Optimal. Leaf size=259 \[ -\frac {3 \sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{8 a^5 \sqrt {a-b} f}-\frac {3 \left (a^2-12 a b+16 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{8 a^5 f}-\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {(7 a-12 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {3 (a-2 b) b \sec (e+f x)}{2 a^4 f \left (a-b+b \sec ^2(e+f x)\right )} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.27, antiderivative size = 259, normalized size of antiderivative = 1.00, number of steps
used = 8, number of rules used = 6, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.261, Rules used = {3745, 481, 541,
536, 213, 211} \begin {gather*} -\frac {3 b (a-2 b) \sec (e+f x)}{2 a^4 f \left (a+b \sec ^2(e+f x)-b\right )}-\frac {b (7 a-12 b) \sec (e+f x)}{8 a^3 f \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a+b \sec ^2(e+f x)-b\right )^2}-\frac {3 \sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \text {ArcTan}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{8 a^5 f \sqrt {a-b}}-\frac {3 \left (a^2-12 a b+16 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{8 a^5 f}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a+b \sec ^2(e+f x)-b\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 211
Rule 213
Rule 481
Rule 536
Rule 541
Rule 3745
Rubi steps
\begin {align*} \int \frac {\csc ^5(e+f x)}{\left (a+b \tan ^2(e+f x)\right )^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^4}{\left (-1+x^2\right )^3 \left (a-b+b x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{f}\\ &=-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-a+b+(-4 a+7 b) x^2}{\left (-1+x^2\right )^2 \left (a-b+b x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{4 a f}\\ &=-\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-(3 a-8 b) (a-b)+5 (5 a-8 b) b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^3} \, dx,x,\sec (e+f x)\right )}{8 a^2 f}\\ &=-\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {(7 a-12 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\text {Subst}\left (\int \frac {-12 (a-4 b) (a-b)^2+12 (7 a-12 b) (a-b) b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )^2} \, dx,x,\sec (e+f x)\right )}{32 a^3 (a-b) f}\\ &=-\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {(7 a-12 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {3 (a-2 b) b \sec (e+f x)}{2 a^4 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {-24 (a-b)^2 \left (a^2-8 a b+8 b^2\right )+96 (a-2 b) (a-b)^2 b x^2}{\left (-1+x^2\right ) \left (a-b+b x^2\right )} \, dx,x,\sec (e+f x)\right )}{64 a^4 (a-b)^2 f}\\ &=-\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {(7 a-12 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {3 (a-2 b) b \sec (e+f x)}{2 a^4 f \left (a-b+b \sec ^2(e+f x)\right )}-\frac {\left (3 b \left (5 a^2-20 a b+16 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{a-b+b x^2} \, dx,x,\sec (e+f x)\right )}{8 a^5 f}+\frac {\left (3 \left (a^2-12 a b+16 b^2\right )\right ) \text {Subst}\left (\int \frac {1}{-1+x^2} \, dx,x,\sec (e+f x)\right )}{8 a^5 f}\\ &=-\frac {3 \sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \tan ^{-1}\left (\frac {\sqrt {b} \sec (e+f x)}{\sqrt {a-b}}\right )}{8 a^5 \sqrt {a-b} f}-\frac {3 \left (a^2-12 a b+16 b^2\right ) \tanh ^{-1}(\cos (e+f x))}{8 a^5 f}-\frac {(5 a-8 b) \cot (e+f x) \csc (e+f x)}{8 a^2 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {\cot ^3(e+f x) \csc (e+f x)}{4 a f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {(7 a-12 b) b \sec (e+f x)}{8 a^3 f \left (a-b+b \sec ^2(e+f x)\right )^2}-\frac {3 (a-2 b) b \sec (e+f x)}{2 a^4 f \left (a-b+b \sec ^2(e+f x)\right )}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 