Optimal. Leaf size=240 \[ -\frac {2 (a \sin (d+e x)+b+c \cos (d+e x))^2 E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \left (a^2-b^2+c^2\right ) \sin ^{\frac {3}{2}}(d+e x) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}-\frac {2 (a \cos (d+e x)-c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))}{e \left (a^2-b^2+c^2\right ) \sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}} \]
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Rubi [A] time = 0.21, antiderivative size = 240, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.121, Rules used = {3164, 3128, 3119, 2653} \[ -\frac {2 (a \sin (d+e x)+b+c \cos (d+e x))^2 E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{e \left (a^2-b^2+c^2\right ) \sin ^{\frac {3}{2}}(d+e x) \sqrt {\frac {a \sin (d+e x)+b+c \cos (d+e x)}{\sqrt {a^2+c^2}+b}} (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}}-\frac {2 (a \cos (d+e x)-c \sin (d+e x)) (a \sin (d+e x)+b+c \cos (d+e x))}{e \left (a^2-b^2+c^2\right ) \sin ^{\frac {3}{2}}(d+e x) (a+b \csc (d+e x)+c \cot (d+e x))^{3/2}} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 3119
Rule 3128
Rule 3164
Rubi steps
\begin {align*} \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx &=\frac {(b+c \cos (d+e x)+a \sin (d+e x))^{3/2} \int \frac {1}{(b+c \cos (d+e x)+a \sin (d+e x))^{3/2}} \, dx}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)}\\ &=-\frac {2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{\left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)}-\frac {(b+c \cos (d+e x)+a \sin (d+e x))^{3/2} \int \sqrt {b+c \cos (d+e x)+a \sin (d+e x)} \, dx}{\left (a^2-b^2+c^2\right ) (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)}\\ &=-\frac {2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{\left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)}-\frac {(b+c \cos (d+e x)+a \sin (d+e x))^2 \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(c,a)\right )}{b+\sqrt {a^2+c^2}}} \, dx}{\left (a^2-b^2+c^2\right ) (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}\\ &=-\frac {2 E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(c,a)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (b+c \cos (d+e x)+a \sin (d+e x))^2}{\left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x) \sqrt {\frac {b+c \cos (d+e x)+a \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}-\frac {2 (b+c \cos (d+e x)+a \sin (d+e x)) (a \cos (d+e x)-c \sin (d+e x))}{\left (a^2-b^2+c^2\right ) e (a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)}\\ \end {align*}
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Mathematica [F] time = 20.78, size = 0, normalized size = 0.00 \[ \int \frac {1}{(a+c \cot (d+e x)+b \csc (d+e x))^{3/2} \sin ^{\frac {3}{2}}(d+e x)} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 1.20, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a} \sqrt {\sin \left (e x + d\right )}}{a^{2} \cos \left (e x + d\right )^{2} + {\left (c^{2} \cos \left (e x + d\right )^{2} - c^{2}\right )} \cot \left (e x + d\right )^{2} + {\left (b^{2} \cos \left (e x + d\right )^{2} - b^{2}\right )} \csc \left (e x + d\right )^{2} - a^{2} + 2 \, {\left (a c \cos \left (e x + d\right )^{2} - a c\right )} \cot \left (e x + d\right ) + 2 \, {\left (a b \cos \left (e x + d\right )^{2} - a b + {\left (b c \cos \left (e x + d\right )^{2} - b c\right )} \cot \left (e x + d\right )\right )} \csc \left (e x + d\right )}, x\right ) \]
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 \[ \int \frac {1}{{\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac {3}{2}} \sin \left (e x + d\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.96, size = 12231, normalized size = 50.96 \[ \text {output too large to display} \]
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 \[ \int \frac {1}{{\left (c \cot \left (e x + d\right ) + b \csc \left (e x + d\right ) + a\right )}^{\frac {3}{2}} \sin \left (e x + d\right )^{\frac {3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{{\sin \left (d+e\,x\right )}^{3/2}\,{\left (a+c\,\mathrm {cot}\left (d+e\,x\right )+\frac {b}{\sin \left (d+e\,x\right )}\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [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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