Optimal. Leaf size=92 \[ \frac {c \cos (e+f x) (a+a \sin (e+f x))^{9/2}}{15 a f \sqrt {c-c \sin (e+f x)}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2} \sqrt {c-c \sin (e+f x)}}{6 a f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.28, antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.079, Rules used = {2920, 2819,
2817} \begin {gather*} \frac {\cos (e+f x) (a \sin (e+f x)+a)^{9/2} \sqrt {c-c \sin (e+f x)}}{6 a f}+\frac {c \cos (e+f x) (a \sin (e+f x)+a)^{9/2}}{15 a f \sqrt {c-c \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2817
Rule 2819
Rule 2920
Rubi steps
\begin {align*} \int \cos ^2(e+f x) (a+a \sin (e+f x))^{7/2} \sqrt {c-c \sin (e+f x)} \, dx &=\frac {\int (a+a \sin (e+f x))^{9/2} (c-c \sin (e+f x))^{3/2} \, dx}{a c}\\ &=\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2} \sqrt {c-c \sin (e+f x)}}{6 a f}+\frac {\int (a+a \sin (e+f x))^{9/2} \sqrt {c-c \sin (e+f x)} \, dx}{3 a}\\ &=\frac {c \cos (e+f x) (a+a \sin (e+f x))^{9/2}}{15 a f \sqrt {c-c \sin (e+f x)}}+\frac {\cos (e+f x) (a+a \sin (e+f x))^{9/2} \sqrt {c-c \sin (e+f x)}}{6 a f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.37, size = 104, normalized size = 1.13 \begin {gather*} \frac {a^3 \sec (e+f x) \sqrt {a (1+\sin (e+f x))} \sqrt {c-c \sin (e+f x)} (-405 \cos (2 (e+f x))-90 \cos (4 (e+f x))+5 \cos (6 (e+f x))+1080 \sin (e+f x)+20 \sin (3 (e+f x))-36 \sin (5 (e+f x)))}{960 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.17, size = 133, normalized size = 1.45
method | result | size |
default | \(-\frac {\sqrt {-c \left (\sin \left (f x +e \right )-1\right )}\, \sin \left (f x +e \right ) \left (a \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}} \left (-5 \left (\cos ^{8}\left (f x +e \right )\right )+3 \left (\cos ^{6}\left (f x +e \right )\right ) \sin \left (f x +e \right )-4 \left (\cos ^{6}\left (f x +e \right )\right )+7 \left (\cos ^{4}\left (f x +e \right )\right ) \sin \left (f x +e \right )+7 \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )+7 \left (\cos ^{2}\left (f x +e \right )\right )+28 \sin \left (f x +e \right )-28\right )}{30 f \cos \left (f x +e \right )^{7}}\) | \(133\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.35, size = 118, normalized size = 1.28 \begin {gather*} \frac {{\left (5 \, a^{3} \cos \left (f x + e\right )^{6} - 30 \, a^{3} \cos \left (f x + e\right )^{4} + 25 \, a^{3} - 2 \, {\left (9 \, a^{3} \cos \left (f x + e\right )^{4} - 8 \, a^{3} \cos \left (f x + e\right )^{2} - 16 \, a^{3}\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}}{30 \, f \cos \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.52, size = 108, normalized size = 1.17 \begin {gather*} \frac {32 \, {\left (5 \, a^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{12} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 6 \, a^{3} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{10} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {a} \sqrt {c}}{15 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 11.49, size = 121, normalized size = 1.32 \begin {gather*} -\frac {a^3\,\sqrt {a\,\left (\sin \left (e+f\,x\right )+1\right )}\,\sqrt {-c\,\left (\sin \left (e+f\,x\right )-1\right )}\,\left (405\,\cos \left (e+f\,x\right )+495\,\cos \left (3\,e+3\,f\,x\right )+85\,\cos \left (5\,e+5\,f\,x\right )-5\,\cos \left (7\,e+7\,f\,x\right )-1100\,\sin \left (2\,e+2\,f\,x\right )+16\,\sin \left (4\,e+4\,f\,x\right )+36\,\sin \left (6\,e+6\,f\,x\right )\right )}{960\,f\,\left (\cos \left (2\,e+2\,f\,x\right )+1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________