Optimal. Leaf size=215 \[ \frac {59 \text {ArcTan}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 d^{5/2} f}-\frac {\text {ArcTan}\left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{2 \sqrt {2} a^3 d^{5/2} f}-\frac {55}{24 a^3 d f (d \tan (e+f x))^{3/2}}+\frac {63}{8 a^3 d^2 f \sqrt {d \tan (e+f x)}}+\frac {11}{8 a^3 d f (d \tan (e+f x))^{3/2} (1+\tan (e+f x))}+\frac {1}{4 a d f (d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^2} \]
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Rubi [A]
time = 0.65, antiderivative size = 215, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.320, Rules used = {3650, 3730,
3731, 3734, 3613, 211, 3715, 65} \begin {gather*} \frac {59 \text {ArcTan}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 d^{5/2} f}-\frac {\text {ArcTan}\left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{2 \sqrt {2} a^3 d^{5/2} f}+\frac {63}{8 a^3 d^2 f \sqrt {d \tan (e+f x)}}+\frac {11}{8 a^3 d f (\tan (e+f x)+1) (d \tan (e+f x))^{3/2}}-\frac {55}{24 a^3 d f (d \tan (e+f x))^{3/2}}+\frac {1}{4 a d f (a \tan (e+f x)+a)^2 (d \tan (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 65
Rule 211
Rule 3613
Rule 3650
Rule 3715
Rule 3730
Rule 3731
Rule 3734
Rubi steps
\begin {align*} \int \frac {1}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^3} \, dx &=\frac {1}{4 a d f (d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^2}+\frac {\int \frac {\frac {11 a^2 d}{2}-2 a^2 d \tan (e+f x)+\frac {7}{2} a^2 d \tan ^2(e+f x)}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^2} \, dx}{4 a^3 d}\\ &=\frac {11}{8 a^3 d f (d \tan (e+f x))^{3/2} (1+\tan (e+f x))}+\frac {1}{4 a d f (d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^2}+\frac {\int \frac {\frac {55 a^4 d^2}{2}-4 a^4 d^2 \tan (e+f x)+\frac {55}{2} a^4 d^2 \tan ^2(e+f x)}{(d \tan (e+f x))^{5/2} (a+a \tan (e+f x))} \, dx}{8 a^6 d^2}\\ &=-\frac {55}{24 a^3 d f (d \tan (e+f x))^{3/2}}+\frac {11}{8 a^3 d f (d \tan (e+f x))^{3/2} (1+\tan (e+f x))}+\frac {1}{4 a d f (d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^2}-\frac {\int \frac {\frac {189 a^5 d^4}{4}+\frac {165}{4} a^5 d^4 \tan ^2(e+f x)}{(d \tan (e+f x))^{3/2} (a+a \tan (e+f x))} \, dx}{12 a^7 d^5}\\ &=-\frac {55}{24 a^3 d f (d \tan (e+f x))^{3/2}}+\frac {63}{8 a^3 d^2 f \sqrt {d \tan (e+f x)}}+\frac {11}{8 a^3 d f (d \tan (e+f x))^{3/2} (1+\tan (e+f x))}+\frac {1}{4 a d f (d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^2}+\frac {\int \frac {\frac {189 a^6 d^6}{8}+3 a^6 d^6 \tan (e+f x)+\frac {189}{8} a^6 d^6 \tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{6 a^8 d^8}\\ &=-\frac {55}{24 a^3 d f (d \tan (e+f x))^{3/2}}+\frac {63}{8 a^3 d^2 f \sqrt {d \tan (e+f x)}}+\frac {11}{8 a^3 d f (d \tan (e+f x))^{3/2} (1+\tan (e+f x))}+\frac {1}{4 a d f (d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^2}+\frac {\int \frac {3 a^7 d^6+3 a^7 d^6 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{12 a^{10} d^8}+\frac {59 \int \frac {1+\tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{16 a^2 d^2}\\ &=-\frac {55}{24 a^3 d f (d \tan (e+f x))^{3/2}}+\frac {63}{8 a^3 d^2 f \sqrt {d \tan (e+f x)}}+\frac {11}{8 a^3 d f (d \tan (e+f x))^{3/2} (1+\tan (e+f x))}+\frac {1}{4 a d f (d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^2}+\frac {59 \text {Subst}\left (\int \frac {1}{\sqrt {d x} (a+a x)} \, dx,x,\tan (e+f x)\right )}{16 a^2 d^2 f}-\frac {\left (3 a^4 d^4\right ) \text {Subst}\left (\int \frac {1}{18 a^{14} d^{12}+d x^2} \, dx,x,\frac {3 a^7 d^6-3 a^7 d^6 \tan (e+f x)}{\sqrt {d \tan (e+f x)}}\right )}{2 f}\\ &=-\frac {\tan ^{-1}\left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{2 \sqrt {2} a^3 d^{5/2} f}-\frac {55}{24 a^3 d f (d \tan (e+f x))^{3/2}}+\frac {63}{8 a^3 d^2 f \sqrt {d \tan (e+f x)}}+\frac {11}{8 a^3 d f (d \tan (e+f x))^{3/2} (1+\tan (e+f x))}+\frac {1}{4 a d f (d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^2}+\frac {59 \text {Subst}\left (\int \frac {1}{a+\frac {a x^2}{d}} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{8 a^2 d^3 f}\\ &=\frac {59 \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{8 a^3 d^{5/2} f}-\frac {\tan ^{-1}\left (\frac {\sqrt {d}-\sqrt {d} \tan (e+f x)}{\sqrt {2} \sqrt {d \tan (e+f x)}}\right )}{2 \sqrt {2} a^3 d^{5/2} f}-\frac {55}{24 a^3 d f (d \tan (e+f x))^{3/2}}+\frac {63}{8 a^3 d^2 f \sqrt {d \tan (e+f x)}}+\frac {11}{8 a^3 d f (d \tan (e+f x))^{3/2} (1+\tan (e+f x))}+\frac {1}{4 a d f (d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^2}\\ \end {align*}
