Optimal. Leaf size=97 \[ \frac{\frac{1}{a^2}-\frac{1}{b^2}}{f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{a^{5/2} f}+\frac{(a+b)^2}{3 a b^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.166095, antiderivative size = 97, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2, Rules used = {4139, 446, 87, 63, 208} \[ \frac{\frac{1}{a^2}-\frac{1}{b^2}}{f \sqrt{a+b \sec ^2(e+f x)}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{a^{5/2} f}+\frac{(a+b)^2}{3 a b^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 4139
Rule 446
Rule 87
Rule 63
Rule 208
Rubi steps
\begin{align*} \int \frac{\tan ^5(e+f x)}{\left (a+b \sec ^2(e+f x)\right )^{5/2}} \, dx &=\frac{\operatorname{Subst}\left (\int \frac{\left (-1+x^2\right )^2}{x \left (a+b x^2\right )^{5/2}} \, dx,x,\sec (e+f x)\right )}{f}\\ &=\frac{\operatorname{Subst}\left (\int \frac{(-1+x)^2}{x (a+b x)^{5/2}} \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=\frac{\operatorname{Subst}\left (\int \left (-\frac{(a+b)^2}{a b (a+b x)^{5/2}}+\frac{a^2-b^2}{a^2 b (a+b x)^{3/2}}+\frac{1}{a^2 x \sqrt{a+b x}}\right ) \, dx,x,\sec ^2(e+f x)\right )}{2 f}\\ &=\frac{(a+b)^2}{3 a b^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{\frac{1}{a^2}-\frac{1}{b^2}}{f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{x \sqrt{a+b x}} \, dx,x,\sec ^2(e+f x)\right )}{2 a^2 f}\\ &=\frac{(a+b)^2}{3 a b^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{\frac{1}{a^2}-\frac{1}{b^2}}{f \sqrt{a+b \sec ^2(e+f x)}}+\frac{\operatorname{Subst}\left (\int \frac{1}{-\frac{a}{b}+\frac{x^2}{b}} \, dx,x,\sqrt{a+b \sec ^2(e+f x)}\right )}{a^2 b f}\\ &=-\frac{\tanh ^{-1}\left (\frac{\sqrt{a+b \sec ^2(e+f x)}}{\sqrt{a}}\right )}{a^{5/2} f}+\frac{(a+b)^2}{3 a b^2 f \left (a+b \sec ^2(e+f x)\right )^{3/2}}+\frac{\frac{1}{a^2}-\frac{1}{b^2}}{f \sqrt{a+b \sec ^2(e+f x)}}\\ \end{align*}
Mathematica [C] time = 8.61337, size = 187, normalized size = 1.93 \[ \frac{4 (a+b) \tan ^6(e+f x) F_1\left (3;\frac{1}{2},\frac{5}{2};4;\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right )}{3 f \left (a+b \sec ^2(e+f x)\right )^{5/2} \left (\sin ^2(e+f x) \left (5 a F_1\left (4;\frac{1}{2},\frac{7}{2};5;\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right )+(a+b) F_1\left (4;\frac{3}{2},\frac{5}{2};5;\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right )\right )+8 (a+b) F_1\left (3;\frac{1}{2},\frac{5}{2};4;\sin ^2(e+f x),\frac{a \sin ^2(e+f x)}{a+b}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
Maple [B] time = 2.303, size = 10947, normalized size = 112.9 \begin{align*} \text{output too large to display} \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 = 3.29382, size = 1332, normalized size = 13.73 \begin{align*} \left [\frac{3 \,{\left (a^{2} b^{2} \cos \left (f x + e\right )^{4} + 2 \, a b^{3} \cos \left (f x + e\right )^{2} + b^{4}\right )} \sqrt{a} \log \left (128 \, a^{4} \cos \left (f x + e\right )^{8} + 256 \, a^{3} b \cos \left (f x + e\right )^{6} + 160 \, a^{2} b^{2} \cos \left (f x + e\right )^{4} + 32 \, a b^{3} \cos \left (f x + e\right )^{2} + b^{4} - 8 \,{\left (16 \, a^{3} \cos \left (f x + e\right )^{8} + 24 \, a^{2} b \cos \left (f x + e\right )^{6} + 10 \, a b^{2} \cos \left (f x + e\right )^{4} + b^{3} \cos \left (f x + e\right )^{2}\right )} \sqrt{a} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}\right ) - 8 \,{\left (2 \,{\left (a^{4} - a^{3} b - 2 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 3 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{24 \,{\left (a^{5} b^{2} f \cos \left (f x + e\right )^{4} + 2 \, a^{4} b^{3} f \cos \left (f x + e\right )^{2} + a^{3} b^{4} f\right )}}, \frac{3 \,{\left (a^{2} b^{2} \cos \left (f x + e\right )^{4} + 2 \, a b^{3} \cos \left (f x + e\right )^{2} + b^{4}\right )} \sqrt{-a} \arctan \left (\frac{{\left (8 \, a^{2} \cos \left (f x + e\right )^{4} + 8 \, a b \cos \left (f x + e\right )^{2} + b^{2}\right )} \sqrt{-a} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{4 \,{\left (2 \, a^{3} \cos \left (f x + e\right )^{4} + 3 \, a^{2} b \cos \left (f x + e\right )^{2} + a b^{2}\right )}}\right ) - 4 \,{\left (2 \,{\left (a^{4} - a^{3} b - 2 \, a^{2} b^{2}\right )} \cos \left (f x + e\right )^{4} + 3 \,{\left (a^{3} b - a b^{3}\right )} \cos \left (f x + e\right )^{2}\right )} \sqrt{\frac{a \cos \left (f x + e\right )^{2} + b}{\cos \left (f x + e\right )^{2}}}}{12 \,{\left (a^{5} b^{2} f \cos \left (f x + e\right )^{4} + 2 \, a^{4} b^{3} f \cos \left (f x + e\right )^{2} + a^{3} b^{4} f\right )}}\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*} \int \frac{\tan ^{5}{\left (e + f x \right )}}{\left (a + b \sec ^{2}{\left (e + f x \right )}\right )^{\frac{5}{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\tan \left (f x + e\right )^{5}}{{\left (b \sec \left (f x + e\right )^{2} + a\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]