Optimal. Leaf size=109 \[ \frac {64 a^2 c^5 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^{5/2}}+\frac {16 a^2 c^4 \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 c^3 \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.18, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.107, Rules used = {2815, 2753,
2752} \begin {gather*} \frac {64 a^2 c^5 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^{5/2}}+\frac {16 a^2 c^4 \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 c^3 \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2752
Rule 2753
Rule 2815
Rubi steps
\begin {align*} \int (a+a \sin (e+f x))^2 (c-c \sin (e+f x))^{5/2} \, dx &=\left (a^2 c^2\right ) \int \cos ^4(e+f x) \sqrt {c-c \sin (e+f x)} \, dx\\ &=\frac {2 a^2 c^3 \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}}+\frac {1}{9} \left (8 a^2 c^3\right ) \int \frac {\cos ^4(e+f x)}{\sqrt {c-c \sin (e+f x)}} \, dx\\ &=\frac {16 a^2 c^4 \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 c^3 \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}}+\frac {1}{63} \left (32 a^2 c^4\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{3/2}} \, dx\\ &=\frac {64 a^2 c^5 \cos ^5(e+f x)}{315 f (c-c \sin (e+f x))^{5/2}}+\frac {16 a^2 c^4 \cos ^5(e+f x)}{63 f (c-c \sin (e+f x))^{3/2}}+\frac {2 a^2 c^3 \cos ^5(e+f x)}{9 f \sqrt {c-c \sin (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 3.35, size = 96, normalized size = 0.88 \begin {gather*} -\frac {a^2 c^2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 (-249+35 \cos (2 (e+f x))+220 \sin (e+f x)) \sqrt {c-c \sin (e+f x)}}{315 f \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 1.73, size = 71, normalized size = 0.65
method | result | size |
default | \(-\frac {2 \left (\sin \left (f x +e \right )-1\right ) c^{3} \left (1+\sin \left (f x +e \right )\right )^{3} a^{2} \left (35 \left (\sin ^{2}\left (f x +e \right )\right )-110 \sin \left (f x +e \right )+107\right )}{315 \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(71\) |
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 [B] Leaf count of result is larger than twice the leaf count of optimal. 214 vs.
\(2 (103) = 206\).
time = 0.34, size = 214, normalized size = 1.96 \begin {gather*} \frac {2 \, {\left (35 \, a^{2} c^{2} \cos \left (f x + e\right )^{5} - 5 \, a^{2} c^{2} \cos \left (f x + e\right )^{4} + 8 \, a^{2} c^{2} \cos \left (f x + e\right )^{3} - 16 \, a^{2} c^{2} \cos \left (f x + e\right )^{2} + 64 \, a^{2} c^{2} \cos \left (f x + e\right ) + 128 \, a^{2} c^{2} + {\left (35 \, a^{2} c^{2} \cos \left (f x + e\right )^{4} + 40 \, a^{2} c^{2} \cos \left (f x + e\right )^{3} + 48 \, a^{2} c^{2} \cos \left (f x + e\right )^{2} + 64 \, a^{2} c^{2} \cos \left (f x + e\right ) + 128 \, a^{2} c^{2}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{315 \, {\left (f \cos \left (f x + e\right ) - f \sin \left (f x + e\right ) + f\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} a^{2} \left (\int c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c}\, dx + \int \left (- 2 c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{2}{\left (e + f x \right )}\right )\, dx + \int c^{2} \sqrt {- c \sin {\left (e + f x \right )} + c} \sin ^{4}{\left (e + f x \right )}\, dx\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 0.57, size = 187, normalized size = 1.72 \begin {gather*} -\frac {\sqrt {2} {\left (1890 \, a^{2} c^{2} \cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 420 \, a^{2} c^{2} \cos \left (-\frac {3}{4} \, \pi + \frac {3}{2} \, f x + \frac {3}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 252 \, a^{2} c^{2} \cos \left (-\frac {5}{4} \, \pi + \frac {5}{2} \, f x + \frac {5}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) - 45 \, a^{2} c^{2} \cos \left (-\frac {7}{4} \, \pi + \frac {7}{2} \, f x + \frac {7}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 35 \, a^{2} c^{2} \cos \left (-\frac {9}{4} \, \pi + \frac {9}{2} \, f x + \frac {9}{2} \, e\right ) \mathrm {sgn}\left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \sqrt {c}}{2520 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int {\left (a+a\,\sin \left (e+f\,x\right )\right )}^2\,{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________