Optimal. Leaf size=105 \[ -\frac {2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}+\frac {2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}+\frac {4 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{3 a d} \]
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Rubi [A] time = 0.20, antiderivative size = 105, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.280, Rules used = {3878, 3872, 2839, 2564, 30, 2567, 2641} \[ -\frac {2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}+\frac {2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}+\frac {4 \sqrt {\sin (c+d x)} F\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{3 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2564
Rule 2567
Rule 2641
Rule 2839
Rule 3872
Rule 3878
Rubi steps
\begin {align*} \int \frac {\sqrt {e \csc (c+d x)}}{a+a \sec (c+d x)} \, dx &=\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{(a+a \sec (c+d x)) \sqrt {\sin (c+d x)}} \, dx\\ &=-\left (\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos (c+d x)}{(-a-a \cos (c+d x)) \sqrt {\sin (c+d x)}} \, dx\right )\\ &=\frac {\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos (c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{a}-\frac {\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x)}{\sin ^{\frac {5}{2}}(c+d x)} \, dx}{a}\\ &=\frac {2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}+\frac {\left (2 \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sqrt {\sin (c+d x)}} \, dx}{3 a}+\frac {\left (\sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{5/2}} \, dx,x,\sin (c+d x)\right )}{a d}\\ &=\frac {2 \cot (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}-\frac {2 \csc (c+d x) \sqrt {e \csc (c+d x)}}{3 a d}+\frac {4 \sqrt {e \csc (c+d x)} F\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{3 a d}\\ \end {align*}
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Mathematica [A] time = 0.38, size = 60, normalized size = 0.57 \[ \frac {2 (e \csc (c+d x))^{3/2} \left (\cos (c+d x)-2 \sin ^{\frac {3}{2}}(c+d x) F\left (\left .\frac {1}{4} (-2 c-2 d x+\pi )\right |2\right )-1\right )}{3 a d e} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.49, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \csc \left (d x + c\right )}}{a \sec \left (d x + c\right ) + a}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e \csc \left (d x + c\right )}}{a \sec \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.34, size = 326, normalized size = 3.10 \[ \frac {\sqrt {\frac {e}{\sin \left (d x +c \right )}}\, \left (1+\cos \left (d x +c \right )\right )^{2} \left (-1+\cos \left (d x +c \right )\right )^{2} \left (2 i \cos \left (d x +c \right ) \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-i-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sin \left (d x +c \right ) \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )+2 i \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}\, \sin \left (d x +c \right ) \sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-i-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \EllipticF \left (\sqrt {\frac {i \cos \left (d x +c \right )-i+\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}, \frac {\sqrt {2}}{2}\right )+\cos \left (d x +c \right ) \sqrt {2}-\sqrt {2}\right ) \sqrt {2}}{3 a d \sin \left (d x +c \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {e \csc \left (d x + c\right )}}{a \sec \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\cos \left (c+d\,x\right )\,\sqrt {\frac {e}{\sin \left (c+d\,x\right )}}}{a\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \frac {\int \frac {\sqrt {e \csc {\left (c + d x \right )}}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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