Optimal. Leaf size=306 \[ -\frac {5 \text {ArcTan}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 d^{3/2} f}+\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 d^{3/2} f}-\frac {\text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 d^{3/2} f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 d^{3/2} f}-\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 d^{3/2} f}-\frac {5}{2 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {1}{2 d f \sqrt {d \tan (e+f x)} \left (a^2+a^2 \tan (e+f x)\right )} \]
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Rubi [A]
time = 0.48, antiderivative size = 306, normalized size of antiderivative = 1.00, number of
steps used = 18, number of rules used = 15, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.600, Rules used
= {3650, 3730, 3734, 12, 3557, 335, 217, 1179, 642, 1176, 631, 210, 3715, 65, 211}
\begin {gather*} -\frac {5 \text {ArcTan}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 d^{3/2} f}+\frac {\text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 d^{3/2} f}-\frac {\text {ArcTan}\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{2 \sqrt {2} a^2 d^{3/2} f}+\frac {\log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{4 \sqrt {2} a^2 d^{3/2} f}-\frac {\log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{4 \sqrt {2} a^2 d^{3/2} f}-\frac {5}{2 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {1}{2 d f \left (a^2 \tan (e+f x)+a^2\right ) \sqrt {d \tan (e+f x)}} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 65
Rule 210
Rule 211
Rule 217
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3557
Rule 3650
Rule 3715
Rule 3730
Rule 3734
Rubi steps
\begin {align*} \int \frac {1}{(d \tan (e+f x))^{3/2} (a+a \tan (e+f x))^2} \, dx &=\frac {1}{2 d f \sqrt {d \tan (e+f x)} \left (a^2+a^2 \tan (e+f x)\right )}+\frac {\int \frac {\frac {5 a^2 d}{2}-a^2 d \tan (e+f x)+\frac {3}{2} a^2 d \tan ^2(e+f x)}{(d \tan (e+f x))^{3/2} (a+a \tan (e+f x))} \, dx}{2 a^3 d}\\ &=-\frac {5}{2 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {1}{2 d f \sqrt {d \tan (e+f x)} \left (a^2+a^2 \tan (e+f x)\right )}-\frac {\int \frac {\frac {7 a^3 d^3}{4}+\frac {1}{2} a^3 d^3 \tan (e+f x)+\frac {5}{4} a^3 d^3 \tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{a^4 d^4}\\ &=-\frac {5}{2 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {1}{2 d f \sqrt {d \tan (e+f x)} \left (a^2+a^2 \tan (e+f x)\right )}-\frac {\int \frac {a^4 d^3}{\sqrt {d \tan (e+f x)}} \, dx}{2 a^6 d^4}-\frac {5 \int \frac {1+\tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{4 a d}\\ &=-\frac {5}{2 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {1}{2 d f \sqrt {d \tan (e+f x)} \left (a^2+a^2 \tan (e+f x)\right )}-\frac {\int \frac {1}{\sqrt {d \tan (e+f x)}} \, dx}{2 a^2 d}-\frac {5 \text {Subst}\left (\int \frac {1}{\sqrt {d x} (a+a x)} \, dx,x,\tan (e+f x)\right )}{4 a d f}\\ &=-\frac {5}{2 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {1}{2 d f \sqrt {d \tan (e+f x)} \left (a^2+a^2 \tan (e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {1}{\sqrt {x} \left (d^2+x^2\right )} \, dx,x,d \tan (e+f x)\right )}{2 a^2 f}-\frac {5 \text {Subst}\left (\int \frac {1}{a+\frac {a x^2}{d}} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{2 a d^2 f}\\ &=-\frac {5 \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 d^{3/2} f}-\frac {5}{2 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {1}{2 d f \sqrt {d \tan (e+f x)} \left (a^2+a^2 \tan (e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {1}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 f}\\ &=-\frac {5 \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 d^{3/2} f}-\frac {5}{2 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {1}{2 d f \sqrt {d \tan (e+f x)} \left (a^2+a^2 \tan (e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{2 a^2 d f}-\frac {\text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{2 a^2 d f}\\ &=-\frac {5 \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 d^{3/2} f}-\frac {5}{2 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {1}{2 d f \sqrt {d \tan (e+f x)} \left (a^2+a^2 \tan (e+f x)\right )}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}+2 x}{-d-\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 d^{3/2} f}+\frac {\text {Subst}\left (\int \frac {\sqrt {2} \sqrt {d}-2 x}{-d+\sqrt {2} \sqrt {d} x-x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 d^{3/2} f}-\frac {\text {Subst}\left (\int \frac {1}{d-\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{4 a^2 d f}-\frac {\text {Subst}\left (\int \frac {1}{d+\sqrt {2} \sqrt {d} x+x^2} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{4 a^2 d f}\\ &=-\frac {5 \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 d^{3/2} f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 d^{3/2} f}-\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 d^{3/2} f}-\frac {5}{2 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {1}{2 d f \sqrt {d \tan (e+f x)} \left (a^2+a^2 \tan (e+f x)\right )}-\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 d^{3/2} f}+\frac {\text {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 d^{3/2} f}\\ &=-\frac {5 \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 d^{3/2} f}+\frac {\tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 d^{3/2} f}-\frac {\tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 d^{3/2} f}+\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 d^{3/2} f}-\frac {\log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 d^{3/2} f}-\frac {5}{2 a^2 d f \sqrt {d \tan (e+f x)}}+\frac {1}{2 d f \sqrt {d \tan (e+f x)} \left (a^2+a^2 \tan (e+f x)\right )}\\ \end {align*}
