Optimal. Leaf size=269 \[ \frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{5/2} c^2 f}-\frac {107 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{64 \sqrt {2} a^{5/2} c^2 f}+\frac {21 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{64 a^3 c^2 f}+\frac {43 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{96 a^4 c^2 f}-\frac {15 \cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{32 a^4 c^2 f}-\frac {\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{16 a^4 c^2 f} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.23, antiderivative size = 269, normalized size of antiderivative = 1.00, number of steps
used = 9, number of rules used = 7, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {3989, 3972,
483, 593, 597, 536, 209} \begin {gather*} \frac {2 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a \sec (e+f x)+a}}\right )}{a^{5/2} c^2 f}-\frac {107 \text {ArcTan}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a \sec (e+f x)+a}}\right )}{64 \sqrt {2} a^{5/2} c^2 f}+\frac {43 \cot ^3(e+f x) (a \sec (e+f x)+a)^{3/2}}{96 a^4 c^2 f}-\frac {\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) (a \sec (e+f x)+a)^{3/2}}{16 a^4 c^2 f}-\frac {15 \cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a \sec (e+f x)+a)^{3/2}}{32 a^4 c^2 f}+\frac {21 \cot (e+f x) \sqrt {a \sec (e+f x)+a}}{64 a^3 c^2 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 209
Rule 483
Rule 536
Rule 593
Rule 597
Rule 3972
Rule 3989
Rubi steps
\begin {align*} \int \frac {1}{(a+a \sec (e+f x))^{5/2} (c-c \sec (e+f x))^2} \, dx &=\frac {\int \frac {\cot ^4(e+f x)}{\sqrt {a+a \sec (e+f x)}} \, dx}{a^2 c^2}\\ &=-\frac {2 \text {Subst}\left (\int \frac {1}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )^3} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^4 c^2 f}\\ &=-\frac {\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{16 a^4 c^2 f}-\frac {\text {Subst}\left (\int \frac {a-7 a^2 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{4 a^5 c^2 f}\\ &=-\frac {15 \cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{32 a^4 c^2 f}-\frac {\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{16 a^4 c^2 f}-\frac {\text {Subst}\left (\int \frac {-43 a^2-75 a^3 x^2}{x^4 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{16 a^6 c^2 f}\\ &=\frac {43 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{96 a^4 c^2 f}-\frac {15 \cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{32 a^4 c^2 f}-\frac {\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{16 a^4 c^2 f}+\frac {\text {Subst}\left (\int \frac {63 a^3-129 a^4 x^2}{x^2 \left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{96 a^6 c^2 f}\\ &=\frac {21 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{64 a^3 c^2 f}+\frac {43 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{96 a^4 c^2 f}-\frac {15 \cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{32 a^4 c^2 f}-\frac {\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{16 a^4 c^2 f}-\frac {\text {Subst}\left (\int \frac {447 a^4+63 a^5 x^2}{\left (1+a x^2\right ) \left (2+a x^2\right )} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{192 a^6 c^2 f}\\ &=\frac {21 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{64 a^3 c^2 f}+\frac {43 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{96 a^4 c^2 f}-\frac {15 \cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{32 a^4 c^2 f}-\frac {\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{16 a^4 c^2 f}+\frac {107 \text {Subst}\left (\int \frac {1}{2+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{64 a^2 c^2 f}-\frac {2 \text {Subst}\left (\int \frac {1}{1+a x^2} \, dx,x,-\frac {\tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^2 c^2 f}\\ &=\frac {2 \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {a+a \sec (e+f x)}}\right )}{a^{5/2} c^2 f}-\frac {107 \tan ^{-1}\left (\frac {\sqrt {a} \tan (e+f x)}{\sqrt {2} \sqrt {a+a \sec (e+f x)}}\right )}{64 \sqrt {2} a^{5/2} c^2 f}+\frac {21 \cot (e+f x) \sqrt {a+a \sec (e+f x)}}{64 a^3 c^2 f}+\frac {43 \cot ^3(e+f x) (a+a \sec (e+f x))^{3/2}}{96 a^4 c^2 f}-\frac {15 \cos (e+f x) \cot ^3(e+f x) \sec ^2\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{32 a^4 c^2 f}-\frac {\cos ^2(e+f x) \cot ^3(e+f x) \sec ^4\left (\frac {1}{2} (e+f x)\right ) (a+a \sec (e+f x))^{3/2}}{16 a^4 c^2 f}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains higher order function than in optimal. Order 4 vs. order 3 in
optimal.
time = 24.26, size = 5650, normalized size = 21.00 \begin {gather*} \text {Result too large to show} \end {gather*}
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(724\) vs.
