Integrand size = 28, antiderivative size = 143 \[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {135 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{8 \sqrt {2} c^{7/2} f}+\frac {9 c^2 \cos ^5(e+f x)}{f (c-c \sin (e+f x))^{11/2}}-\frac {45 \cos ^3(e+f x)}{4 f (c-c \sin (e+f x))^{7/2}}+\frac {135 \cos (e+f x)}{8 c^2 f (c-c \sin (e+f x))^{3/2}} \]
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Time = 0.23 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.10, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2815, 2759, 2728, 212} \[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{7/2}} \, dx=-\frac {5 a^3 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{8 \sqrt {2} c^{7/2} f}+\frac {a^3 c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^{11/2}}+\frac {5 a^3 \cos (e+f x)}{8 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac {5 a^3 \cos ^3(e+f x)}{12 f (c-c \sin (e+f x))^{7/2}} \]
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Rule 212
Rule 2728
Rule 2759
Rule 2815
Rubi steps \begin{align*} \text {integral}& = \left (a^3 c^3\right ) \int \frac {\cos ^6(e+f x)}{(c-c \sin (e+f x))^{13/2}} \, dx \\ & = \frac {a^3 c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^{11/2}}-\frac {1}{6} \left (5 a^3 c\right ) \int \frac {\cos ^4(e+f x)}{(c-c \sin (e+f x))^{9/2}} \, dx \\ & = \frac {a^3 c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 \cos ^3(e+f x)}{12 f (c-c \sin (e+f x))^{7/2}}+\frac {\left (5 a^3\right ) \int \frac {\cos ^2(e+f x)}{(c-c \sin (e+f x))^{5/2}} \, dx}{8 c} \\ & = \frac {a^3 c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 \cos ^3(e+f x)}{12 f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 \cos (e+f x)}{8 c^2 f (c-c \sin (e+f x))^{3/2}}-\frac {\left (5 a^3\right ) \int \frac {1}{\sqrt {c-c \sin (e+f x)}} \, dx}{16 c^3} \\ & = \frac {a^3 c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 \cos ^3(e+f x)}{12 f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 \cos (e+f x)}{8 c^2 f (c-c \sin (e+f x))^{3/2}}+\frac {\left (5 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 )}{8 c^3 f} \\ & = -\frac {5 a^3 \text {arctanh}\left (\frac {\sqrt {c} \cos (e+f x)}{\sqrt {2} \sqrt {c-c \sin (e+f x)}}\right )}{8 \sqrt {2} c^{7/2} f}+\frac {a^3 c^2 \cos ^5(e+f x)}{3 f (c-c \sin (e+f x))^{11/2}}-\frac {5 a^3 \cos ^3(e+f x)}{12 f (c-c \sin (e+f x))^{7/2}}+\frac {5 a^3 \cos (e+f x)}{8 c^2 f (c-c \sin (e+f x))^{3/2}} \\ \end{align*}
Result contains complex when optimal does not.
