Optimal. Leaf size=191 \[ \frac {45 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 a^{5/2} f}-\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{a^{5/2} f}-\frac {19 \cot (e+f x)}{8 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {13 \cot (e+f x) \csc (e+f x)}{12 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a^2 f \sqrt {a+a \sin (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.64, antiderivative size = 191, normalized size of antiderivative = 1.00, number of steps
used = 16, number of rules used = 8, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.348, Rules used = {2796, 2858,
3063, 3064, 2728, 212, 2852, 3123} \begin {gather*} \frac {45 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{8 a^{5/2} f}-\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{a^{5/2} f}-\frac {19 \cot (e+f x)}{8 a^2 f \sqrt {a \sin (e+f x)+a}}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a^2 f \sqrt {a \sin (e+f x)+a}}+\frac {13 \cot (e+f x) \csc (e+f x)}{12 a^2 f \sqrt {a \sin (e+f x)+a}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 2728
Rule 2796
Rule 2852
Rule 2858
Rule 3063
Rule 3064
Rule 3123
Rubi steps
\begin {align*} \int \frac {\cot ^4(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx &=\frac {\int \frac {\csc ^4(e+f x) \left (1+\sin ^2(e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{a^2}-\frac {2 \int \frac {\csc ^3(e+f x)}{\sqrt {a+a \sin (e+f x)}} \, dx}{a^2}\\ &=\frac {\cot (e+f x) \csc (e+f x)}{a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {\int \frac {\csc ^3(e+f x) \left (-\frac {a}{2}+\frac {11}{2} a \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{3 a^3}+\frac {\int \frac {\csc ^2(e+f x) (a-3 a \sin (e+f x))}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^3}\\ &=-\frac {\cot (e+f x)}{2 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {13 \cot (e+f x) \csc (e+f x)}{12 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {\int \frac {\csc ^2(e+f x) \left (\frac {45 a^2}{4}-\frac {3}{4} a^2 \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{6 a^4}+\frac {\int \frac {\csc (e+f x) \left (-\frac {7 a^2}{2}+\frac {1}{2} a^2 \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^4}\\ &=-\frac {19 \cot (e+f x)}{8 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {13 \cot (e+f x) \csc (e+f x)}{12 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {\int \frac {\csc (e+f x) \left (-\frac {51 a^3}{8}+\frac {45}{8} a^3 \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{6 a^5}-\frac {7 \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \, dx}{4 a^3}+\frac {2 \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{a^2}\\ &=-\frac {19 \cot (e+f x)}{8 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {13 \cot (e+f x) \csc (e+f x)}{12 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {17 \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \, dx}{16 a^3}+\frac {2 \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{a^2}+\frac {7 \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{2 a^2 f}-\frac {4 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{a^2 f}\\ &=\frac {7 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{2 a^{5/2} f}-\frac {2 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{a^{5/2} f}-\frac {19 \cot (e+f x)}{8 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {13 \cot (e+f x) \csc (e+f x)}{12 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {17 \text {Subst}\left (\int \frac {1}{a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 a^2 f}-\frac {4 \text {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\frac {a \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{a^2 f}\\ &=\frac {45 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{8 a^{5/2} f}-\frac {4 \sqrt {2} \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{a^{5/2} f}-\frac {19 \cot (e+f x)}{8 a^2 f \sqrt {a+a \sin (e+f x)}}+\frac {13 \cot (e+f x) \csc (e+f x)}{12 a^2 f \sqrt {a+a \sin (e+f x)}}-\frac {\cot (e+f x) \csc ^2(e+f x)}{3 a^2 f \sqrt {a+a \sin (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 1.58, size = 332, normalized size = 1.74 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^5 \left ((1536+1536 i) (-1)^{3/4} \tanh ^{-1}\left (\left (\frac {1}{2}+\frac {i}{2}\right ) (-1)^{3/4} \left (-1+\tan \left (\frac {1}{4} (e+f x)\right )\right )\right )-\frac {8 \csc ^9\left (\frac {1}{2} (e+f x)\right ) \left (396 \cos \left (\frac {1}{2} (e+f x)\right )-218 \cos \left (\frac {3}{2} (e+f x)\right )-114 \cos \left (\frac {5}{2} (e+f x)\right )-396 \sin \left (\frac {1}{2} (e+f x)\right )-405 \log \left (1+\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)+405 \log \left (1-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (e+f x)-218 \sin \left (\frac {3}{2} (e+f x)\right )+114 \sin \left (\frac {5}{2} (e+f x)\right )+135 \log \left (1+\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))-135 \log \left (1-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \sin (3 (e+f x))\right )}{\left (\csc ^2\left (\frac {1}{4} (e+f x)\right )-\sec ^2\left (\frac {1}{4} (e+f x)\right )\right )^3}\right )}{192 f (a (1+\sin (e+f x)))^{5/2}} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 2.59, size = 182, normalized size = 0.95
method | result | size |
default | \(-\frac {\left (1+\sin \left (f x +e \right )\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \left (-135 a^{5} \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{\sqrt {a}}\right ) \left (\sin ^{3}\left (f x +e \right )\right )+57 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {5}{2}} a^{\frac {5}{2}}+96 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a^{5} \left (\sin ^{3}\left (f x +e \right )\right )-88 \left (-a \left (\sin \left (f x +e \right )-1\right )\right )^{\frac {3}{2}} a^{\frac {7}{2}}+39 \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, a^{\frac {9}{2}}\right )}{24 a^{\frac {15}{2}} \sin \left (f x +e \right )^{3} \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(182\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [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]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 616 vs.
