4.1 Problem number 19

$$\int \frac{{\tan }^7(d+ex)}{{(a+b\tan (d+ex)+c{\tan }^2(d+ex))}^{3/2}}dx$$

Optimal antiderivative $$\frac{3b\mathrm{arctanh}\left(\frac{b+2c\tan (ex+d)}{2\sqrt{c}\sqrt{a+b\tan (ex+d)+c({\tan }^2(ex+d))}}\right)}{2c^{\frac{5}{2}}e}-\frac{5b(-12ac+7b^2)\mathrm{arctanh}\left(\frac{b+2c\tan (ex+d)}{2\sqrt{c}\sqrt{a+b\tan (ex+d)+c({\tan }^2(ex+d))}}\right)}{16c^{\frac{9}{2}}e}+\frac{\mathrm{arctanh}\left(\frac{(b^2-(a-c)(a-c-\sqrt{a^2-2ac+b^2+c^2})-b(2a-2c+\sqrt{a^2-2ac+b^2+c^2})\tan (ex+d))\sqrt{2}}{2\sqrt{2a-2c+\sqrt{a^2-2ac+b^2+c^2}}\sqrt{a^2-b^2-2ac+c^2-(a-c)\sqrt{a^2-2ac+b^2+c^2}}\sqrt{a+b\tan (ex+d)+c({\tan }^2(ex+d))}}\right)\sqrt{2a-2c+\sqrt{a^2-2ac+b^2+c^2}}\sqrt{a^2-b^2-2ac+c^2-(a-c)\sqrt{a^2-2ac+b^2+c^2}}\sqrt{2}}{2{(a^2-2ac+b^2+c^2)}^{\frac{3}{2}}e}-\frac{\mathrm{arctanh}\left(\frac{(b^2-(a-c)(a-c+\sqrt{a^2-2ac+b^2+c^2})-b(2a-2c-\sqrt{a^2-2ac+b^2+c^2})\tan (ex+d))\sqrt{2}}{2\sqrt{2a-2c-\sqrt{a^2-2ac+b^2+c^2}}\sqrt{a^2-b^2-2ac+c^2+(a-c)\sqrt{a^2-2ac+b^2+c^2}}\sqrt{a+b\tan (ex+d)+c({\tan }^2(ex+d))}}\right)\sqrt{2a-2c-\sqrt{a^2-2ac+b^2+c^2}}\sqrt{a^2-b^2-2ac+c^2+(a-c)\sqrt{a^2-2ac+b^2+c^2}}\sqrt{2}}{2{(a^2-2ac+b^2+c^2)}^{\frac{3}{2}}e}+\frac{(-16ac+7b^2)\sqrt{a+b\tan (ex+d)+c({\tan }^2(ex+d))}({\tan }^2(ex+d))}{3c^2(-4ac+b^2)e}-\frac{2b\sqrt{a+b\tan (ex+d)+c({\tan }^2(ex+d))}({\tan }^3(ex+d))}{c(-4ac+b^2)e}+\frac{4a+2b\tan (ex+d)}{(-4ac+b^2)e\sqrt{a+b\tan (ex+d)+c({\tan }^2(ex+d))}}-\frac{2({\tan }^2(ex+d))(2a+b\tan (ex+d))}{(-4ac+b^2)e\sqrt{a+b\tan (ex+d)+c({\tan }^2(ex+d))}}+\frac{2({\tan }^4(ex+d))(2a+b\tan (ex+d))}{(-4ac+b^2)e\sqrt{a+b\tan (ex+d)+c({\tan }^2(ex+d))}}-\frac{\sqrt{a+b\tan (ex+d)+c({\tan }^2(ex+d))}(3b^2-8ac-2bc\tan (ex+d))}{c^2(-4ac+b^2)e}-\frac{2(a(b^2-2(a-c)c)+bc(a+c)\tan (ex+d))}{(b^2+{(a-c)}^2)(-4ac+b^2)e\sqrt{a+b\tan (ex+d)+c({\tan }^2(ex+d))}}+\frac{\sqrt{a+b\tan (ex+d)+c({\tan }^2(ex+d))}(105b^4-460a{b}^{2}c+256a^2{c}^2-2bc(-116ac+35b^2)\tan (ex+d))}{24c^4(-4ac+b^2)e}$$

command

int(tan(e*x+d)^7/(a+b*tan(e*x+d)+c*tan(e*x+d)^2)^(3/2),x)

Maple 2022.1 output

hanged

Maple 2021.1 output

$$\text{output too large to display}$$