Integrand size = 27, antiderivative size = 188 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^3} \, dx=-\frac {9 \sqrt {3} \left (3 c^2+10 c d+19 d^2\right ) \text {arctanh}\left (\frac {\sqrt {3} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {3+3 \sin (e+f x)}}\right )}{4 d^{5/2} (c+d)^{5/2} f}+\frac {9 (c-d) \cos (e+f x) \sqrt {3+3 \sin (e+f x)}}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {81 (c-d) (c+3 d) \cos (e+f x)}{4 d^2 (c+d)^2 f \sqrt {3+3 \sin (e+f x)} (c+d \sin (e+f x))} \]
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Time = 0.31 (sec) , antiderivative size = 194, normalized size of antiderivative = 1.03, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2841, 3059, 2852, 214} \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^3} \, dx=-\frac {a^{5/2} \left (3 c^2+10 c d+19 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a \sin (e+f x)+a}}\right )}{4 d^{5/2} f (c+d)^{5/2}}+\frac {3 a^3 (c-d) (c+3 d) \cos (e+f x)}{4 d^2 f (c+d)^2 \sqrt {a \sin (e+f x)+a} (c+d \sin (e+f x))}+\frac {a^2 (c-d) \cos (e+f x) \sqrt {a \sin (e+f x)+a}}{2 d f (c+d) (c+d \sin (e+f x))^2} \]
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Rule 214
Rule 2841
Rule 2852
Rule 3059
Rubi steps \begin{align*} \text {integral}& = \frac {a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 d (c+d) f (c+d \sin (e+f x))^2}-\frac {a \int \frac {\sqrt {a+a \sin (e+f x)} \left (\frac {1}{2} a (c-9 d)-\frac {1}{2} a (3 c+5 d) \sin (e+f x)\right )}{(c+d \sin (e+f x))^2} \, dx}{2 d (c+d)} \\ & = \frac {a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {3 a^3 (c-d) (c+3 d) \cos (e+f x)}{4 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}+\frac {\left (a^2 \left (3 c^2+10 c d+19 d^2\right )\right ) \int \frac {\sqrt {a+a \sin (e+f x)}}{c+d \sin (e+f x)} \, dx}{8 d^2 (c+d)^2} \\ & = \frac {a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {3 a^3 (c-d) (c+3 d) \cos (e+f x)}{4 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))}-\frac {\left (a^3 \left (3 c^2+10 c d+19 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{a c+a d-d x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{4 d^2 (c+d)^2 f} \\ & = -\frac {a^{5/2} \left (3 c^2+10 c d+19 d^2\right ) \text {arctanh}\left (\frac {\sqrt {a} \sqrt {d} \cos (e+f x)}{\sqrt {c+d} \sqrt {a+a \sin (e+f x)}}\right )}{4 d^{5/2} (c+d)^{5/2} f}+\frac {a^2 (c-d) \cos (e+f x) \sqrt {a+a \sin (e+f x)}}{2 d (c+d) f (c+d \sin (e+f x))^2}+\frac {3 a^3 (c-d) (c+3 d) \cos (e+f x)}{4 d^2 (c+d)^2 f \sqrt {a+a \sin (e+f x)} (c+d \sin (e+f x))} \\ \end{align*}
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 8.65 (sec) , antiderivative size = 923, normalized size of antiderivative = 4.91 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^3} \, dx=\frac {9 \sqrt {3} (1+\sin (e+f x))^{5/2} \left (\frac {\left (3 c^2+10 c d+19 d^2\right ) \left ((c+d) \left (e+f x-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )\right )+\sqrt {c+d} \text {RootSum}\left [c+4 d \text {$\#$1}+2 c \text {$\#$1}^2-4 d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {-c \sqrt {d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right )-d^{3/2} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right )-d \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right )-2 c \sqrt {d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}-2 d^{3/2} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}-c \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}+c \sqrt {d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^2+d^{3/2} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^2+3 d \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^2-c \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^3}{-d-c \text {$\#$1}+3 d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]\right )}{(c+d)^{7/2}}+\frac {\left (3 c^2+10 c d+19 d^2\right ) \left (-\left ((c+d) \left (e+f x-2 \log \left (\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )\right )\right )+\sqrt {c+d} \text {RootSum}\left [c+4 d \text {$\#$1}+2 c \text {$\#$1}^2-4 d \text {$\#$1}^3+c \text {$\#$1}^4\&,\frac {-c \sqrt {d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right )-d^{3/2} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right )+d \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right )-2 c \sqrt {d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}-2 d^{3/2} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}+c \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}+c \sqrt {d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^2+d^{3/2} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^2-3 d \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^2+c \sqrt {c+d} \log \left (-\text {$\#$1}+\tan \left (\frac {1}{4} (e+f x)\right )\right ) \text {$\#$1}^3}{-d-c \text {$\#$1}+3 d \text {$\#$1}^2-c \text {$\#$1}^3}\&\right ]\right )}{(c+d)^{7/2}}-\frac {8 (c-d)^2 \sqrt {d} \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{(c+d) (c+d \sin (e+f x))^2}-\frac {4 \sqrt {d} \left (-5 c^2-6 c d+11 d^2\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right )}{(c+d)^2 (c+d \sin (e+f x))}\right )}{16 d^{5/2} f \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(566\) vs. \(2(170)=340\).
