Optimal. Leaf size=285 \[ -\frac {80 c^5 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {40 c^4 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a^2 f \sqrt {a \sin (e+f x)+a}}-\frac {10 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt {a \sin (e+f x)+a}}-\frac {10 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 f \sqrt {a \sin (e+f x)+a}}-\frac {5 c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a^2 f \sqrt {a \sin (e+f x)+a}}-\frac {\cos (e+f x) (c-c \sin (e+f x))^{9/2}}{a f (a \sin (e+f x)+a)^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.86, antiderivative size = 285, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {2841, 2739, 2740, 2737, 2667, 31} \[ -\frac {40 c^4 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a^2 f \sqrt {a \sin (e+f x)+a}}-\frac {10 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt {a \sin (e+f x)+a}}-\frac {10 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 f \sqrt {a \sin (e+f x)+a}}-\frac {80 c^5 \cos (e+f x) \log (\sin (e+f x)+1)}{a^2 f \sqrt {a \sin (e+f x)+a} \sqrt {c-c \sin (e+f x)}}-\frac {5 c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a^2 f \sqrt {a \sin (e+f x)+a}}-\frac {\cos (e+f x) (c-c \sin (e+f x))^{9/2}}{a f (a \sin (e+f x)+a)^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 2667
Rule 2737
Rule 2739
Rule 2740
Rule 2841
Rubi steps
\begin {align*} \int \frac {\cos ^2(e+f x) (c-c \sin (e+f x))^{9/2}}{(a+a \sin (e+f x))^{5/2}} \, dx &=\frac {\int \frac {(c-c \sin (e+f x))^{11/2}}{(a+a \sin (e+f x))^{3/2}} \, dx}{a c}\\ &=-\frac {\cos (e+f x) (c-c \sin (e+f x))^{9/2}}{a f (a+a \sin (e+f x))^{3/2}}-\frac {5 \int \frac {(c-c \sin (e+f x))^{9/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a^2}\\ &=-\frac {5 c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cos (e+f x) (c-c \sin (e+f x))^{9/2}}{a f (a+a \sin (e+f x))^{3/2}}-\frac {(10 c) \int \frac {(c-c \sin (e+f x))^{7/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a^2}\\ &=-\frac {10 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {5 c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cos (e+f x) (c-c \sin (e+f x))^{9/2}}{a f (a+a \sin (e+f x))^{3/2}}-\frac {\left (20 c^2\right ) \int \frac {(c-c \sin (e+f x))^{5/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a^2}\\ &=-\frac {10 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {10 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {5 c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cos (e+f x) (c-c \sin (e+f x))^{9/2}}{a f (a+a \sin (e+f x))^{3/2}}-\frac {\left (40 c^3\right ) \int \frac {(c-c \sin (e+f x))^{3/2}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a^2}\\ &=-\frac {40 c^4 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {10 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {10 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {5 c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cos (e+f x) (c-c \sin (e+f x))^{9/2}}{a f (a+a \sin (e+f x))^{3/2}}-\frac {\left (80 c^4\right ) \int \frac {\sqrt {c-c \sin (e+f x)}}{\sqrt {a+a \sin (e+f x)}} \, dx}{a^2}\\ &=-\frac {40 c^4 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {10 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {10 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {5 c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cos (e+f x) (c-c \sin (e+f x))^{9/2}}{a f (a+a \sin (e+f x))^{3/2}}-\frac {\left (80 c^5 \cos (e+f x)\right ) \int \frac {\cos (e+f x)}{a+a \sin (e+f x)} \, dx}{a \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=-\frac {40 c^4 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {10 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {10 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {5 c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cos (e+f x) (c-c \sin (e+f x))^{9/2}}{a f (a+a \sin (e+f x))^{3/2}}-\frac {\left (80 c^5 \cos (e+f x)\right ) \operatorname {Subst}\left (\int \frac {1}{a+x} \, dx,x,a \sin (e+f x)\right )}{a^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}\\ &=-\frac {80 c^5 \cos (e+f x) \log (1+\sin (e+f x))}{a^2 f \sqrt {a+a \sin (e+f x)} \sqrt {c-c \sin (e+f x)}}-\frac {40 c^4 \cos (e+f x) \sqrt {c-c \sin (e+f x)}}{a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {10 c^3 \cos (e+f x) (c-c \sin (e+f x))^{3/2}}{a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {10 c^2 \cos (e+f x) (c-c \sin (e+f x))^{5/2}}{3 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {5 c \cos (e+f x) (c-c \sin (e+f x))^{7/2}}{4 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cos (e+f x) (c-c \sin (e+f x))^{9/2}}{a f (a+a \sin (e+f x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 6.64, size = 553, normalized size = 1.94 \[ \frac {203 \sin (e+f x) (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5}{4 f (a (\sin (e+f x)+1))^{5/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}+\frac {47 \cos (2 (e+f x)) (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5}{8 f (a (\sin (e+f x)+1))^{5/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}-\frac {\cos (4 (e+f x)) (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5}{32 f (a (\sin (e+f x)+1))^{5/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}-\frac {7 \sin (3 (e+f x)) (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5}{12 f (a (\sin (e+f x)+1))^{5/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}-\frac {32 (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^3}{f (a (\sin (e+f x)+1))^{5/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9}-\frac {160 (c-c \sin (e+f x))^{9/2} \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )^5 \log \left (\sin \left (\frac {1}{2} (e+f x)\right )+\cos \left (\frac {1}{2} (e+f x)\right )\right )}{f (a (\sin (e+f x)+1))^{5/2} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [F] time = 2.26, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {{\left (c^{4} \cos \left (f x + e\right )^{6} - 8 \, c^{4} \cos \left (f x + e\right )^{4} + 8 \, c^{4} \cos \left (f x + e\right )^{2} + 4 \, {\left (c^{4} \cos \left (f x + e\right )^{4} - 2 \, c^{4} \cos \left (f x + e\right )^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {-c \sin \left (f x + e\right ) + c}}{3 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3}\right )} \sin \left (f x + e\right )}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.36, size = 349, normalized size = 1.22 \[ \frac {\left (3 \sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )-25 \left (\cos ^{4}\left (f x +e \right )\right )-116 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+500 \left (\cos ^{2}\left (f x +e \right )\right )-960 \sin \left (f x +e \right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+1920 \sin \left (f x +e \right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-859 \sin \left (f x +e \right )-960 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+1920 \ln \left (-\frac {-1+\cos \left (f x +e \right )-\sin \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-475\right ) \left (-c \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {9}{2}} \left (\cos ^{2}\left (f x +e \right )+\sin \left (f x +e \right ) \cos \left (f x +e \right )+\cos \left (f x +e \right )-2 \sin \left (f x +e \right )-2\right )}{12 f \left (\sin \left (f x +e \right ) \left (\cos ^{4}\left (f x +e \right )\right )+\cos ^{5}\left (f x +e \right )+4 \sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right )-5 \left (\cos ^{4}\left (f x +e \right )\right )-12 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-8 \left (\cos ^{3}\left (f x +e \right )\right )-8 \sin \left (f x +e \right ) \cos \left (f x +e \right )+20 \left (\cos ^{2}\left (f x +e \right )\right )+16 \sin \left (f x +e \right )+8 \cos \left (f x +e \right )-16\right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {9}{2}} \cos \left (f x + e\right )^{2}}{{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {{\cos \left (e+f\,x\right )}^2\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{9/2}}{{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
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.
[In]
[Out]
________________________________________________________________________________________