Optimal. Leaf size=225 \[ -\frac {3 a^3 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{512 \sqrt {2} c^{11/2} f}+\frac {a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{15/2}}-\frac {a^3 \cos ^3(e+f x)}{8 f (c-c \sin (e+f x))^{11/2}}+\frac {a^3 \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{7/2}}-\frac {a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a^3 \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.26, antiderivative size = 225, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 5, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2815, 2759,
2729, 2728, 212} \begin {gather*} -\frac {3 a^3 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{512 \sqrt {2} c^{11/2} f}-\frac {3 a^3 \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}-\frac {a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}+\frac {a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{15/2}}+\frac {a^3 \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{7/2}}-\frac {a^3 \cos ^3(e+f x)}{8 f (c-c \sin (e+f x))^{11/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 2728
Rule 2729
Rule 2759
Rule 2815
Rubi steps
\begin {align*} \int \frac {(a+a \sin (e+f x))^3}{(c-c \sin (e+f x))^{11/2}} \, dx &=\left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{17/2}} \, dx\\ &=\frac {a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{15/2}}-\frac {1}{2} \left (a^3 c\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{13/2}} \, dx\\ &=\frac {a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{15/2}}-\frac {a^3 \cos ^3(e+f x)}{8 f (c-c \sin (e+f x))^{11/2}}+\frac {\left (3 a^3\right ) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{9/2}} \, dx}{16 c}\\ &=\frac {a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{15/2}}-\frac {a^3 \cos ^3(e+f x)}{8 f (c-c \sin (e+f x))^{11/2}}+\frac {a^3 \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{7/2}}-\frac {a^3 \int \frac {1}{(c-c \sin (e+f x))^{5/2}} \, dx}{32 c^3}\\ &=\frac {a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{15/2}}-\frac {a^3 \cos ^3(e+f x)}{8 f (c-c \sin (e+f x))^{11/2}}+\frac {a^3 \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{7/2}}-\frac {a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {\left (3 a^3\right ) \int \frac {1}{(c-c \sin (e+f x))^{3/2}} \, dx}{256 c^4}\\ &=\frac {a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{15/2}}-\frac {a^3 \cos ^3(e+f x)}{8 f (c-c \sin (e+f x))^{11/2}}+\frac {a^3 \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{7/2}}-\frac {a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a^3 \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}-\frac {\left (3 a^3\right ) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{1024 c^5}\\ &=\frac {a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{15/2}}-\frac {a^3 \cos ^3(e+f x)}{8 f (c-c \sin (e+f x))^{11/2}}+\frac {a^3 \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{7/2}}-\frac {a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a^3 \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}+\frac {\left (3 a^3\right ) \text {Subst}\left (\int \frac {1}{2 c-x^2} \, dx,x,-\frac {c \cos (e+f x)}{\sqrt {c-c \sin (e+f x)}}\right )}{512 c^5 f}\\ &=-\frac {3 a^3 \tanh ^{-1}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{512 \sqrt {2} c^{11/2} f}+\frac {a^3 c^2 \cos ^5(e+f x)}{5 f (c-c \sin (e+f x))^{15/2}}-\frac {a^3 \cos ^3(e+f x)}{8 f (c-c \sin (e+f x))^{11/2}}+\frac {a^3 \cos (e+f x)}{16 c^2 f (c-c \sin (e+f x))^{7/2}}-\frac {a^3 \cos (e+f x)}{128 c^3 f (c-c \sin (e+f x))^{5/2}}-\frac {3 a^3 \cos (e+f x)}{512 c^4 f (c-c \sin (e+f x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 2.68, size = 435, normalized size = 1.93 \begin {gather*} \frac {a^3 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (2048 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-2688 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+992 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5-20 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^7-15 