Time = 62.66 (sec) , antiderivative size = 567, normalized size of antiderivative = 3.02
method | result | size |
default | \(-\frac {a \left (-\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} d^{2} \left (3 c^{2}+10 c d +19 d^{2}\right ) \left (\cos ^{2}\left (f x +e \right )\right )+2 \sin \left (f x +e \right ) \operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} c d \left (3 c^{2}+10 c d +19 d^{2}\right )+5 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, c^{2} d +6 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, c \,d^{2}-11 \left (a -a \sin \left (f x +e \right )\right )^{\frac {3}{2}} \sqrt {a \left (c +d \right ) d}\, d^{3}+3 \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} c^{4}+10 \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} c^{3} d +22 \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} c^{2} d^{2}+10 \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} c \,d^{3}+19 \,\operatorname {arctanh}\left (\frac {\sqrt {a -a \sin \left (f x +e \right )}\, d}{\sqrt {a c d +a \,d^{2}}}\right ) a^{2} d^{4}-3 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a \,c^{3}-13 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a \,c^{2} d +3 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a c \,d^{2}+13 \sqrt {a -a \sin \left (f x +e \right )}\, \sqrt {a \left (c +d \right ) d}\, a \,d^{3}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (\sin \left (f x +e \right )+1\right )}{4 \sqrt {a \left (c +d \right ) d}\, \left (c +d \sin \left (f x +e \right )\right )^{2} \left (c +d \right )^{2} d^{2} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(567\) |
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Leaf count of result is larger than twice the leaf count of optimal. 839 vs. \(2 (170) = 340\).
Time = 0.47 (sec) , antiderivative size = 1994, normalized size of antiderivative = 10.61 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^3} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^3} \, dx=\text {Timed out} \]
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\[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^3} \, dx=\int { \frac {{\left (a \sin \left (f x + e\right ) + a\right )}^{\frac {5}{2}}}{{\left (d \sin \left (f x + e\right ) + c\right )}^{3}} \,d x } \]
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Leaf count of result is larger than twice the leaf count of optimal. 429 vs. \(2 (170) = 340\).
Time = 0.37 (sec) , antiderivative size = 429, normalized size of antiderivative = 2.28 \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^3} \, dx=-\frac {\sqrt {2} \sqrt {a} {\left (\frac {\sqrt {2} {\left (3 \, a^{2} c^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 10 \, a^{2} c d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) + 19 \, a^{2} d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )\right )} \arctan \left (\frac {\sqrt {2} d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )}{\sqrt {-c d - d^{2}}}\right )}{{\left (c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right )} \sqrt {-c d - d^{2}}} - \frac {2 \, {\left (10 \, a^{2} c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 12 \, a^{2} c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 22 \, a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, a^{2} c^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) - 13 \, a^{2} c^{2} d \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 3 \, a^{2} c d^{2} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 13 \, a^{2} d^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right ) \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (c^{2} d^{2} + 2 \, c d^{3} + d^{4}\right )} {\left (2 \, d \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - c - d\right )}^{2}}\right )}}{8 \, f} \]
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Timed out. \[ \int \frac {(3+3 \sin (e+f x))^{5/2}}{(c+d \sin (e+f x))^3} \, dx=\int \frac {{\left (a+a\,\sin \left (e+f\,x\right )\right )}^{5/2}}{{\left (c+d\,\sin \left (e+f\,x\right )\right )}^3} \,d x \]
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