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Mathematica [A]
time = 6.30, size = 368, normalized size = 1.71 \begin {gather*} \frac {\sec ^3(e+f x) (\cos (e+f x)+\sin (e+f x))^3 \left (\frac {8}{3}+6 \cot (e+f x)-\frac {2}{3} \csc ^2(e+f x)+\frac {1}{8 (\cos (e+f x)+\sin (e+f x))^2}-\frac {17 \sin (e+f x)}{8 (\cos (e+f x)+\sin (e+f x))}\right ) \tan ^3(e+f x)}{f (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^3}+\frac {\sec ^3(e+f x) (\cos (e+f x)+\sin (e+f x))^3 \tan ^{\frac {5}{2}}(e+f x) \left (\frac {126 \text {ArcTan}\left (\sqrt {\tan (e+f x)}\right ) \csc (e+f x) \sec ^3(e+f x) (1+\tan (e+f x))}{(1+\cot (e+f x)) \left (1+\tan ^2(e+f x)\right )^2}+\frac {2 \left (\sqrt {2} \left (-\text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )+\text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right )\right )-2 \text {ArcTan}\left (\sqrt {\tan (e+f x)}\right )\right ) \csc ^2(e+f x) \sec ^2(e+f x) \sin (2 (e+f x)) (1+\tan (e+f x))}{(1+\cot (e+f x)) \left (1+\tan ^2(e+f x)\right )}\right )}{16 f (d \tan (e+f x))^{5/2} (a+a \tan (e+f x))^3} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(378\) vs.
\(2(178)=356\).
time = 0.20, size = 379, normalized size = 1.76
method | result | size |
derivativedivides | \(\frac {2 d^{4} \left (-\frac {1}{3 d^{5} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {3}{d^{6} \sqrt {d \tan \left (f x +e \right )}}+\frac {\frac {\frac {15 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{4}+\frac {17 d \sqrt {d \tan \left (f x +e \right )}}{4}}{\left (d \tan \left (f x +e \right )+d \right )^{2}}+\frac {59 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{4 \sqrt {d}}}{4 d^{6}}+\frac {\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}}{4 d^{6}}\right )}{f \,a^{3}}\) | \(379\) |
default | \(\frac {2 d^{4} \left (-\frac {1}{3 d^{5} \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}+\frac {3}{d^{6} \sqrt {d \tan \left (f x +e \right )}}+\frac {\frac {\frac {15 \left (d \tan \left (f x +e \right )\right )^{\frac {3}{2}}}{4}+\frac {17 d \sqrt {d \tan \left (f x +e \right )}}{4}}{\left (d \tan \left (f x +e \right )+d \right )^{2}}+\frac {59 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{4 \sqrt {d}}}{4 d^{6}}+\frac {\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{8 \left (d^{2}\right )^{\frac {1}{4}}}}{4 d^{6}}\right )}{f \,a^{3}}\) | \(379\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 219, normalized size = 1.02 \begin {gather*} \frac {\frac {189 \, d^{3} \tan \left (f x + e\right )^{3} + 323 \, d^{3} \tan \left (f x + e\right )^{2} + 112 \, d^{3} \tan \left (f x + e\right ) - 16 \, d^{3}}{\left (d \tan \left (f x + e\right )\right )^{\frac {7}{2}} a^{3} d + 2 \, \left (d \tan \left (f x + e\right )\right )^{\frac {5}{2}} a^{3} d^{2} + \left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a^{3} d^{3}} + \frac {6 \, {\left (\frac {\sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}} + \frac {\sqrt {2} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {d} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {d}}\right )}{\sqrt {d}}\right )}}{a^{3} d} + \frac {177 \, \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a^{3} d^{\frac {3}{2}}}}{24 \, d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.85, size = 524, normalized size = 2.44 \begin {gather*} \left [-\frac {6 \, \sqrt {2} {\left (\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{3} + \tan \left (f x + e\right )^{2}\right )} \sqrt {-d} \log \left (\frac {d \tan \left (f x + e\right )^{2} - 2 \, \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {-d} {\left (\tan \left (f x + e\right ) - 1\right )} - 4 \, d \tan \left (f x + e\right ) + d}{\tan \left (f x + e\right )^{2} + 1}\right ) + 177 \, {\left (\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{3} + \tan \left (f x + e\right )^{2}\right )} \sqrt {-d} \log \left (\frac {d \tan \left (f x + e\right ) - 2 \, \sqrt {d \tan \left (f x + e\right )} \sqrt {-d} - d}{\tan \left (f x + e\right ) + 1}\right ) - 2 \, {\left (189 \, \tan \left (f x + e\right )^{3} + 323 \, \tan \left (f x + e\right )^{2} + 112 \, \tan \left (f x + e\right ) - 16\right )} \sqrt {d \tan \left (f x + e\right )}}{48 \, {\left (a^{3} d^{3} f \tan \left (f x + e\right )^{4} + 2 \, a^{3} d^{3} f \tan \left (f x + e\right )^{3} + a^{3} d^{3} f \tan \left (f x + e\right )^{2}\right )}}, \frac {6 \, \sqrt {2} {\left (\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{3} + \tan \left (f x + e\right )^{2}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {2} \sqrt {d \tan \left (f x + e\right )} {\left (\tan \left (f x + e\right ) - 1\right )}}{2 \, \sqrt {d} \tan \left (f x + e\right )}\right ) + 177 \, {\left (\tan \left (f x + e\right )^{4} + 2 \, \tan \left (f x + e\right )^{3} + \tan \left (f x + e\right )^{2}\right )} \sqrt {d} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right ) + {\left (189 \, \tan \left (f x + e\right )^{3} + 323 \, \tan \left (f x + e\right )^{2} + 112 \, \tan \left (f x + e\right ) - 16\right )} \sqrt {d \tan \left (f x + e\right )}}{24 \, {\left (a^{3} d^{3} f \tan \left (f x + e\right )^{4} + 2 \, a^{3} d^{3} f \tan \left (f x + e\right )^{3} + a^{3} d^{3} f \tan \left (f x + e\right )^{2}\right )}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {1}{\left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \tan ^{3}{\left (e + f x \right )} + 3 \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \tan ^{2}{\left (e + f x \right )} + 3 \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}} \tan {\left (e + f x \right )} + \left (d \tan {\left (e + f x \right )}\right )^{\frac {5}{2}}}\, dx}{a^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.00, size = 366, normalized size = 1.70 \begin {gather*} \frac {\sqrt {2} {\left (d \sqrt {{\left | d \right |}} + {\left | d \right |}^{\frac {3}{2}}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} + 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{8 \, a^{3} d^{4} f} + \frac {\sqrt {2} {\left (d \sqrt {{\left | d \right |}} + {\left | d \right |}^{\frac {3}{2}}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \sqrt {{\left | d \right |}} - 2 \, \sqrt {d \tan \left (f x + e\right )}\right )}}{2 \, \sqrt {{\left | d \right |}}}\right )}{8 \, a^{3} d^{4} f} + \frac {59 \, \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{8 \, a^{3} d^{\frac {5}{2}} f} + \frac {\sqrt {2} {\left (d \sqrt {{\left | d \right |}} - {\left | d \right |}^{\frac {3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{16 \, a^{3} d^{4} f} - \frac {\sqrt {2} {\left (d \sqrt {{\left | d \right |}} - {\left | d \right |}^{\frac {3}{2}}\right )} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {{\left | d \right |}} + {\left | d \right |}\right )}{16 \, a^{3} d^{4} f} + \frac {15 \, \sqrt {d \tan \left (f x + e\right )} d \tan \left (f x + e\right ) + 17 \, \sqrt {d \tan \left (f x + e\right )} d}{8 \, {\left (d \tan \left (f x + e\right ) + d\right )}^{2} a^{3} d^{2} f} + \frac {2 \, {\left (9 \, d \tan \left (f x + e\right ) - d\right )}}{3 \, \sqrt {d \tan \left (f x + e\right )} a^{3} d^{3} f \tan \left (f x + e\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.02, size = 192, normalized size = 0.89 \begin {gather*} \frac {\frac {63\,d\,{\mathrm {tan}\left (e+f\,x\right )}^3}{8}+\frac {323\,d\,{\mathrm {tan}\left (e+f\,x\right )}^2}{24}+\frac {14\,d\,\mathrm {tan}\left (e+f\,x\right )}{3}-\frac {2\,d}{3}}{a^3\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{7/2}+2\,a^3\,d\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{5/2}+a^3\,d^2\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}+\frac {59\,\mathrm {atan}\left (\frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{\sqrt {d}}\right )}{8\,a^3\,d^{5/2}\,f}+\frac {\sqrt {2}\,\left (2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}\right )+2\,\mathrm {atan}\left (\frac {\sqrt {2}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\sqrt {d}}+\frac {\sqrt {2}\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}}{2\,d^{3/2}}\right )\right )}{8\,a^3\,d^{5/2}\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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