Optimal. Leaf size=194 \[ -\frac {a^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{9/2}}-\frac {a^3 \tan (e+f x)}{2 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{c^3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {a^3 \log (1-\cos (e+f x)) \tan (e+f x)}{c^4 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.25, antiderivative size = 194, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {3995, 3992,
3996, 31} \begin {gather*} \frac {a^3 \tan (e+f x) \log (1-\cos (e+f x))}{c^4 f \sqrt {a \sec (e+f x)+a} \sqrt {c-c \sec (e+f x)}}-\frac {a^3 \tan (e+f x)}{c^3 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{3/2}}-\frac {a^3 \tan (e+f x)}{2 c^2 f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{f \sqrt {a \sec (e+f x)+a} (c-c \sec (e+f x))^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 31
Rule 3992
Rule 3995
Rule 3996
Rubi steps
\begin {align*} \int \frac {(a+a \sec (e+f x))^{5/2}}{(c-c \sec (e+f x))^{9/2}} \, dx &=-\frac {a^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{9/2}}+\frac {a^2 \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{5/2}} \, dx}{c^2}\\ &=-\frac {a^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{9/2}}-\frac {a^3 \tan (e+f x)}{2 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}+\frac {a^2 \int \frac {\sqrt {a+a \sec (e+f x)}}{(c-c \sec (e+f x))^{3/2}} \, dx}{c^3}\\ &=-\frac {a^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{9/2}}-\frac {a^3 \tan (e+f x)}{2 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{c^3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {a^2 \int \frac {\sqrt {a+a \sec (e+f x)}}{\sqrt {c-c \sec (e+f x)}} \, dx}{c^4}\\ &=-\frac {a^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{9/2}}-\frac {a^3 \tan (e+f x)}{2 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{c^3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {\left (a^3 \tan (e+f x)\right ) \text {Subst}\left (\int \frac {1}{-c+c x} \, dx,x,\cos (e+f x)\right )}{c^3 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ &=-\frac {a^3 \tan (e+f x)}{f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{9/2}}-\frac {a^3 \tan (e+f x)}{2 c^2 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{5/2}}-\frac {a^3 \tan (e+f x)}{c^3 f \sqrt {a+a \sec (e+f x)} (c-c \sec (e+f x))^{3/2}}+\frac {a^3 \log (1-\cos (e+f x)) \tan (e+f x)}{c^4 f \sqrt {a+a \sec (e+f x)} \sqrt {c-c \sec (e+f x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 5.77, size = 285, normalized size = 1.47 \begin {gather*} \frac {\sec ^{\frac {9}{2}}(e+f x) (a (1+\sec (e+f x)))^{5/2} \left (\frac {16 \sqrt {2} e^{\frac {1}{2} i (e+f x)} \sqrt {\frac {\left (1+e^{i (e+f x)}\right )^2}{1+e^{2 i (e+f x)}}} \left (-i f x+2 \log \left (1-e^{i (e+f x)}\right )\right )}{\left (1+e^{i (e+f x)}\right ) \sqrt {\frac {e^{i (e+f x)}}{1+e^{2 i (e+f x)}}} f}+\frac {(-54+89 \cos (e+f x)-60 \cos (2 (e+f x))+23 \cos (3 (e+f x))-6 \cos (4 (e+f x))) \csc ^8\left (\frac {1}{2} (e+f x)\right ) \sec \left (\frac {1}{2} (e+f x)\right ) \sqrt {\sec (e+f x)} \sqrt {1+\sec (e+f x)}}{8 f}\right ) \sin ^9\left (\frac {1}{2} (e+f x)\right )}{(1+\sec (e+f x))^{5/2} (c-c \sec (e+f x))^{9/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.28, size = 353, normalized size = 1.82
method | result | size |
default | \(-\frac {\left (-1+\cos \left (f x +e \right )\right ) \left (32 \left (\cos ^{4}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-16 \left (\cos ^{4}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-29 \left (\cos ^{4}\left (f x +e \right )\right )-128 \left (\cos ^{3}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+64 \left (\cos ^{3}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+20 \left (\cos ^{3}\left (f x +e \right )\right )+192 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-96 \left (\cos ^{2}\left (f x +e \right )\right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+10 \left (\cos ^{2}\left (f x +e \right )\right )-128 \cos \left (f x +e \right ) \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )+64 \cos \left (f x +e \right ) \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )-28 \cos \left (f x +e \right )+32 \ln \left (-\frac {-1+\cos \left (f x +e \right )}{\sin \left (f x +e \right )}\right )-16 \ln \left (\frac {2}{\cos \left (f x +e \right )+1}\right )+11\right ) \sqrt {\frac {a \left (\cos \left (f x +e \right )+1\right )}{\cos \left (f x +e \right )}}\, a^{2}}{16 f \sin \left (f x +e \right ) \cos \left (f x +e \right )^{4} \left (\frac {c \left (-1+\cos \left (f x +e \right )\right )}{\cos \left (f x +e \right )}\right )^{\frac {9}{2}}}\) | \(353\) |
risch | \(\frac {a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) x}{c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}}-\frac {2 a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \left (f x +e \right )}{c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}+\frac {2 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left (6 \,{\mathrm e}^{7 i \left (f x +e \right )}-23 \,{\mathrm e}^{6 i \left (f x +e \right )}+54 \,{\mathrm e}^{5 i \left (f x +e \right )}-66 \,{\mathrm e}^{4 i \left (f x +e \right )}+54 \,{\mathrm e}^{3 i \left (f x +e \right )}-23 \,{\mathrm e}^{2 i \left (f x +e \right )}+6 \,{\mathrm e}^{i \left (f x +e \right )}\right )}{c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{7} \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}-\frac {2 i a^{2} \sqrt {\frac {a \left ({\mathrm e}^{i \left (f x +e \right )}+1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, \left ({\mathrm e}^{i \left (f x +e \right )}-1\right ) \ln \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )}{c^{4} \left ({\mathrm e}^{i \left (f x +e \right )}+1\right ) \sqrt {\frac {c \left ({\mathrm e}^{i \left (f x +e \right )}-1\right )^{2}}{{\mathrm e}^{2 i \left (f x +e \right )}+1}}\, f}\) | \(478\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 6623 vs.
\(2 (189) = 378\).
time = 23.48, size = 6623, normalized size = 34.14 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [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]
________________________________________________________________________________________
Giac [A]
time = 2.16, size = 230, normalized size = 1.19 \begin {gather*} -\frac {\frac {48 \, \sqrt {-a c} a^{3} \log \left ({\left | a \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2}\right )}{c^{5} {\left | a \right |}} - \frac {48 \, \sqrt {-a c} a^{3} \log \left ({\left | -a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a \right |}\right )}{c^{5} {\left | a \right |}} - \frac {100 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{4} \sqrt {-a c} a^{3} + 352 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{3} \sqrt {-a c} a^{4} + 480 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )}^{2} \sqrt {-a c} a^{5} + 292 \, {\left (a \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{2} - a\right )} \sqrt {-a c} a^{6} + 67 \, \sqrt {-a c} a^{7}}{a^{4} c^{5} {\left | a \right |} \tan \left (\frac {1}{2} \, f x + \frac {1}{2} \, e\right )^{8}}}{48 \, 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 {{\left (a+\frac {a}{\cos \left (e+f\,x\right )}\right )}^{5/2}}{{\left (c-\frac {c}{\cos \left (e+f\,x\right )}\right )}^{9/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________