Optimal. Leaf size=234 \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} b^{7/2} d}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right )}{\sqrt{2} b^{7/2} d}+\frac{2}{b^3 d \sqrt{b \tan (c+d x)}}+\frac{\log \left (\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} b^{7/2} d}-\frac{\log \left (\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} b^{7/2} d}-\frac{2}{5 b d (b \tan (c+d x))^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.180953, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 9, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.75, Rules used = {3474, 3476, 329, 297, 1162, 617, 204, 1165, 628} \[ -\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} b^{7/2} d}+\frac{\tan ^{-1}\left (\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}+1\right )}{\sqrt{2} b^{7/2} d}+\frac{2}{b^3 d \sqrt{b \tan (c+d x)}}+\frac{\log \left (\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} b^{7/2} d}-\frac{\log \left (\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}+\sqrt{b}\right )}{2 \sqrt{2} b^{7/2} d}-\frac{2}{5 b d (b \tan (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 3474
Rule 3476
Rule 329
Rule 297
Rule 1162
Rule 617
Rule 204
Rule 1165
Rule 628
Rubi steps
\begin{align*} \int \frac{1}{(b \tan (c+d x))^{7/2}} \, dx &=-\frac{2}{5 b d (b \tan (c+d x))^{5/2}}-\frac{\int \frac{1}{(b \tan (c+d x))^{3/2}} \, dx}{b^2}\\ &=-\frac{2}{5 b d (b \tan (c+d x))^{5/2}}+\frac{2}{b^3 d \sqrt{b \tan (c+d x)}}+\frac{\int \sqrt{b \tan (c+d x)} \, dx}{b^4}\\ &=-\frac{2}{5 b d (b \tan (c+d x))^{5/2}}+\frac{2}{b^3 d \sqrt{b \tan (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{x}}{b^2+x^2} \, dx,x,b \tan (c+d x)\right )}{b^3 d}\\ &=-\frac{2}{5 b d (b \tan (c+d x))^{5/2}}+\frac{2}{b^3 d \sqrt{b \tan (c+d x)}}+\frac{2 \operatorname{Subst}\left (\int \frac{x^2}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{b^3 d}\\ &=-\frac{2}{5 b d (b \tan (c+d x))^{5/2}}+\frac{2}{b^3 d \sqrt{b \tan (c+d x)}}-\frac{\operatorname{Subst}\left (\int \frac{b-x^2}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{b^3 d}+\frac{\operatorname{Subst}\left (\int \frac{b+x^2}{b^2+x^4} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{b^3 d}\\ &=-\frac{2}{5 b d (b \tan (c+d x))^{5/2}}+\frac{2}{b^3 d \sqrt{b \tan (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{b}+2 x}{-b-\sqrt{2} \sqrt{b} x-x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} b^{7/2} d}+\frac{\operatorname{Subst}\left (\int \frac{\sqrt{2} \sqrt{b}-2 x}{-b+\sqrt{2} \sqrt{b} x-x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} b^{7/2} d}+\frac{\operatorname{Subst}\left (\int \frac{1}{b-\sqrt{2} \sqrt{b} x+x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 b^3 d}+\frac{\operatorname{Subst}\left (\int \frac{1}{b+\sqrt{2} \sqrt{b} x+x^2} \, dx,x,\sqrt{b \tan (c+d x)}\right )}{2 b^3 d}\\ &=\frac{\log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} b^{7/2} d}-\frac{\log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} b^{7/2} d}-\frac{2}{5 b d (b \tan (c+d x))^{5/2}}+\frac{2}{b^3 d \sqrt{b \tan (c+d x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} b^{7/2} d}-\frac{\operatorname{Subst}\left (\int \frac{1}{-1-x^2} \, dx,x,1+\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} b^{7/2} d}\\ &=-\frac{\tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} b^{7/2} d}+\frac{\tan ^{-1}\left (1+\frac{\sqrt{2} \sqrt{b \tan (c+d x)}}{\sqrt{b}}\right )}{\sqrt{2} b^{7/2} d}+\frac{\log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)-\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} b^{7/2} d}-\frac{\log \left (\sqrt{b}+\sqrt{b} \tan (c+d x)+\sqrt{2} \sqrt{b \tan (c+d x)}\right )}{2 \sqrt{2} b^{7/2} d}-\frac{2}{5 b d (b \tan (c+d x))^{5/2}}+\frac{2}{b^3 d \sqrt{b \tan (c+d x)}}\\ \end{align*}
Mathematica [C] time = 0.102988, size = 40, normalized size = 0.17 \[ -\frac{2 \, _2F_1\left (-\frac{5}{4},1;-\frac{1}{4};-\tan ^2(c+d x)\right )}{5 b d (b \tan (c+d x))^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A] time = 0.019, size = 202, normalized size = 0.9 \begin{align*}{\frac{\sqrt{2}}{4\,d{b}^{3}}\ln \left ({ \left ( b\tan \left ( dx+c \right ) -\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( dx+c \right ) }\sqrt{2}+\sqrt{{b}^{2}} \right ) \left ( b\tan \left ( dx+c \right ) +\sqrt [4]{{b}^{2}}\sqrt{b\tan \left ( dx+c \right ) }\sqrt{2}+\sqrt{{b}^{2}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{b}^{2}}}}}+{\frac{\sqrt{2}}{2\,d{b}^{3}}\arctan \left ({\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{b}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{b}^{2}}}}}-{\frac{\sqrt{2}}{2\,d{b}^{3}}\arctan \left ( -{\sqrt{2}\sqrt{b\tan \left ( dx+c \right ) }{\frac{1}{\sqrt [4]{{b}^{2}}}}}+1 \right ){\frac{1}{\sqrt [4]{{b}^{2}}}}}-{\frac{2}{5\,bd} \left ( b\tan \left ( dx+c \right ) \right ) ^{-{\frac{5}{2}}}}+2\,{\frac{1}{d{b}^{3}\sqrt{b\tan \left ( dx+c \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 [B] time = 1.77854, size = 1985, normalized size = 8.48 \begin{align*} \text{result too large to display} \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*} \int \frac{1}{\left (b \tan{\left (c + d x \right )}\right )^{\frac{7}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [A] time = 1.50618, size = 313, normalized size = 1.34 \begin{align*} \frac{1}{20} \, b{\left (\frac{10 \, \sqrt{2}{\left | b \right |}^{\frac{3}{2}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | b \right |}} + 2 \, \sqrt{b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt{{\left | b \right |}}}\right )}{b^{6} d} + \frac{10 \, \sqrt{2}{\left | b \right |}^{\frac{3}{2}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{{\left | b \right |}} - 2 \, \sqrt{b \tan \left (d x + c\right )}\right )}}{2 \, \sqrt{{\left | b \right |}}}\right )}{b^{6} d} - \frac{5 \, \sqrt{2}{\left | b \right |}^{\frac{3}{2}} \log \left (b \tan \left (d x + c\right ) + \sqrt{2} \sqrt{b \tan \left (d x + c\right )} \sqrt{{\left | b \right |}} +{\left | b \right |}\right )}{b^{6} d} + \frac{5 \, \sqrt{2}{\left | b \right |}^{\frac{3}{2}} \log \left (b \tan \left (d x + c\right ) - \sqrt{2} \sqrt{b \tan \left (d x + c\right )} \sqrt{{\left | b \right |}} +{\left | b \right |}\right )}{b^{6} d} + \frac{8 \,{\left (5 \, b^{2} \tan \left (d x + c\right )^{2} - b^{2}\right )}}{\sqrt{b \tan \left (d x + c\right )} b^{6} d \tan \left (d x + c\right )^{2}}\right )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]