6.38, size = 468, normalized size = 1.81 \begin {gather*} -\frac {3 \sqrt {a-b} \sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \text {ArcTan}\left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (\sqrt {a-b} \cos \left (\frac {1}{2} (e+f x)\right )-\sqrt {a} \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {b}}\right )}{8 a^5 (-a+b) f}-\frac {3 \sqrt {a-b} \sqrt {b} \left (5 a^2-20 a b+16 b^2\right ) \text {ArcTan}\left (\frac {\sec \left (\frac {1}{2} (e+f x)\right ) \left (\sqrt {a-b} \cos \left (\frac {1}{2} (e+f x)\right )+\sqrt {a} \sin \left (\frac {1}{2} (e+f x)\right )\right )}{\sqrt {b}}\right )}{8 a^5 (-a+b) f}+\frac {b^2 \cos (e+f x)}{a^3 f (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))^2}-\frac {3 \left (3 a b \cos (e+f x)-4 b^2 \cos (e+f x)\right )}{4 a^4 f (a+b+a \cos (2 (e+f x))-b \cos (2 (e+f x)))}-\frac {3 (a-4 b) \csc ^2\left (\frac {1}{2} (e+f x)\right )}{32 a^4 f}-\frac {\csc ^4\left (\frac {1}{2} (e+f x)\right )}{64 a^3 f}-\frac {3 \left (a^2-12 a b+16 b^2\right ) \log \left (\cos \left (\frac {1}{2} (e+f x)\right )\right )}{8 a^5 f}+\frac {3 \left (a^2-12 a b+16 b^2\right ) \log \left (\sin \left (\frac {1}{2} (e+f x)\right )\right )}{8 a^5 f}+\frac {3 (a-4 b) \sec ^2\left (\frac {1}{2} (e+f x)\right )}{32 a^4 f}+\frac {\sec ^4\left (\frac {1}{2} (e+f x)\right )}{64 a^3 f} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.49, size = 265, normalized size = 1.02
method | result | size |
derivativedivides | \(\frac {\frac {1}{16 a^{3} \left (\cos \left (f x +e \right )+1\right )^{2}}-\frac {-3 a +12 b}{16 a^{4} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-3 a^{2}+36 a b -48 b^{2}\right ) \ln \left (\cos \left (f x +e \right )+1\right )}{16 a^{5}}-\frac {1}{16 a^{3} \left (\cos \left (f x +e \right )-1\right )^{2}}-\frac {-3 a +12 b}{16 a^{4} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (3 a^{2}-36 a b +48 b^{2}\right ) \ln \left (\cos \left (f x +e \right )-1\right )}{16 a^{5}}+\frac {b \left (\frac {-\frac {3 a \left (3 a^{2}-7 a b +4 b^{2}\right ) \left (\cos ^{3}\left (f x +e \right )\right )}{8}+\left (-\frac {7}{8} a^{2} b +\frac {3}{2} a \,b^{2}\right ) \cos \left (f x +e \right )}{\left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )^{2}}+\frac {3 \left (5 a^{2}-20 a b +16 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{8 \sqrt {b \left (a -b \right )}}\right )}{a^{5}}}{f}\) | \(265\) |
default | \(\frac {\frac {1}{16 a^{3} \left (\cos \left (f x +e \right )+1\right )^{2}}-\frac {-3 a +12 b}{16 a^{4} \left (\cos \left (f x +e \right )+1\right )}+\frac {\left (-3 a^{2}+36 a b -48 b^{2}\right ) \ln \left (\cos \left (f x +e \right )+1\right )}{16 a^{5}}-\frac {1}{16 a^{3} \left (\cos \left (f x +e \right )-1\right )^{2}}-\frac {-3 a +12 b}{16 a^{4} \left (\cos \left (f x +e \right )-1\right )}+\frac {\left (3 a^{2}-36 a b +48 b^{2}\right ) \ln \left (\cos \left (f x +e \right )-1\right )}{16 a^{5}}+\frac {b \left (\frac {-\frac {3 a \left (3 a^{2}-7 a b +4 b^{2}\right ) \left (\cos ^{3}\left (f x +e \right )\right )}{8}+\left (-\frac {7}{8} a^{2} b +\frac {3}{2} a \,b^{2}\right ) \cos \left (f x +e \right )}{\left (a \left (\cos ^{2}\left (f x +e \right )\right )-\left (\cos ^{2}\left (f x +e \right )\right ) b +b \right )^{2}}+\frac {3 \left (5 a^{2}-20 a b +16 b^{2}\right ) \arctan \left (\frac {\left (a -b \right ) \cos \left (f x +e \right )}{\sqrt {b \left (a -b \right )}}\right )}{8 \sqrt {b \left (a -b \right )}}\right )}{a^{5}}}{f}\) | \(265\) |
risch | \(\text {Expression too large to display}\) | \(1036\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \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. 851 vs.
