Optimal. Leaf size=279 \[ -\frac {d^{3/2} \text {ArcTan}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 f}-\frac {d^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 f}+\frac {d^{3/2} \text {ArcTan}\left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 f}+\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 f}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 f}+\frac {d \sqrt {d \tan (e+f x)}}{2 f \left (a^2+a^2 \tan (e+f x)\right )} \]
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Rubi [A]
time = 0.35, antiderivative size = 279, 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 = {3648, 3734, 12, 16, 3557, 335, 303, 1176, 631, 210, 1179, 642, 3715, 65, 211}
\begin {gather*} -\frac {d^{3/2} \text {ArcTan}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 f}-\frac {d^{3/2} \text {ArcTan}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 f}+\frac {d^{3/2} \text {ArcTan}\left (\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}+1\right )}{2 \sqrt {2} a^2 f}+\frac {d^{3/2} \log \left (\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{4 \sqrt {2} a^2 f}-\frac {d^{3/2} \log \left (\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}+\sqrt {d}\right )}{4 \sqrt {2} a^2 f}+\frac {d \sqrt {d \tan (e+f x)}}{2 f \left (a^2 \tan (e+f x)+a^2\right )} \end {gather*}
Antiderivative was successfully verified.
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Rule 12
Rule 16
Rule 65
Rule 210
Rule 211
Rule 303
Rule 335
Rule 631
Rule 642
Rule 1176
Rule 1179
Rule 3557
Rule 3648
Rule 3715
Rule 3734
Rubi steps
\begin {align*} \int \frac {(d \tan (e+f x))^{3/2}}{(a+a \tan (e+f x))^2} \, dx &=\frac {d \sqrt {d \tan (e+f x)}}{2 f \left (a^2+a^2 \tan (e+f x)\right )}-\frac {\int \frac {\frac {a d^2}{2}-a d^2 \tan (e+f x)-\frac {1}{2} a d^2 \tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{2 a^2}\\ &=\frac {d \sqrt {d \tan (e+f x)}}{2 f \left (a^2+a^2 \tan (e+f x)\right )}-\frac {\int -\frac {2 a^2 d^2 \tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{4 a^4}-\frac {d^2 \int \frac {1+\tan ^2(e+f x)}{\sqrt {d \tan (e+f x)} (a+a \tan (e+f x))} \, dx}{4 a}\\ &=\frac {d \sqrt {d \tan (e+f x)}}{2 f \left (a^2+a^2 \tan (e+f x)\right )}+\frac {d^2 \int \frac {\tan (e+f x)}{\sqrt {d \tan (e+f x)}} \, dx}{2 a^2}-\frac {d^2 \text {Subst}\left (\int \frac {1}{\sqrt {d x} (a+a x)} \, dx,x,\tan (e+f x)\right )}{4 a f}\\ &=\frac {d \sqrt {d \tan (e+f x)}}{2 f \left (a^2+a^2 \tan (e+f x)\right )}+\frac {d \int \sqrt {d \tan (e+f x)} \, dx}{2 a^2}-\frac {d \text {Subst}\left (\int \frac {1}{a+\frac {a x^2}{d}} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{2 a f}\\ &=-\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 f}+\frac {d \sqrt {d \tan (e+f x)}}{2 f \left (a^2+a^2 \tan (e+f x)\right )}+\frac {d^2 \text {Subst}\left (\int \frac {\sqrt {x}}{d^2+x^2} \, dx,x,d \tan (e+f x)\right )}{2 a^2 f}\\ &=-\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 f}+\frac {d \sqrt {d \tan (e+f x)}}{2 f \left (a^2+a^2 \tan (e+f x)\right )}+\frac {d^2 \text {Subst}\left (\int \frac {x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{a^2 f}\\ &=-\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 f}+\frac {d \sqrt {d \tan (e+f x)}}{2 f \left (a^2+a^2 \tan (e+f x)\right )}-\frac {d^2 \text {Subst}\left (\int \frac {d-x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{2 a^2 f}+\frac {d^2 \text {Subst}\left (\int \frac {d+x^2}{d^2+x^4} \, dx,x,\sqrt {d \tan (e+f x)}\right )}{2 a^2 f}\\ &=-\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 f}+\frac {d \sqrt {d \tan (e+f x)}}{2 f \left (a^2+a^2 \tan (e+f x)\right )}+\frac {d^{3/2} \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 f}+\frac {d^{3/2} \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 f}+\frac {d^2 \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 f}+\frac {d^2 \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 f}\\ &=-\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 f}+\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 f}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 f}+\frac {d \sqrt {d \tan (e+f x)}}{2 f \left (a^2+a^2 \tan (e+f x)\right )}+\frac {d^{3/2} \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 f}-\frac {d^{3/2} \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 f}\\ &=-\frac {d^{3/2} \tan ^{-1}\left (\frac {\sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 a^2 f}-\frac {d^{3/2} \tan ^{-1}\left (1-\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 f}+\frac {d^{3/2} \tan ^{-1}\left (1+\frac {\sqrt {2} \sqrt {d \tan (e+f x)}}{\sqrt {d}}\right )}{2 \sqrt {2} a^2 f}+\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)-\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 f}-\frac {d^{3/2} \log \left (\sqrt {d}+\sqrt {d} \tan (e+f x)+\sqrt {2} \sqrt {d \tan (e+f x)}\right )}{4 \sqrt {2} a^2 f}+\frac {d \sqrt {d \tan (e+f x)}}{2 f \left (a^2+a^2 \tan (e+f x)\right )}\\ \end {align*}
