Integrand size = 25, antiderivative size = 410 \[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \cos (c+d x)} \, dx=\frac {\sqrt [4]{-a^2+b^2} e^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{3/2} d}+\frac {\sqrt [4]{-a^2+b^2} e^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{3/2} d}+\frac {2 a e^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{b^2 d \sqrt {e \sin (c+d x)}}-\frac {a \left (a^2-b^2\right ) e^2 \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{b^2 \left (a^2-b \left (b-\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \sin (c+d x)}}-\frac {a \left (a^2-b^2\right ) e^2 \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {-a^2+b^2}},\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{b^2 \left (a^2-b \left (b+\sqrt {-a^2+b^2}\right )\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 e \sqrt {e \sin (c+d x)}}{b d} \]
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Time = 1.06 (sec) , antiderivative size = 410, 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, 2720, 2781, 2886, 2884, 335, 218, 214, 211} \[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \cos (c+d x)} \, dx=\frac {e^{3/2} \sqrt [4]{b^2-a^2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{b^{3/2} d}+\frac {e^{3/2} \sqrt [4]{b^2-a^2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt {e} \sqrt [4]{b^2-a^2}}\right )}{b^{3/2} d}-\frac {a e^2 \left (a^2-b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b-\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b^2 d \left (a^2-b \left (b-\sqrt {b^2-a^2}\right )\right ) \sqrt {e \sin (c+d x)}}-\frac {a e^2 \left (a^2-b^2\right ) \sqrt {\sin (c+d x)} \operatorname {EllipticPi}\left (\frac {2 b}{b+\sqrt {b^2-a^2}},\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b^2 d \left (a^2-b \left (\sqrt {b^2-a^2}+b\right )\right ) \sqrt {e \sin (c+d x)}}+\frac {2 a e^2 \sqrt {\sin (c+d x)} \operatorname {EllipticF}\left (\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right ),2\right )}{b^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e \sqrt {e \sin (c+d x)}}{b d} \]
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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 2946
Rubi steps \begin{align*} \text {integral}& = -\frac {2 e \sqrt {e \sin (c+d x)}}{b d}-\frac {e^2 \int \frac {-b-a \cos (c+d x)}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{b} \\ & = -\frac {2 e \sqrt {e \sin (c+d x)}}{b d}+\frac {\left (a e^2\right ) \int \frac {1}{\sqrt {e \sin (c+d x)}} \, dx}{b^2}+\frac {\left (\left (-a^2+b^2\right ) e^2\right ) \int \frac {1}{(a+b \cos (c+d x)) \sqrt {e \sin (c+d x)}} \, dx}{b^2} \\ & = -\frac {2 e \sqrt {e \sin (c+d x)}}{b d}-\frac {\left (a \sqrt {-a^2+b^2} e^2\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^2}-\frac {\left (a \sqrt {-a^2+b^2} e^2\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^2}+\frac {\left (\left (a^2-b^2\right ) e^3\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 d}+\frac {\left (a e^2 \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{b^2 \sqrt {e \sin (c+d x)}} \\ & = \frac {2 a e^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{b^2 d \sqrt {e \sin (c+d x)}}-\frac {2 e \sqrt {e \sin (c+d x)}}{b d}+\frac {\left (2 \left (a^2-b^2\right ) e^3\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 d}-\frac {\left (a \sqrt {-a^2+b^2} e^2 \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^2 \sqrt {e \sin (c+d x)}}-\frac {\left (a \sqrt {-a^2+b^2} e^2 \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^2 \sqrt {e \sin (c+d x)}} \\ & = \frac {2 a e^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{b^2 d \sqrt {e \sin (c+d x)}}+\frac {a \sqrt {-a^2+b^2} e^2 \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^2 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {a \sqrt {-a^2+b^2} e^2 \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^2 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 e \sqrt {e \sin (c+d x)}}{b d}+\frac {\left (\sqrt {-a^2+b^2} e^2\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 d}+\frac {\left (\sqrt {-a^2+b^2} e^2\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 d} \\ & = \frac {\sqrt [4]{-a^2+b^2} e^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{3/2} d}+\frac {\sqrt [4]{-a^2+b^2} e^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {e \sin (c+d x)}}{\sqrt [4]{-a^2+b^2} \sqrt {e}}\right )}{b^{3/2} d}+\frac {2 a e^2 \operatorname {EllipticF}\left (\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right ),2\right ) \sqrt {\sin (c+d x)}}{b^2 d \sqrt {e \sin (c+d x)}}+\frac {a \sqrt {-a^2+b^2} e^2 \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^2 \left (b-\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {a \sqrt {-a^2+b^2} e^2 \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^2 \left (b+\sqrt {-a^2+b^2}\right ) d \sqrt {e \sin (c+d x)}}-\frac {2 e \sqrt {e \sin (c+d x)}}{b d} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 14.11 (sec) , antiderivative size = 434, normalized size of antiderivative = 1.06 \[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \cos (c+d x)} \, dx=-\frac {\left (\frac {1}{20}-\frac {i}{20}\right ) \cos (c+d x) \left (a+b \sqrt {\cos ^2(c+d x)}\right ) (e \sin (c+d x))^{3/2} \left (-5 \left (a^2-b^2\right ) \left (2 \sqrt [4]{-a^2+b^2} \arctan \left (1-\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )-2 \sqrt [4]{-a^2+b^2} \arctan \left (1+\frac {(1+i) \sqrt {b} \sqrt {\sin (c+d x)}}{\sqrt [4]{-a^2+b^2}}\right )+\sqrt [4]{-a^2+b^2} \log \left (\sqrt {-a^2+b^2}-(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )-\sqrt [4]{-a^2+b^2} \log \left (\sqrt {-a^2+b^2}+(1+i) \sqrt {b} \sqrt [4]{-a^2+b^2} \sqrt {\sin (c+d x)}+i b \sin (c+d x)\right )+(4+4 i) \sqrt {b} \sqrt {\sin (c+d x)}\right )+(4+4 i) a b^{3/2} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\sin ^2(c+d x),\frac {b^2 \sin ^2(c+d x)}{-a^2+b^2}\right ) \sin ^{\frac {5}{2}}(c+d x)\right )}{b^{3/2} \left (-a^2+b^2\right ) d \sqrt {\cos ^2(c+d x)} (a+b \cos (c+d x)) \sin ^{\frac {3}{2}}(c+d x)} \]
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Time = 2.77 (sec) , antiderivative size = 654, normalized size of antiderivative = 1.60
method | result | size |
default | \(\frac {-2 e b \left (\frac {\sqrt {e \sin \left (d x +c \right )}}{b^{2}}-\frac {e^{2} \left (a^{2}-b^{2}\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^{2} \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^{2} a \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 ) F\left (\sqrt {1-\sin \left (d x +c \right )}, \frac {\sqrt {2}}{2}\right )}{b^{2} \sqrt {\left (\cos ^{2}\left (d x +c \right )\right ) e \sin \left (d x +c \right )}}+\frac {\left (-a^{2}+b^{2}\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^{2}}\right )}{\cos \left (d x +c \right ) \sqrt {e \sin \left (d x +c \right )}}}{d}\) | \(654\) |
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Timed out. \[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \cos (c+d x)} \, dx=\text {Timed out} \]
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\[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \cos (c+d x)} \, dx=\int \frac {\left (e \sin {\left (c + d x \right )}\right )^{\frac {3}{2}}}{a + b \cos {\left (c + d x \right )}}\, dx \]
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\[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \cos (c+d x)} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
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\[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \cos (c+d x)} \, dx=\int { \frac {\left (e \sin \left (d x + c\right )\right )^{\frac {3}{2}}}{b \cos \left (d x + c\right ) + a} \,d x } \]
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Timed out. \[ \int \frac {(e \sin (c+d x))^{3/2}}{a+b \cos (c+d x)} \, dx=\int \frac {{\left (e\,\sin \left (c+d\,x\right )\right )}^{3/2}}{a+b\,\cos \left (c+d\,x\right )} \,d x \]
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