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^9+(15+15 i) \sqrt [4]{-1} \tan ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^{10}+4096 \sin \left (\frac {1}{2} (e+f x)\right )-5376 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {1}{2} (e+f x)\right )+1984 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sin \left (\frac {1}{2} (e+f x)\right )-40 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 \sin \left (\frac {1}{2} (e+f x)\right )-30 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^8 \sin \left (\frac {1}{2} (e+f x)\right )\right ) (1+\sin (e+f x))^3}{2560 f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^6 (c-c \sin (e+f x))^{11/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 3.27, size = 353, normalized size = 1.57
method | result | size |
default | \(\frac {a^{3} \left (480 \sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, c^{\frac {13}{2}}-1120 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {3}{2}} c^{\frac {11}{2}}+1024 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {5}{2}} c^{\frac {9}{2}}+280 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {7}{2}} c^{\frac {7}{2}}-30 \left (c \left (1+\sin \left (f x +e \right )\right )\right )^{\frac {9}{2}} c^{\frac {5}{2}}+15 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \left (\sin ^{5}\left (f x +e \right )\right ) c^{7}-75 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \left (\sin ^{4}\left (f x +e \right )\right ) c^{7}+150 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \left (\sin ^{3}\left (f x +e \right )\right ) c^{7}-150 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \left (\sin ^{2}\left (f x +e \right )\right ) c^{7}+75 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) c^{7}-15 \sqrt {2}\, \arctanh \left (\frac {\sqrt {c \left (1+\sin \left (f x +e \right )\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{7}\right ) \sqrt {c \left (1+\sin \left (f x +e \right )\right )}}{5120 c^{\frac {25}{2}} \left (\sin \left (f x +e \right )-1\right )^{4} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(353\) |
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. 646 vs.
\(2 (206) = 412\).
time = 0.38, size = 646, normalized size = 2.87 \begin {gather*} \frac {15 \, \sqrt {2} {\left (a^{3} \cos \left (f x + e\right )^{6} - 5 \, a^{3} \cos \left (f x + e\right )^{5} - 18 \, a^{3} \cos \left (f x + e\right )^{4} + 20 \, a^{3} \cos \left (f x + e\right )^{3} + 48 \, a^{3} \cos \left (f x + e\right )^{2} - 16 \, a^{3} \cos \left (f x + e\right ) - 32 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{5} + 6 \, a^{3} \cos \left (f x + e\right )^{4} - 12 \, a^{3} \cos \left (f x + e\right )^{3} - 32 \, a^{3} \cos \left (f x + e\right )^{2} + 16 \, a^{3} \cos \left (f x + e\right ) + 32 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) + 4 \, {\left (15 \, a^{3} \cos \left (f x + e\right )^{5} - 65 \, a^{3} \cos \left (f x + e\right )^{4} + 812 \, a^{3} \cos \left (f x + e\right )^{3} + 1796 \, a^{3} \cos \left (f x + e\right )^{2} - 1144 \, a^{3} \cos \left (f x + e\right ) - 2048 \, a^{3} + {\left (15 \, a^{3} \cos \left (f x + e\right )^{4} + 80 \, a^{3} \cos \left (f x + e\right )^{3} + 892 \, a^{3} \cos \left (f x + e\right )^{2} - 904 \, a^{3} \cos \left (f x + e\right ) - 2048 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{10240 \, {\left (c^{6} f \cos \left (f x + e\right )^{6} - 5 \, c^{6} f \cos \left (f x + e\right )^{5} - 18 \, c^{6} f \cos \left (f x + e\right )^{4} + 20 \, c^{6} f \cos \left (f x + e\right )^{3} + 48 \, c^{6} f \cos \left (f x + e\right )^{2} - 16 \, c^{6} f \cos \left (f x + e\right ) - 32 \, c^{6} f + {\left (c^{6} f \cos \left (f x + e\right )^{5} + 6 \, c^{6} f \cos \left (f x + e\right )^{4} - 12 \, c^{6} f \cos \left (f x + e\right )^{3} - 32 \, c^{6} f \cos \left (f x + e\right )^{2} + 16 \, c^{6} f \cos \left (f x + e\right ) + 32 \, c^{6} f\right )} \sin \left (f x + e\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{11/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________