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Mathematica [A]
time = 2.17, size = 229, normalized size = 0.82 \begin {gather*} \frac {\sec (e+f x) (\cos (e+f x)+\sin (e+f x))^2 \left (\frac {2 \cot (e+f x)}{\cos (e+f x)+\sin (e+f x)}-\frac {\csc (e+f x) \left (2 \sqrt {2} \text {ArcTan}\left (1-\sqrt {2} \sqrt {\tan (e+f x)}\right )-2 \sqrt {2} \text {ArcTan}\left (1+\sqrt {2} \sqrt {\tan (e+f x)}\right )+4 \text {ArcTan}\left (\sqrt {\tan (e+f x)}\right )-\sqrt {2} \log \left (-1+\sqrt {2} \sqrt {\tan (e+f x)}-\tan (e+f x)\right )+\sqrt {2} \log \left (1+\sqrt {2} \sqrt {\tan (e+f x)}+\tan (e+f x)\right )\right )}{2 \sqrt {\tan (e+f x)}}\right ) (d \tan (e+f x))^{3/2}}{4 a^2 f (1+\tan (e+f x))^2} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.14, size = 197, normalized size = 0.71
method | result | size |
derivativedivides | \(\frac {2 d^{3} \left (-\frac {-\frac {\sqrt {d \tan \left (f x +e \right )}}{2 \left (d \tan \left (f x +e \right )+d \right )}+\frac {\arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{2 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 d \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f \,a^{2}}\) | \(197\) |
default | \(\frac {2 d^{3} \left (-\frac {-\frac {\sqrt {d \tan \left (f x +e \right )}}{2 \left (d \tan \left (f x +e \right )+d \right )}+\frac {\arctan \left (\frac {\sqrt {d \tan \left (f x +e \right )}}{\sqrt {d}}\right )}{2 \sqrt {d}}}{2 d}+\frac {\sqrt {2}\, \left (\ln \left (\frac {d \tan \left (f x +e \right )-\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}{d \tan \left (f x +e \right )+\left (d^{2}\right )^{\frac {1}{4}} \sqrt {d \tan \left (f x +e \right )}\, \sqrt {2}+\sqrt {d^{2}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )-2 \arctan \left (-\frac {\sqrt {2}\, \sqrt {d \tan \left (f x +e \right )}}{\left (d^{2}\right )^{\frac {1}{4}}}+1\right )\right )}{16 d \left (d^{2}\right )^{\frac {1}{4}}}\right )}{f \,a^{2}}\) | \(197\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.51, size = 229, normalized size = 0.82 \begin {gather*} \frac {\frac {4 \, \sqrt {d \tan \left (f x + e\right )} d^{3}}{a^{2} d \tan \left (f x + e\right ) + a^{2} d} + \frac {d^{3} {\left (\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}}\right )}}{a^{2}} - \frac {4 \, d^{\frac {5}{2}} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a^{2}}}{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. 887 vs.
\(2 (224) = 448\).
time = 1.64, size = 1855, normalized size = 6.65 \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 {\left (d \tan {\left (e + f x \right )}\right )^{\frac {3}{2}}}{\tan ^{2}{\left (e + f x \right )} + 2 \tan {\left (e + f x \right )} + 1}\, dx}{a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.72, size = 264, normalized size = 0.95 \begin {gather*} \frac {1}{8} \, d {\left (\frac {2 \, \sqrt {2} {\left | d \right |}^{\frac {3}{2}} \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} {\left | d \right |}^{\frac {3}{2}} \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} {\left | d \right |}^{\frac {3}{2}} \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} {\left | d \right |}^{\frac {3}{2}} \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 {4 \, \sqrt {d} \arctan \left (\frac {\sqrt {d \tan \left (f x + e\right )}}{\sqrt {d}}\right )}{a^{2} f} + \frac {4 \, \sqrt {d \tan \left (f x + e\right )} d}{{\left (d \tan \left (f x + e\right ) + d\right )} a^{2} f}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.61, size = 375, normalized size = 1.34 \begin {gather*} \frac {\mathrm {atan}\left (\frac {4\,d^{16}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {d^6}{a^8\,f^4}\right )}^{1/4}}{\frac {4\,d^{18}}{a^2\,f}+4\,a^2\,d^{15}\,f\,\sqrt {-\frac {d^6}{a^8\,f^4}}}+\frac {4\,d^{13}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {d^6}{a^8\,f^4}\right )}^{3/4}}{\frac {4\,d^{18}}{a^6\,f^3}+\frac {4\,d^{15}\,\sqrt {-\frac {d^6}{a^8\,f^4}}}{a^2\,f}}\right )\,{\left (-\frac {d^6}{a^8\,f^4}\right )}^{1/4}}{2}+\mathrm {atan}\left (\frac {d^{16}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {d^6}{256\,a^8\,f^4}\right )}^{1/4}\,16{}\mathrm {i}}{\frac {4\,d^{18}}{a^2\,f}-64\,a^2\,d^{15}\,f\,\sqrt {-\frac {d^6}{256\,a^8\,f^4}}}-\frac {d^{13}\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}\,{\left (-\frac {d^6}{256\,a^8\,f^4}\right )}^{3/4}\,256{}\mathrm {i}}{\frac {4\,d^{18}}{a^6\,f^3}-\frac {64\,d^{15}\,\sqrt {-\frac {d^6}{256\,a^8\,f^4}}}{a^2\,f}}\right )\,{\left (-\frac {d^6}{256\,a^8\,f^4}\right )}^{1/4}\,2{}\mathrm {i}+\frac {d^2\,\sqrt {d\,\mathrm {tan}\left (e+f\,x\right )}}{2\,\left (a^2\,d\,f+a^2\,d\,f\,\mathrm {tan}\left (e+f\,x\right )\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}\,1{}\mathrm {i}}{2\,a^2\,f} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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