Optimal. Leaf size=145 \[ -\frac {2 e \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{5 a d}-\frac {4 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{5 a d}+\frac {2 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{5 a d}-\frac {4 e \sqrt {\sin (c+d x)} E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{5 a d} \]
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Rubi [A] time = 0.23, antiderivative size = 145, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3878, 3872, 2839, 2564, 30, 2567, 2636, 2639} \[ -\frac {2 e \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{5 a d}-\frac {4 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{5 a d}+\frac {2 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{5 a d}-\frac {4 e \sqrt {\sin (c+d x)} E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{5 a d} \]
Antiderivative was successfully verified.
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Rule 30
Rule 2564
Rule 2567
Rule 2636
Rule 2639
Rule 2839
Rule 3872
Rule 3878
Rubi steps
\begin {align*} \int \frac {(e \csc (c+d x))^{3/2}}{a+a \sec (c+d x)} \, dx &=\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{(a+a \sec (c+d x)) \sin ^{\frac {3}{2}}(c+d x)} \, dx\\ &=-\left (\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos (c+d x)}{(-a-a \cos (c+d x)) \sin ^{\frac {3}{2}}(c+d x)} \, dx\right )\\ &=\frac {\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos (c+d x)}{\sin ^{\frac {7}{2}}(c+d x)} \, dx}{a}-\frac {\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x)}{\sin ^{\frac {7}{2}}(c+d x)} \, dx}{a}\\ &=\frac {2 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{5 a d}+\frac {\left (2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sin ^{\frac {3}{2}}(c+d x)} \, dx}{5 a}+\frac {\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1}{x^{7/2}} \, dx,x,\sin (c+d x)\right )}{a d}\\ &=-\frac {4 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{5 a d}+\frac {2 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{5 a d}-\frac {2 e \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{5 a d}-\frac {\left (2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{5 a}\\ &=-\frac {4 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{5 a d}+\frac {2 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{5 a d}-\frac {2 e \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{5 a d}-\frac {4 e \sqrt {e \csc (c+d x)} E\left (\left .\frac {1}{2} \left (c-\frac {\pi }{2}+d x\right )\right |2\right ) \sqrt {\sin (c+d x)}}{5 a d}\\ \end {align*}
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Mathematica [C] time = 1.39, size = 230, normalized size = 1.59 \[ \frac {\cos ^2\left (\frac {1}{2} (c+d x)\right ) (e \csc (c+d x))^{3/2} \left (-\frac {6 \tan (c+d x) \left (\sec ^2\left (\frac {1}{2} (c+d x)\right )+4 \sec (c) \cos (d x)\right )}{d}+\frac {8 \sqrt {2} e^{i (c-d x)} \sqrt {\frac {i e^{i (c+d x)}}{-1+e^{2 i (c+d x)}}} \left (\left (1+e^{2 i c}\right ) e^{2 i d x} \sqrt {1-e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};e^{2 i (c+d x)}\right )-3 e^{2 i (c+d x)}+3\right ) \sec (c+d x)}{\left (1+e^{2 i c}\right ) d \csc ^{\frac {3}{2}}(c+d x)}\right )}{15 a (\sec (c+d x)+1)} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.45, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \csc \left (d x + c\right )} e \csc \left (d x + c\right )}{a \sec \left (d x + c\right ) + a}, x\right ) \]
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 \[ \int \frac {\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}}}{a \sec \left (d x + c\right ) + a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.29, size = 799, normalized size = 5.51 \[ -\frac {\left (-1+\cos \left (d x +c \right )\right ) \left (4 \left (\cos ^{2}\left (d x +c \right )\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 )}}\, \sqrt {-\frac {i \cos \left (d x +c \right )-i-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \EllipticE \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 \left (\cos ^{2}\left (d x +c \right )\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 )}}\, \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 )+8 \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 \cos \left (d x +c \right )-i-\sin \left (d x +c \right )}{\sin \left (d x +c \right )}}\, \EllipticE \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 ) \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}-4 \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 \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 ) \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}+4 \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 )}}\, \EllipticE \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 ) \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}-2 \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 ) \sqrt {-\frac {i \left (-1+\cos \left (d x +c \right )\right )}{\sin \left (d x +c \right )}}-2 \cos \left (d x +c \right ) \sqrt {2}-3 \sqrt {2}\right ) \left (\frac {e}{\sin \left (d x +c \right )}\right )^{\frac {3}{2}} \sqrt {2}}{5 a d \sin \left (d x +c \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
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 \[ \int \frac {\cos \left (c+d\,x\right )\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2}}{a\,\left (\cos \left (c+d\,x\right )+1\right )} \,d x \]
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 \[ \frac {\int \frac {\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\sec {\left (c + d x \right )} + 1}\, dx}{a} \]
Verification of antiderivative is not currently implemented for this CAS.
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