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Mathematica [A]
time = 1.38, size = 203, normalized size = 0.66 \begin {gather*} \frac {\left (\sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )-\sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right )-10 \text {ArcTan}\left (\sqrt {\tan (e+f x)}\right )+\frac {\log \left (-1+\sqrt {2} \sqrt {\tan (e+f x)}-\tan (e+f x)\right )-\log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )}{\sqrt {2}}-\frac {2 (4 \cos (e+f x)+5 \sin (e+f x))}{(\cos (e+f x)+\sin (e+f x)) \sqrt {\tan (e+f x)}}\right ) \tan ^{\frac {3}{2}}(e+f x)}{4 a^2 f (d \tan (e+f x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.17, size = 212, normalized size = 0.69
method | result | size |
derivativedivides | \(\frac {2 d^{3} \left (-\frac {1}{d^{4} \sqrt {d \tan \left (f x +e \right )}}-\frac {\frac {\sqrt {d \tan \left (f x +e \right )}}{2 d \tan \left (f x +e \right )+2 d}+\frac {5 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{2 d^{4}}-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 d^{5}}\right )}{f \,a^{2}}\) | \(212\) |
default | \(\frac {2 d^{3} \left (-\frac {1}{d^{4} \sqrt {d \tan \left (f x +e \right )}}-\frac {\frac {\sqrt {d \tan \left (f x +e \right )}}{2 d \tan \left (f x +e \right )+2 d}+\frac {5 \arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{2 d^{4}}-\frac {\left (d^{2}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 d^{5}}\right )}{f \,a^{2}}\) | \(212\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.52, size = 240, normalized size = 0.78 \begin {gather*} -\frac {\frac {4 \, {\left (5 \, d \tan \left (f x + e\right ) + 4 \, d\right )}}{\left (d \tan \left (f x + e\right )\right )^{\frac {3}{2}} a^{2} + \sqrt {d \tan \left (f x + e\right )} a^{2} d} + \frac {\frac {2 \, \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 {2 \, \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} \log \left (d \tan \left (f x + e\right ) + \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}} - \frac {\sqrt {2} \log \left (d \tan \left (f x + e\right ) - \sqrt {2} \sqrt {d \tan \left (f x + e\right )} \sqrt {d} + d\right )}{\sqrt {d}}}{a^{2}} + \frac {20 \, \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a^{2} \sqrt {d}}}{8 \, d f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1151 vs.
\(2 (248) = 496\).
time = 2.18, size = 2384, normalized size = 7.79 \begin {gather*} \text {Too large to display} \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 {3}{2}} \tan ^{2}{\left (e + f x \right )} + 2 \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}} \tan {\left (e + f x \right )} + \left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}\, dx}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.75, size = 291, normalized size = 0.95 \begin {gather*} -\frac {\frac {2 \, \sqrt {2} \sqrt {{\left | d \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 )}{a^{2} d f} + \frac {2 \, \sqrt {2} \sqrt {{\left | d \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 )}{a^{2} d f} + \frac {\sqrt {2} \sqrt {{\left | d \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 )}{a^{2} d f} - \frac {\sqrt {2} \sqrt {{\left | d \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 )}{a^{2} d f} + \frac {20 \, \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a^{2} \sqrt {d} f} + \frac {4 \, {\left (5 \, d \tan \left (f x + e\right ) + 4 \, d\right )}}{{\left (\sqrt {d \tan \left (f x + e\right )} d \tan \left (f x + e\right ) + \sqrt {d \tan \left (f x + e\right )} d\right )} a^{2} f}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.72, size = 415, normalized size = 1.36 \begin {gather*} \frac {\mathrm {atan}\left (\frac {2048\,a^{10}\,d^{13}\,f^5\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {1}{a^8\,d^6\,f^4}\right )}^{1/4}}{51200\,a^8\,d^{12}\,f^4-2048\,a^{12}\,d^{15}\,f^6\,\sqrt {-\frac {1}{a^8\,d^6\,f^4}}}+\frac {51200\,a^{14}\,d^{16}\,f^7\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {1}{a^8\,d^6\,f^4}\right )}^{3/4}}{51200\,a^8\,d^{12}\,f^4-2048\,a^{12}\,d^{15}\,f^6\,\sqrt {-\frac {1}{a^8\,d^6\,f^4}}}\right )\,{\left (-\frac {1}{a^8\,d^6\,f^4}\right )}^{1/4}}{2}+\mathrm {atan}\left (\frac {a^{10}\,d^{13}\,f^5\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {1}{256\,a^8\,d^6\,f^4}\right )}^{1/4}\,8192{}\mathrm {i}}{51200\,a^8\,d^{12}\,f^4+32768\,a^{12}\,d^{15}\,f^6\,\sqrt {-\frac {1}{256\,a^8\,d^6\,f^4}}}-\frac {a^{14}\,d^{16}\,f^7\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {1}{256\,a^8\,d^6\,f^4}\right )}^{3/4}\,3276800{}\mathrm {i}}{51200\,a^8\,d^{12}\,f^4+32768\,a^{12}\,d^{15}\,f^6\,\sqrt {-\frac {1}{256\,a^8\,d^6\,f^4}}}\right )\,{\left (-\frac {1}{256\,a^8\,d^6\,f^4}\right )}^{1/4}\,2{}\mathrm {i}-\frac {\frac {5\,\mathrm {tan}\left (e+f\,x\right )}{2}+2}{a^2\,f\,{\left (d\,\mathrm {tan}\left (e+f\,x\right )\right )}^{3/2}+a^2\,d\,f\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}-\frac {\mathrm {atan}\left (\frac {\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,\sqrt {-d^3}\,1{}\mathrm {i}}{d^2}\right )\,\sqrt {-d^3}\,5{}\mathrm {i}}{2\,a^2\,d^3\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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