Optimal. Leaf size=121 \[ \frac{2 a}{d^3 f \sqrt{d \tan (e+f x)}}-\frac{2 a}{3 d^2 f (d \tan (e+f x))^{3/2}}-\frac{\sqrt{2} a \tanh ^{-1}\left (\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac{2 a}{5 d f (d \tan (e+f x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.165502, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13, Rules used = {3529, 3532, 208} \[ \frac{2 a}{d^3 f \sqrt{d \tan (e+f x)}}-\frac{2 a}{3 d^2 f (d \tan (e+f x))^{3/2}}-\frac{\sqrt{2} a \tanh ^{-1}\left (\frac{\sqrt{d} \tan (e+f x)+\sqrt{d}}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac{2 a}{5 d f (d \tan (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3529
Rule 3532
Rule 208
Rubi steps
\begin{align*} \int \frac{a+a \tan (e+f x)}{(d \tan (e+f x))^{7/2}} \, dx &=-\frac{2 a}{5 d f (d \tan (e+f x))^{5/2}}+\frac{\int \frac{a d-a d \tan (e+f x)}{(d \tan (e+f x))^{5/2}} \, dx}{d^2}\\ &=-\frac{2 a}{5 d f (d \tan (e+f x))^{5/2}}-\frac{2 a}{3 d^2 f (d \tan (e+f x))^{3/2}}+\frac{\int \frac{-a d^2-a d^2 \tan (e+f x)}{(d \tan (e+f x))^{3/2}} \, dx}{d^4}\\ &=-\frac{2 a}{5 d f (d \tan (e+f x))^{5/2}}-\frac{2 a}{3 d^2 f (d \tan (e+f x))^{3/2}}+\frac{2 a}{d^3 f \sqrt{d \tan (e+f x)}}+\frac{\int \frac{-a d^3+a d^3 \tan (e+f x)}{\sqrt{d \tan (e+f x)}} \, dx}{d^6}\\ &=-\frac{2 a}{5 d f (d \tan (e+f x))^{5/2}}-\frac{2 a}{3 d^2 f (d \tan (e+f x))^{3/2}}+\frac{2 a}{d^3 f \sqrt{d \tan (e+f x)}}-\frac{\left (2 a^2\right ) \operatorname{Subst}\left (\int \frac{1}{-2 a^2 d^6+d x^2} \, dx,x,\frac{-a d^3-a d^3 \tan (e+f x)}{\sqrt{d \tan (e+f x)}}\right )}{f}\\ &=-\frac{\sqrt{2} a \tanh ^{-1}\left (\frac{\sqrt{d}+\sqrt{d} \tan (e+f x)}{\sqrt{2} \sqrt{d \tan (e+f x)}}\right )}{d^{7/2} f}-\frac{2 a}{5 d f (d \tan (e+f x))^{5/2}}-\frac{2 a}{3 d^2 f (d \tan (e+f x))^{3/2}}+\frac{2 a}{d^3 f \sqrt{d \tan (e+f x)}}\\ \end{align*}
Mathematica [C] time = 0.209191, size = 68, normalized size = 0.56 \[ -\frac{\left (\frac{1}{5}+\frac{i}{5}\right ) a \left (\, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};-i \tan (e+f x)\right )-i \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};i \tan (e+f x)\right )\right )}{d f (d \tan (e+f x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 0.02, size = 393, normalized size = 3.3 \begin{align*} -{\frac{a\sqrt{2}}{4\,f{d}^{4}}\sqrt [4]{{d}^{2}}\ln \left ({ \left ( d\tan \left ( fx+e \right ) +\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) \left ( d\tan \left ( fx+e \right ) -\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) ^{-1}} \right ) }-{\frac{a\sqrt{2}}{2\,f{d}^{4}}\sqrt [4]{{d}^{2}}\arctan \left ({\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }+{\frac{a\sqrt{2}}{2\,f{d}^{4}}\sqrt [4]{{d}^{2}}\arctan \left ( -{\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ) }+{\frac{a\sqrt{2}}{4\,f{d}^{3}}\ln \left ({ \left ( d\tan \left ( fx+e \right ) -\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) \left ( d\tan \left ( fx+e \right ) +\sqrt [4]{{d}^{2}}\sqrt{d\tan \left ( fx+e \right ) }\sqrt{2}+\sqrt{{d}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}}+{\frac{a\sqrt{2}}{2\,f{d}^{3}}\arctan \left ({\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}}-{\frac{a\sqrt{2}}{2\,f{d}^{3}}\arctan \left ( -{\sqrt{2}\sqrt{d\tan \left ( fx+e \right ) }{\frac{1}{\sqrt [4]{{d}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{d}^{2}}}}}-{\frac{2\,a}{3\,{d}^{2}f} \left ( d\tan \left ( fx+e \right ) \right ) ^{-{\frac{3}{2}}}}-{\frac{2\,a}{5\,df} \left ( d\tan \left ( fx+e \right ) \right ) ^{-{\frac{5}{2}}}}+2\,{\frac{a}{f{d}^{3}\sqrt{d\tan \left ( fx+e \right ) }}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A] time = 1.87575, size = 668, normalized size = 5.52 \begin{align*} \left [\frac{15 \, \sqrt{2} a \sqrt{d} \log \left (\frac{\tan \left (f x + e\right )^{2} - \frac{2 \, \sqrt{2} \sqrt{d \tan \left (f x + e\right )}{\left (\tan \left (f x + e\right ) + 1\right )}}{\sqrt{d}} + 4 \, \tan \left (f x + e\right ) + 1}{\tan \left (f x + e\right )^{2} + 1}\right ) \tan \left (f x + e\right )^{3} + 4 \,{\left (15 \, a \tan \left (f x + e\right )^{2} - 5 \, a \tan \left (f x + e\right ) - 3 \, a\right )} \sqrt{d \tan \left (f x + e\right )}}{30 \, d^{4} f \tan \left (f x + e\right )^{3}}, \frac{15 \, \sqrt{2} a d \sqrt{-\frac{1}{d}} \arctan \left (\frac{\sqrt{2} \sqrt{d \tan \left (f x + e\right )} \sqrt{-\frac{1}{d}}{\left (\tan \left (f x + e\right ) + 1\right )}}{2 \, \tan \left (f x + e\right )}\right ) \tan \left (f x + e\right )^{3} + 2 \,{\left (15 \, a \tan \left (f x + e\right )^{2} - 5 \, a \tan \left (f x + e\right ) - 3 \, a\right )} \sqrt{d \tan \left (f x + e\right )}}{15 \, d^{4} f \tan \left (f x + e\right )^{3}}\right ] \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} a \left (\int \frac{1}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{7}{2}}}\, dx + \int \frac{\tan{\left (e + f x \right )}}{\left (d \tan{\left (e + f x \right )}\right )^{\frac{7}{2}}}\, dx\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] time = 1.24128, size = 398, normalized size = 3.29 \begin{align*} -\frac{\sqrt{2}{\left (a d \sqrt{{\left | d \right |}} - a{\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 )}{2 \, d^{5} f} - \frac{\sqrt{2}{\left (a d \sqrt{{\left | d \right |}} - a{\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 )}{2 \, d^{5} f} - \frac{\sqrt{2}{\left (a d \sqrt{{\left | d \right |}} + a{\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 )}{4 \, d^{5} f} + \frac{\sqrt{2}{\left (a d \sqrt{{\left | d \right |}} + a{\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 )}{4 \, d^{5} f} + \frac{2 \,{\left (15 \, a d^{2} \tan \left (f x + e\right )^{2} - 5 \, a d^{2} \tan \left (f x + e\right ) - 3 \, a d^{2}\right )}}{15 \, \sqrt{d \tan \left (f x + e\right )} d^{5} f \tan \left (f x + e\right )^{2}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]