\(2 (251) = 502\).
time = 1.67, size = 1741, normalized size = 6.72 \begin {gather*} \text {Too large to display} \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 [B] Leaf count of result is larger than twice the leaf count of optimal. 922 vs.
\(2 (251) = 502\).
time = 1.08, size = 922, normalized size = 3.56 \begin {gather*} \frac {\frac {12 \, {\left (a^{2} - 12 \, a b + 16 \, b^{2}\right )} \log \left (\frac {{\left | -\cos \left (f x + e\right ) + 1 \right |}}{{\left | \cos \left (f x + e\right ) + 1 \right |}}\right )}{a^{5}} - \frac {24 \, {\left (5 \, a^{2} b - 20 \, a b^{2} + 16 \, b^{3}\right )} \arctan \left (-\frac {a \cos \left (f x + e\right ) - b \cos \left (f x + e\right ) - b}{\sqrt {a b - b^{2}} \cos \left (f x + e\right ) + \sqrt {a b - b^{2}}}\right )}{\sqrt {a b - b^{2}} a^{5}} - \frac {\frac {8 \, a^{3} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {24 \, a^{2} b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {a^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}}}{a^{6}} - \frac {a^{4} - \frac {4 \, a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {16 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} - \frac {20 \, a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {216 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {304 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {20 \, a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {360 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} - \frac {1024 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {896 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}} + \frac {5 \, a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {64 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {192 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {256 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} - \frac {256 \, b^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{4}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{4}} + \frac {16 \, a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {168 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {384 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} - \frac {256 \, a b^{3} {\left (\cos \left (f x + e\right ) - 1\right )}^{5}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{5}} + \frac {6 \, a^{4} {\left (\cos \left (f x + e\right ) - 1\right )}^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} - \frac {72 \, a^{3} b {\left (\cos \left (f x + e\right ) - 1\right )}^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}} + \frac {96 \, a^{2} b^{2} {\left (\cos \left (f x + e\right ) - 1\right )}^{6}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{6}}}{a^{5} {\left (\frac {a {\left (\cos \left (f x + e\right ) - 1\right )}}{\cos \left (f x + e\right ) + 1} + \frac {2 \, a {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} - \frac {4 \, b {\left (\cos \left (f x + e\right ) - 1\right )}^{2}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{2}} + \frac {a {\left (\cos \left (f x + e\right ) - 1\right )}^{3}}{{\left (\cos \left (f x + e\right ) + 1\right )}^{3}}\right )}^{2}}}{64 \, f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 12.82, size = 1357, normalized size = 5.24 \begin {gather*} \frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4}{64\,a^3\,f}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {3\,\left (a-2\,b\right )}{16\,a^4}-\frac {1}{16\,a^3}\right )}{f}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4\,\left (\frac {13\,a^3}{2}-72\,a^2\,b+100\,a\,b^2\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (2\,a^3-42\,a^2\,b+144\,a\,b^2-128\,b^3\right )-{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (11\,a^3-174\,a^2\,b+496\,a\,b^2-416\,b^3\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (4\,a^2\,b-a^3\right )-\frac {a^3}{4}+\frac {{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (31\,a^4-592\,a^3\,b+2016\,a^2\,b^2-2944\,a\,b^3+1792\,b^4\right )}{4\,a}}{f\,\left (16\,a^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^4+16\,a^6\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{12}+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^8\,\left (96\,a^6-256\,a^5\,b+256\,a^4\,b^2\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^6\,\left (128\,a^5\,b-64\,a^6\right )+{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^{10}\,\left (128\,a^5\,b-64\,a^6\right )\right )}+\frac {\ln \left (\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )\right )\,\left (3\,a^2-36\,a\,b+48\,b^2\right )}{8\,a^5\,f}+\frac {3\,\sqrt {b}\,\mathrm {atan}\left (\frac {a^{13}\,{\left (a-b\right )}^{3/2}\,\left (\frac {256\,\left (\frac {\frac {675\,a^6\,b^2}{32}-\frac {3375\,a^5\,b^3}{8}+2835\,a^4\,b^4-8910\,a^3\,b^5+14256\,a^2\,b^6-11232\,a\,b^7+3456\,b^8}{a^{12}}-\frac {9\,b\,{\left (5\,a^2-20\,a\,b+16\,b^2\right )}^2\,\left (192\,a^{14}-4992\,a^{13}\,b+24576\,a^{12}\,b^2-43008\,a^{11}\,b^3+24576\,a^{10}\,b^4\right )}{8192\,a^{22}\,\left (a-b\right )}\right )\,\left (a^5-45\,a^4\,b+420\,a^3\,b^2-1344\,a^2\,b^3+1728\,a\,b^4-768\,b^5\right )}{\sqrt {b}\,\left (b\,\left (b\,\left (b\,\left (1680\,a^7+b\,\left (768\,a^5\,b-1920\,a^6\right )\right )-600\,a^8\right )+75\,a^9\right )-4\,a^{10}\right )}-256\,{\mathrm {tan}\left (\frac {e}{2}+\frac {f\,x}{2}\right )}^2\,\left (\frac {\left (\frac {\frac {675\,a^5\,b^2}{16}-\frac {2025\,a^4\,b^3}{4}+2295\,a^3\,b^4-4860\,a^2\,b^5+4752\,a\,b^6-1728\,b^7}{a^{11}}+\frac {9\,b\,{\left (5\,a^2-20\,a\,b+16\,b^2\right )}^2\,\left (-96\,a^{13}+3552\,a^{12}\,b-36480\,a^{11}\,b^2+125952\,a^{10}\,b^3-165888\,a^9\,b^4+73728\,a^8\,b^5\right )}{4096\,a^{21}\,\left (a-b\right )}\right )\,\left (a^5-45\,a^4\,b+420\,a^3\,b^2-1344\,a^2\,b^3+1728\,a\,b^4-768\,b^5\right )}{\sqrt {b}\,\left (b\,\left (b\,\left (b\,\left (1680\,a^7+b\,\left (768\,a^5\,b-1920\,a^6\right )\right )-600\,a^8\right )+75\,a^9\right )-4\,a^{10}\right )}-\frac {\left (\frac {b^{3/2}\,{\left (5\,a^2-20\,a\,b+16\,b^2\right )}^3\,\left (-\frac {27\,a^{16}}{256}+\frac {351\,a^{15}\,b}{128}-\frac {243\,a^{14}\,b^2}{32}+\frac {81\,a^{13}\,b^3}{16}\right )}{a^{26}\,{\left (a-b\right )}^{3/2}}-\frac {3\,\sqrt {b}\,\left (5\,a^2-20\,a\,b+16\,b^2\right )\,\left (540\,a^9\,b-9720\,a^8\,b^2+67248\,a^7\,b^3-225792\,a^6\,b^4+389376\,a^5\,b^5-331776\,a^4\,b^6+110592\,a^3\,b^7\right )}{256\,a^{16}\,\sqrt {a-b}}\right )\,\left (17\,a^5-330\,a^4\,b+1848\,a^3\,b^2-4224\,a^2\,b^3+4224\,a\,b^4-1536\,b^5\right )}{2\,a^5\,\sqrt {a-b}\,\left (4\,a^5-75\,a^4\,b+600\,a^3\,b^2-1680\,a^2\,b^3+1920\,a\,b^4-768\,b^5\right )}\right )+\frac {128\,\left (\frac {b^{3/2}\,{\left (5\,a^2-20\,a\,b+16\,b^2\right )}^3\,\left (\frac {27\,a^{17}}{256}-\frac {27\,a^{16}\,b}{64}+\frac {27\,a^{15}\,b^2}{64}\right )}{a^{27}\,{\left (a-b\right )}^{3/2}}+\frac {3\,\sqrt {b}\,\left (5\,a^2-20\,a\,b+16\,b^2\right )\,\left (720\,a^{10}\,b-14760\,a^9\,b^2+85824\,a^8\,b^3-210816\,a^7\,b^4+230400\,a^6\,b^5-92160\,a^5\,b^6\right )}{512\,a^{17}\,\sqrt {a-b}}\right )\,\left (17\,a^5-330\,a^4\,b+1848\,a^3\,b^2-4224\,a^2\,b^3+4224\,a\,b^4-1536\,b^5\right )}{a^5\,\sqrt {a-b}\,\left (4\,a^5-75\,a^4\,b+600\,a^3\,b^2-1680\,a^2\,b^3+1920\,a\,b^4-768\,b^5\right )}\right )}{675\,a^4\,b-5400\,a^3\,b^2+15120\,a^2\,b^3-17280\,a\,b^4+6912\,b^5}\right )\,\left (5\,a^2-20\,a\,b+16\,b^2\right )}{8\,a^5\,f\,\sqrt {a-b}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________