Time = 1.04 (sec) , antiderivative size = 271, normalized size of antiderivative = 1.90 \[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {9 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (32 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )-52 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^3+33 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^5+(15+15 i) \sqrt [4]{-1} \arctan \left (\left (\frac {1}{2}+\frac {i}{2}\right ) \sqrt [4]{-1} \left (1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^6+64 \sin \left (\frac {1}{2} (e+f x)\right )-104 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \sin \left (\frac {1}{2} (e+f x)\right )+66 \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )^4 \sin \left (\frac {1}{2} (e+f x)\right )\right )}{8 f (c-c \sin (e+f x))^{7/2}} \]
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Time = 3.51 (sec) , antiderivative size = 245, normalized size of antiderivative = 1.71
method | result | size |
default | \(\frac {a^{3} \left (15 \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \left (\sin ^{3}\left (f x +e \right )\right ) \sqrt {2}\, c^{3}-45 \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \left (\sin ^{2}\left (f x +e \right )\right ) \sqrt {2}\, c^{3}+66 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {5}{2}} \sqrt {c}+45 \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) \sin \left (f x +e \right ) \sqrt {2}\, c^{3}-160 \left (c \left (\sin \left (f x +e \right )+1\right )\right )^{\frac {3}{2}} c^{\frac {3}{2}}-15 \sqrt {2}\, \operatorname {arctanh}\left (\frac {\sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, \sqrt {2}}{2 \sqrt {c}}\right ) c^{3}+120 \sqrt {c \left (\sin \left (f x +e \right )+1\right )}\, c^{\frac {5}{2}}\right ) \sqrt {c \left (\sin \left (f x +e \right )+1\right )}}{48 c^{\frac {13}{2}} \left (\sin \left (f x +e \right )-1\right )^{2} \cos \left (f x +e \right ) \sqrt {c -c \sin \left (f x +e \right )}\, f}\) | \(245\) |
parts | \(\text {Expression too large to display}\) | \(992\) |
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Leaf count of result is larger than twice the leaf count of optimal. 440 vs. \(2 (134) = 268\).
Time = 0.27 (sec) , antiderivative size = 440, normalized size of antiderivative = 3.08 \[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{7/2}} \, dx=\frac {15 \, \sqrt {2} {\left (a^{3} \cos \left (f x + e\right )^{4} - 3 \, a^{3} \cos \left (f x + e\right )^{3} - 8 \, a^{3} \cos \left (f x + e\right )^{2} + 4 \, a^{3} \cos \left (f x + e\right ) + 8 \, a^{3} + {\left (a^{3} \cos \left (f x + e\right )^{3} + 4 \, a^{3} \cos \left (f x + e\right )^{2} - 4 \, a^{3} \cos \left (f x + e\right ) - 8 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {c} \log \left (-\frac {c \cos \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {-c \sin \left (f x + e\right ) + c} \sqrt {c} {\left (\cos \left (f x + e\right ) + \sin \left (f x + e\right ) + 1\right )} + 3 \, c \cos \left (f x + e\right ) + {\left (c \cos \left (f x + e\right ) - 2 \, c\right )} \sin \left (f x + e\right ) + 2 \, c}{\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2}\right ) - 4 \, {\left (33 \, a^{3} \cos \left (f x + e\right )^{3} + 19 \, a^{3} \cos \left (f x + e\right )^{2} - 46 \, a^{3} \cos \left (f x + e\right ) - 32 \, a^{3} + {\left (33 \, a^{3} \cos \left (f x + e\right )^{2} + 14 \, a^{3} \cos \left (f x + e\right ) - 32 \, a^{3}\right )} \sin \left (f x + e\right )\right )} \sqrt {-c \sin \left (f x + e\right ) + c}}{96 \, {\left (c^{4} f \cos \left (f x + e\right )^{4} - 3 \, c^{4} f \cos \left (f x + e\right )^{3} - 8 \, c^{4} f \cos \left (f x + e\right )^{2} + 4 \, c^{4} f \cos \left (f x + e\right ) + 8 \, c^{4} f + {\left (c^{4} f \cos \left (f x + e\right )^{3} + 4 \, c^{4} f \cos \left (f x + e\right )^{2} - 4 \, c^{4} f \cos \left (f x + e\right ) - 8 \, c^{4} f\right )} \sin \left (f x + e\right )\right )}} \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Timed out} \]
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\[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{7/2}} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{3}}{{\left (-c \sin \left (f x + e\right ) + c\right )}^{\frac {7}{2}}} \,d x } \]
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Exception generated. \[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{7/2}} \, dx=\text {Exception raised: TypeError} \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^3}{(c-c \sin (e+f x))^{7/2}} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^3}{{\left (c-c\,\sin \left (e+f\,x\right )\right )}^{7/2}} \,d x \]
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