Optimal. Leaf size=250 \[ \frac {4 e \csc ^4(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {4 e \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{5 a^2 d}-\frac {4 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{15 a^2 d}-\frac {2 e \cot ^3(c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {2 e \cot (c+d x) \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}+\frac {16 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{45 a^2 d}-\frac {4 e \sqrt {\sin (c+d x)} E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{15 a^2 d} \]
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Rubi [A] time = 0.49, antiderivative size = 250, normalized size of antiderivative = 1.00, number of steps used = 16, number of rules used = 9, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.360, Rules used = {3878, 3872, 2875, 2873, 2567, 2636, 2639, 2564, 14} \[ \frac {4 e \csc ^4(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {4 e \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{5 a^2 d}-\frac {4 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{15 a^2 d}-\frac {2 e \cot ^3(c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {2 e \cot (c+d x) \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}+\frac {16 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{45 a^2 d}-\frac {4 e \sqrt {\sin (c+d x)} E\left (\left .\frac {1}{2} \left (c+d x-\frac {\pi }{2}\right )\right |2\right ) \sqrt {e \csc (c+d x)}}{15 a^2 d} \]
Antiderivative was successfully verified.
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Rule 14
Rule 2564
Rule 2567
Rule 2636
Rule 2639
Rule 2873
Rule 2875
Rule 3872
Rule 3878
Rubi steps
\begin {align*} \int \frac {(e \csc (c+d x))^{3/2}}{(a+a \sec (c+d x))^2} \, dx &=\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{(a+a \sec (c+d x))^2 \sin ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x)}{(-a-a \cos (c+d x))^2 \sin ^{\frac {3}{2}}(c+d x)} \, dx\\ &=\frac {\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x) (-a+a \cos (c+d x))^2}{\sin ^{\frac {11}{2}}(c+d x)} \, dx}{a^4}\\ &=\frac {\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \left (\frac {a^2 \cos ^2(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)}-\frac {2 a^2 \cos ^3(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)}+\frac {a^2 \cos ^4(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)}\right ) \, dx}{a^4}\\ &=\frac {\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^2(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)} \, dx}{a^2}+\frac {\left (e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^4(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)} \, dx}{a^2}-\frac {\left (2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {\cos ^3(c+d x)}{\sin ^{\frac {11}{2}}(c+d x)} \, dx}{a^2}\\ &=-\frac {2 e \cot ^3(c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {2 e \cot (c+d x) \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {\left (2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sin ^{\frac {7}{2}}(c+d x)} \, dx}{9 a^2}-\frac {\left (2 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}{3 a^2}-\frac {\left (2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \operatorname {Subst}\left (\int \frac {1-x^2}{x^{11/2}} \, dx,x,\sin (c+d x)\right )}{a^2 d}\\ &=\frac {16 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{45 a^2 d}-\frac {2 e \cot ^3(c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {2 e \cot (c+d x) \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 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}{15 a^2}+\frac {\left (4 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \frac {1}{\sin ^{\frac {3}{2}}(c+d x)} \, dx}{15 a^2}-\frac {\left (2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \operatorname {Subst}\left (\int \left (\frac {1}{x^{11/2}}-\frac {1}{x^{7/2}}\right ) \, dx,x,\sin (c+d x)\right )}{a^2 d}\\ &=-\frac {4 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{15 a^2 d}+\frac {16 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{45 a^2 d}-\frac {2 e \cot ^3(c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {4 e \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{5 a^2 d}-\frac {2 e \cot (c+d x) \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}+\frac {4 e \csc ^4(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}+\frac {\left (2 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{15 a^2}-\frac {\left (4 e \sqrt {e \csc (c+d x)} \sqrt {\sin (c+d x)}\right ) \int \sqrt {\sin (c+d x)} \, dx}{15 a^2}\\ &=-\frac {4 e \cos (c+d x) \sqrt {e \csc (c+d x)}}{15 a^2 d}+\frac {16 e \cot (c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{45 a^2 d}-\frac {2 e \cot ^3(c+d x) \csc (c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}-\frac {4 e \csc ^2(c+d x) \sqrt {e \csc (c+d x)}}{5 a^2 d}-\frac {2 e \cot (c+d x) \csc ^3(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 d}+\frac {4 e \csc ^4(c+d x) \sqrt {e \csc (c+d x)}}{9 a^2 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)}}{15 a^2 d}\\ \end {align*}
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Mathematica [C] time = 1.83, size = 247, normalized size = 0.99 \[ \frac {\cos ^4\left (\frac {1}{2} (c+d x)\right ) \sec (c+d x) (e \csc (c+d x))^{3/2} \left (-\frac {2 \tan (c+d x) \left ((13 \cos (c+d x)+8) \sec ^4\left (\frac {1}{2} (c+d x)\right )+24 \sec (c) \cos (d x)\right )}{d}+\frac {16 \sqrt {2} e^{i (c-d x)} \sqrt {\frac {i e^{i (c+d x)}}{-1+e^{2 i (c+d x)}}} \left (\left (1+e^{2 i c}\right ) e^{2 i d x} \sqrt {1-e^{2 i (c+d x)}} \, _2F_1\left (\frac {1}{2},\frac {3}{4};\frac {7}{4};e^{2 i (c+d x)}\right )-3 e^{2 i (c+d x)}+3\right ) \sec (c+d x)}{\left (1+e^{2 i c}\right ) d \csc ^{\frac {3}{2}}(c+d x)}\right )}{45 a^2 (\sec (c+d x)+1)^2} \]
Antiderivative was successfully verified.
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fricas [F] time = 0.61, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {\sqrt {e \csc \left (d x + c\right )} e \csc \left (d x + c\right )}{a^{2} \sec \left (d x + c\right )^{2} + 2 \, a^{2} \sec \left (d x + c\right ) + a^{2}}, x\right ) \]
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 \[ \int \frac {\left (e \csc \left (d x + c\right )\right )^{\frac {3}{2}}}{{\left (a \sec \left (d x + c\right ) + a\right )}^{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.34, size = 1044, normalized size = 4.18 \[ \text {result too large to display} \]
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 = -0.00 \[ \int \frac {{\cos \left (c+d\,x\right )}^2\,{\left (\frac {e}{\sin \left (c+d\,x\right )}\right )}^{3/2}}{a^2\,{\left (\cos \left (c+d\,x\right )+1\right )}^2} \,d x \]
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 \[ \frac {\int \frac {\left (e \csc {\left (c + d x \right )}\right )^{\frac {3}{2}}}{\sec ^{2}{\left (c + d x \right )} + 2 \sec {\left (c + d x \right )} + 1}\, dx}{a^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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