Optimal. Leaf size=141 \[ \frac {5 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{a^{5/2} f}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {2} a^{5/2} f}-\frac {2 \cos (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}-\frac {\cot (e+f x)}{a f (a+a \sin (e+f x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.23, antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps
used = 7, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {2794, 3057,
3064, 2728, 212, 2852} \begin {gather*} \frac {5 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a \sin (e+f x)+a}}\right )}{a^{5/2} f}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a \sin (e+f x)+a}}\right )}{\sqrt {2} a^{5/2} f}-\frac {2 \cos (e+f x)}{a f (a \sin (e+f x)+a)^{3/2}}-\frac {\cot (e+f x)}{a f (a \sin (e+f x)+a)^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 212
Rule 2728
Rule 2794
Rule 2852
Rule 3057
Rule 3064
Rubi steps
\begin {align*} \int \frac {\cot ^2(e+f x)}{(a+a \sin (e+f x))^{5/2}} \, dx &=-\frac {\cot (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {\csc (e+f x) \left (-\frac {5 a}{2}+\frac {3}{2} a \sin (e+f x)\right )}{(a+a \sin (e+f x))^{3/2}} \, dx}{a^2}\\ &=-\frac {2 \cos (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}-\frac {\cot (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}+\frac {\int \frac {\csc (e+f x) \left (-5 a^2+2 a^2 \sin (e+f x)\right )}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^4}\\ &=-\frac {2 \cos (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}-\frac {\cot (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}-\frac {5 \int \csc (e+f x) \sqrt {a+a \sin (e+f x)} \, dx}{2 a^3}+\frac {7 \int \frac {1}{\sqrt {a+a \sin (e+f x)}} \, dx}{2 a^2}\\ &=-\frac {2 \cos (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}-\frac {\cot (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}+\frac {5 \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 )}{a^2 f}-\frac {7 \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 {5 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {a+a \sin (e+f x)}}\right )}{a^{5/2} f}-\frac {7 \tanh ^{-1}\left (\frac {\sqrt {a} \cos (e+f x)}{\sqrt {2} \sqrt {a+a \sin (e+f x)}}\right )}{\sqrt {2} a^{5/2} f}-\frac {2 \cos (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}-\frac {\cot (e+f x)}{a f (a+a \sin (e+f x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 0.50, size = 451, normalized size = 3.20 \begin {gather*} \frac {\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^3 \left (8 \sin \left (\frac {1}{2} (e+f x)\right )-4 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )+2 \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+(28+28 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 ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-\cot \left (\frac {1}{4} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+10 \log \left (1+\cos \left (\frac {1}{2} (e+f x)\right )-\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2-10 \log \left (1-\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2+\frac {2 \sin \left (\frac {1}{4} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}{\cos \left (\frac {1}{4} (e+f x)\right )-\sin \left (\frac {1}{4} (e+f x)\right )}-\frac {2 \sin \left (\frac {1}{4} (e+f x)\right ) \left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2}{\cos \left (\frac {1}{4} (e+f x)\right )+\sin \left (\frac {1}{4} (e+f x)\right )}-\left (\cos \left (\frac {1}{2} (e+f x)\right )+\sin \left (\frac {1}{2} (e+f x)\right )\right )^2 \tan \left (\frac {1}{4} (e+f x)\right )\right )}{4 f (a (1+\sin (e+f x)))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 2.23, size = 219, normalized size = 1.55
method | result | size |
default | \(-\frac {\left (7 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) \left (\sin ^{2}\left (f x +e \right )\right ) a -10 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{\sqrt {a}}\right ) \left (\sin ^{2}\left (f x +e \right )\right ) a +7 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {2}}{2 \sqrt {a}}\right ) a \sin \left (f x +e \right )+4 \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a}\, \sin \left (f x +e \right )-10 \arctanh \left (\frac {\sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{\sqrt {a}}\right ) a \sin \left (f x +e \right )+2 \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}\, \sqrt {a}\right ) \sqrt {-a \left (\sin \left (f x +e \right )-1\right )}}{2 a^{\frac {7}{2}} \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {a +a \sin \left (f x +e \right )}\, f}\) | \(219\) |
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. 589 vs.
\(2 (130) = 260\).
time = 0.39, size = 589, normalized size = 4.18 \begin {gather*} \frac {5 \, {\left (\cos \left (f x + e\right )^{3} + 2 \, \cos \left (f x + e\right )^{2} + {\left (\cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) - 2\right )} \sin \left (f x + e\right ) - \cos \left (f x + e\right ) - 2\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 {7 \, \sqrt {2} {\left (a \cos \left (f x + e\right )^{3} + 2 \, a \cos \left (f x + e\right )^{2} - a \cos \left (f x + e\right ) + {\left (a \cos \left (f x + e\right )^{2} - a \cos \left (f x + e\right ) - 2 \, a\right )} \sin \left (f x + e\right ) - 2 \, 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 (2 \, \cos \left (f x + e\right )^{2} + {\left (2 \, \cos \left (f x + e\right ) + 1\right )} \sin \left (f x + e\right ) + \cos \left (f x + e\right ) - 1\right )} \sqrt {a \sin \left (f x + e\right ) + a}}{4 \, {\left (a^{3} f \cos \left (f x + e\right )^{3} + 2 \, a^{3} f \cos \left (f x + e\right )^{2} - a^{3} f \cos \left (f x + e\right ) - 2 \, a^{3} f + {\left (a^{3} f \cos \left (f x + e\right )^{2} - a^{3} f \cos \left (f x + e\right ) - 2 \, 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 ^{2}{\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 [B] Leaf count of result is larger than twice the leaf count of optimal. 275 vs.
\(2 (130) = 260\).
time = 9.94, size = 275, normalized size = 1.95 \begin {gather*} \frac {\sqrt {a} {\left (\frac {7 \, \sqrt {2} \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 {7 \, \sqrt {2} \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 {10 \, \log \left ({\left | \sqrt {2} + 2 \, \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 {10 \, \log \left ({\left | -\sqrt {2} + 2 \, \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 {2 \, {\left (4 \, \sqrt {2} \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{3} - 3 \, \sqrt {2} \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 )^{4} - 3 \, \sin \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} + 1\right )} a^{3} \mathrm {sgn}\left (\cos \left (-\frac {1}{4} \, \pi + \frac {1}{2} \, f x + \frac {1}{2} \, e\right )\right )}\right )}}{4 \, 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 )}^2}{{\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]
________________________________________________________________________________________