Optimal. Leaf size=371 \[ \frac {2 \left (a^2-b^2+c^2\right ) \cos ^{\frac {3}{2}}(d+e x) \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}} (a+b \sec (d+e x)+c \tan (d+e x))^{3/2} F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{3 e (a \cos (d+e x)+b+c \sin (d+e x))^2}+\frac {8 b \cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{3 e (a \cos (d+e x)+b+c \sin (d+e x)) \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}}}-\frac {2 \cos ^{\frac {3}{2}}(d+e x) (c \cos (d+e x)-a \sin (d+e x)) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}{3 e (a \cos (d+e x)+b+c \sin (d+e x))} \]
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Rubi [A] time = 0.39, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.212, Rules used = {3163, 3120, 3149, 3119, 2653, 3127, 2661} \[ \frac {2 \left (a^2-b^2+c^2\right ) \cos ^{\frac {3}{2}}(d+e x) \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}} (a+b \sec (d+e x)+c \tan (d+e x))^{3/2} F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{3 e (a \cos (d+e x)+b+c \sin (d+e x))^2}+\frac {8 b \cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2} E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right )}{3 e (a \cos (d+e x)+b+c \sin (d+e x)) \sqrt {\frac {a \cos (d+e x)+b+c \sin (d+e x)}{\sqrt {a^2+c^2}+b}}}-\frac {2 \cos ^{\frac {3}{2}}(d+e x) (c \cos (d+e x)-a \sin (d+e x)) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}{3 e (a \cos (d+e x)+b+c \sin (d+e x))} \]
Antiderivative was successfully verified.
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Rule 2653
Rule 2661
Rule 3119
Rule 3120
Rule 3127
Rule 3149
Rule 3163
Rubi steps
\begin {align*} \int \cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2} \, dx &=\frac {\left (\cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}\right ) \int (b+a \cos (d+e x)+c \sin (d+e x))^{3/2} \, dx}{(b+a \cos (d+e x)+c \sin (d+e x))^{3/2}}\\ &=-\frac {2 \cos ^{\frac {3}{2}}(d+e x) (c \cos (d+e x)-a \sin (d+e x)) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}{3 e (b+a \cos (d+e x)+c \sin (d+e x))}+\frac {\left (2 \cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}\right ) \int \frac {\frac {1}{2} \left (a^2+3 b^2+c^2\right )+2 a b \cos (d+e x)+2 b c \sin (d+e x)}{\sqrt {b+a \cos (d+e x)+c \sin (d+e x)}} \, dx}{3 (b+a \cos (d+e x)+c \sin (d+e x))^{3/2}}\\ &=-\frac {2 \cos ^{\frac {3}{2}}(d+e x) (c \cos (d+e x)-a \sin (d+e x)) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}{3 e (b+a \cos (d+e x)+c \sin (d+e x))}+\frac {\left (4 b \cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}\right ) \int \sqrt {b+a \cos (d+e x)+c \sin (d+e x)} \, dx}{3 (b+a \cos (d+e x)+c \sin (d+e x))^{3/2}}+\frac {\left (\left (a^2-b^2+c^2\right ) \cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}\right ) \int \frac {1}{\sqrt {b+a \cos (d+e x)+c \sin (d+e x)}} \, dx}{3 (b+a \cos (d+e x)+c \sin (d+e x))^{3/2}}\\ &=-\frac {2 \cos ^{\frac {3}{2}}(d+e x) (c \cos (d+e x)-a \sin (d+e x)) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}{3 e (b+a \cos (d+e x)+c \sin (d+e x))}+\frac {\left (4 b \cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}\right ) \int \sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(a,c)\right )}{b+\sqrt {a^2+c^2}}} \, dx}{3 (b+a \cos (d+e x)+c \sin (d+e x)) \sqrt {\frac {b+a \cos (d+e x)+c \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}+\frac {\left (\left (a^2-b^2+c^2\right ) \cos ^{\frac {3}{2}}(d+e x) \sqrt {\frac {b+a \cos (d+e x)+c \sin (d+e x)}{b+\sqrt {a^2+c^2}}} (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}\right ) \int \frac {1}{\sqrt {\frac {b}{b+\sqrt {a^2+c^2}}+\frac {\sqrt {a^2+c^2} \cos \left (d+e x-\tan ^{-1}(a,c)\right )}{b+\sqrt {a^2+c^2}}}} \, dx}{3 (b+a \cos (d+e x)+c \sin (d+e x))^2}\\ &=-\frac {2 \cos ^{\frac {3}{2}}(d+e x) (c \cos (d+e x)-a \sin (d+e x)) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}{3 e (b+a \cos (d+e x)+c \sin (d+e x))}+\frac {8 b \cos ^{\frac {3}{2}}(d+e x) E\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}{3 e (b+a \cos (d+e x)+c \sin (d+e x)) \sqrt {\frac {b+a \cos (d+e x)+c \sin (d+e x)}{b+\sqrt {a^2+c^2}}}}+\frac {2 \left (a^2-b^2+c^2\right ) \cos ^{\frac {3}{2}}(d+e x) F\left (\frac {1}{2} \left (d+e x-\tan ^{-1}(a,c)\right )|\frac {2 \sqrt {a^2+c^2}}{b+\sqrt {a^2+c^2}}\right ) \sqrt {\frac {b+a \cos (d+e x)+c \sin (d+e x)}{b+\sqrt {a^2+c^2}}} (a+b \sec (d+e x)+c \tan (d+e x))^{3/2}}{3 e (b+a \cos (d+e x)+c \sin (d+e x))^2}\\ \end {align*}
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Mathematica [F] time = 150.81, size = 0, normalized size = 0.00 \[ \int \cos ^{\frac {3}{2}}(d+e x) (a+b \sec (d+e x)+c \tan (d+e x))^{3/2} \, dx \]
Verification is Not applicable to the result.
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fricas [F] time = 0.93, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b \cos \left (e x + d\right ) \sec \left (e x + d\right ) + c \cos \left (e x + d\right ) \tan \left (e x + d\right ) + a \cos \left (e x + d\right )\right )} \sqrt {b \sec \left (e x + d\right ) + c \tan \left (e x + d\right ) + a} \sqrt {\cos \left (e x + d\right )}, x\right ) \]
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 \[ \int {\left (b \sec \left (e x + d\right ) + c \tan \left (e x + d\right ) + a\right )}^{\frac {3}{2}} \cos \left (e x + d\right )^{\frac {3}{2}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 1.29, size = 20776, normalized size = 56.00 \[ \text {output too large to display} \]
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 \[ \int {\left (b \sec \left (e x + d\right ) + c \tan \left (e x + d\right ) + a\right )}^{\frac {3}{2}} \cos \left (e x + d\right )^{\frac {3}{2}}\,{d x} \]
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 \[ \int {\cos \left (d+e\,x\right )}^{3/2}\,{\left (a+c\,\mathrm {tan}\left (d+e\,x\right )+\frac {b}{\cos \left (d+e\,x\right )}\right )}^{3/2} \,d x \]
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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