left (a^2-b^2\right )^{3/2} e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a-\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^5 \left (a-\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {b^2 \left (a^2-b^2\right )^{3/2} e^4 \operatorname {EllipticPi}\left (\frac {2 a}{a+\sqrt {a^2-b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{a^5 \left (a+\sqrt {a^2-b^2}\right ) d \sqrt {e \sin (c+d x)}}+\frac {2 e^3 \left (21 b \left (a^2-b^2\right )-a \left (5 a^2-7 b^2\right ) \cos (c+d x)\right ) \sqrt {e \sin (c+d x)}}{21 a^4 d}+\frac {2 e (7 b-5 a \cos (c+d x)) (e \sin (c+d x))^{5/2}}{35 a^2 d} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 47.84 (sec) , antiderivative size = 2049, normalized size of antiderivative = 3.97 \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \sec (c+d x)} \, dx=\text {Result too large to show} \]
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Time = 16.19 (sec) , antiderivative size = 772, normalized size of antiderivative = 1.50
method | result | size |
default | \(\frac {2 e b \left (-\frac {\sqrt {e \sin \left (d x +c \right )}\, e^{2} \left (\cos \left (d x +c \right )^{2} a^{2}-6 a^{2}+5 b^{2}\right )}{5 a^{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 )}{a^{2}}\right )^{\frac {1}{4}} \left (\ln \left (\frac {\sqrt {e \sin \left (d x +c \right )}+\left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{a^{2}}\right )^{\frac {1}{4}}}{\sqrt {e \sin \left (d x +c \right )}-\left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{a^{2}}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{\left (\frac {e^{2} \left (a^{2}-b^{2}\right )}{a^{2}}\right )^{\frac {1}{4}}}\right )\right )}{4 a^{4} \left (-a^{2} e^{2}+b^{2} e^{2}\right )}\right )+\frac {\sqrt {\cos \left (d x +c \right )^{2} e \sin \left (d x +c \right )}\, a \,e^{4} \left (-\frac {-6 a^{4} \cos \left (d x +c \right )^{4} \sin \left (d x +c \right )+5 \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right ) a^{4}-28 a^{2} b^{2} \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+21 b^{4} \sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticF}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {\sqrt {2}}{2}\right )+16 a^{4} \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )-14 a^{2} b^{2} \cos \left (d x +c \right )^{2} \sin \left (d x +c \right )}{21 a^{6} \sqrt {\cos \left (d x +c \right )^{2} e \sin \left (d x +c \right )}}-\frac {b^{2} \left (a^{4}-2 a^{2} b^{2}+b^{4}\right ) \left (-\frac {\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {1}{1-\frac {\sqrt {a^{2}-b^{2}}}{a}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a^{2}-b^{2}}\, a \sqrt {\cos \left (d x +c \right )^{2} e \sin \left (d x +c \right )}\, \left (1-\frac {\sqrt {a^{2}-b^{2}}}{a}\right )}+\frac {\sqrt {-\sin \left (d x +c \right )+1}\, \sqrt {2 \sin \left (d x +c \right )+2}\, \sqrt {\sin \left (d x +c \right )}\, \operatorname {EllipticPi}\left (\sqrt {-\sin \left (d x +c \right )+1}, \frac {1}{1+\frac {\sqrt {a^{2}-b^{2}}}{a}}, \frac {\sqrt {2}}{2}\right )}{2 \sqrt {a^{2}-b^{2}}\, a \sqrt {\cos \left (d x +c \right )^{2} e \sin \left (d x +c \right )}\, \left (1+\frac {\sqrt {a^{2}-b^{2}}}{a}\right )}\right )}{a^{6}}\right )}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(772\) |
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Timed out. \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \sec (c+d x)} \, dx=\text {Timed out} \]
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Timed out. \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \sec (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \sec (c+d x)} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}}{b \sec \left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \sec (c+d x)} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {7}{2}}}{b \sec \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e \sin (c+d x))^{7/2}}{a+b \sec (c+d x)} \, dx=\int \frac {\cos \left (c+d\,x\right )\,{\left (e\,\sin \left (c+d\,x\right )\right )}^{7/2}}{b+a\,\cos \left (c+d\,x\right )} \,d x \]
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