\(2 (174) = 348\).
time = 0.42, size = 616, normalized size = 3.23 \begin {gather*} \frac {135 \, {\left (\cos \left (f x + e\right )^{4} - 2 \, \cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) - 1\right )} \sin \left (f x + e\right ) + 1\right )} \sqrt {a} \log \left (\frac {a \cos \left (f x + e\right )^{3} - 7 \, a \cos \left (f x + e\right )^{2} + 4 \, {\left (\cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right ) + 3\right )} \sin \left (f x + e\right ) - 2 \, \cos \left (f x + e\right ) - 3\right )} \sqrt {a \sin \left (f x + e\right ) + a} \sqrt {a} - 9 \, a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} + 8 \, a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) - a}{\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - 1\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 1}\right ) + \frac {192 \, \sqrt {2} {\left (a \cos \left (f x + e\right )^{4} - 2 \, a \cos \left (f x + e\right )^{2} - {\left (a \cos \left (f x + e\right )^{3} + a \cos \left (f x + e\right )^{2} - a \cos \left (f x + e\right ) - a\right )} \sin \left (f x + e\right ) + a\right )} \log \left (-\frac {\cos \left (f x + e\right )^{2} - {\left (\cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) - \frac {2 \, \sqrt {2} \sqrt {a \sin \left (f x + e\right ) + a} {\left (\cos \left (f x + e\right ) - \sin \left (f x + e\right ) + 1\right )}}{\sqrt {a}} + 3 \, \cos \left (f x + e\right ) + 2}{\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 )}{\sqrt {a}} + 4 \, {\left (57 \, \cos \left (f x + e\right )^{3} + 83 \, \cos \left (f x + e\right )^{2} - {\left (57 \, \cos \left (f x + e\right )^{2} - 26 \, \cos \left (f x + e\right ) - 91\right )} \sin \left (f x + e\right ) - 65 \, \cos \left (f x + e\right ) - 91\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{96 \, {\left (a^{3} f \cos \left (f x + e\right )^{4} - 2 \, a^{3} f \cos \left (f x + e\right )^{2} + a^{3} f - {\left (a^{3} f \cos \left (f x + e\right )^{3} + a^{3} f \cos \left (f x + e\right )^{2} - a^{3} f \cos \left (f x + e\right ) - a^{3} 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]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\cot ^{4}{\left (e + f x \right )}}{\left (a \left (\sin {\left (e + f x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A]
time = 50.44, size = 252, normalized size = 1.32 \begin {gather*} \frac {\sqrt {2} \sqrt {a} {\left (\frac {135 \, \sqrt {2} \log \left (\frac {{\left | -2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}{{\left | 2 \, \sqrt {2} + 4 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) \right |}}\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} + \frac {192 \, \log \left (\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {192 \, \log \left (-\sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right ) + 1\right )}{a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )} - \frac {4 \, {\left (228 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{5} - 176 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} + 39 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}}{{\left (2 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - 1\right )}^{3} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )}}{96 \, 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 \frac {{\mathrm {cot}\left (e+f\,x\right )}^4}{{\left (a+a\,\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]
________________________________________________________________________________________