\(2(234)=468\).
time = 0.30, size = 725, normalized size = 2.70
method | result | size |
default | \(\frac {\sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, \left (\cos \left (f x +e \right )+1\right ) \left (-1+\cos \left (f x +e \right )\right )^{2} \left (384 \left (\cos ^{3}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}+321 \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\cos \left (f x +e \right )+1}{\sin \left (f x +e \right )}\right ) \sin \left (f x +e \right ) \left (\cos ^{3}\left (f x +e \right )\right )+384 \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right ) \sqrt {2}+321 \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\cos \left (f x +e \right )+1}{\sin \left (f x +e \right )}\right ) \left (\cos ^{2}\left (f x +e \right )\right ) \sin \left (f x +e \right )-384 \sin \left (f x +e \right ) \cos \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {2}-321 \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\cos \left (f x +e \right )+1}{\sin \left (f x +e \right )}\right ) \cos \left (f x +e \right ) \sin \left (f x +e \right )-384 \sqrt {2}\, \arctanh \left (\frac {\sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right ) \sqrt {2}}{2 \cos \left (f x +e \right )}\right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \sin \left (f x +e \right )-410 \left (\cos ^{4}\left (f x +e \right )\right )-321 \sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}\, \ln \left (\frac {\sin \left (f x +e \right ) \sqrt {-\frac {2 \cos \left (f x +e \right )}{\cos \left (f x +e \right )+1}}-\cos \left (f x +e \right )+1}{\sin \left (f x +e \right )}\right )-142 \left (\cos ^{3}\left (f x +e \right )\right )+298 \left (\cos ^{2}\left (f x +e \right )\right )+126 \cos \left (f x +e \right )\right )}{384 c^{2} f \sin \left (f x +e \right )^{7} a^{3}}\) | \(725\) |
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 [A]
time = 4.66, size = 768, normalized size = 2.86 \begin {gather*} \left [-\frac {321 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) - 1\right )} \sqrt {-a} \log \left (-\frac {2 \, \sqrt {2} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right ) - 3 \, a \cos \left (f x + e\right )^{2} - 2 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )^{2} + 2 \, \cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) + 384 \, {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) - 1\right )} \sqrt {-a} \log \left (-\frac {8 \, a \cos \left (f x + e\right )^{3} + 4 \, {\left (2 \, \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right )\right )} \sqrt {-a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \sin \left (f x + e\right ) - 7 \, a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right ) + 1}\right ) \sin \left (f x + e\right ) - 4 \, {\left (205 \, \cos \left (f x + e\right )^{4} + 71 \, \cos \left (f x + e\right )^{3} - 149 \, \cos \left (f x + e\right )^{2} - 63 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{768 \, {\left (a^{3} c^{2} f \cos \left (f x + e\right )^{3} + a^{3} c^{2} f \cos \left (f x + e\right )^{2} - a^{3} c^{2} f \cos \left (f x + e\right ) - a^{3} c^{2} f\right )} \sin \left (f x + e\right )}, \frac {321 \, \sqrt {2} {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) - 1\right )} \sqrt {a} \arctan \left (\frac {\sqrt {2} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right )}{\sqrt {a} \sin \left (f x + e\right )}\right ) \sin \left (f x + e\right ) + 384 \, {\left (\cos \left (f x + e\right )^{3} + \cos \left (f x + e\right )^{2} - \cos \left (f x + e\right ) - 1\right )} \sqrt {a} \arctan \left (\frac {2 \, \sqrt {a} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}} \cos \left (f x + e\right ) \sin \left (f x + e\right )}{2 \, a \cos \left (f x + e\right )^{2} + a \cos \left (f x + e\right ) - a}\right ) \sin \left (f x + e\right ) + 2 \, {\left (205 \, \cos \left (f x + e\right )^{4} + 71 \, \cos \left (f x + e\right )^{3} - 149 \, \cos \left (f x + e\right )^{2} - 63 \, \cos \left (f x + e\right )\right )} \sqrt {\frac {a \cos \left (f x + e\right ) + a}{\cos \left (f x + e\right )}}}{384 \, {\left (a^{3} c^{2} f \cos \left (f x + e\right )^{3} + a^{3} c^{2} f \cos \left (f x + e\right )^{2} - a^{3} c^{2} f \cos \left (f x + e\right ) - a^{3} c^{2} 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*} \frac {\int \frac {1}{a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{4}{\left (e + f x \right )} - 2 a^{2} \sqrt {a \sec {\left (e + f x \right )} + a} \sec ^{2}{\left (e + f x \right )} + a^{2} \sqrt {a \sec {\left (e + f x \right )} + a}}\, dx}{c^{2}} \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 {1}{{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}\,{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________