3.2 Fricas 1.3.6 and Fricas 1.3.5

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3.2.1 Test number 14

Test folder name

test_cases/1_Algebraic_functions/1.1_Binomial_products/1.1.1_Linear/1.1.1.3-a+b_x-^m-c+d_x-^n-e+f_x-^p

3.2.1.1 Problem number 625

$$\int \frac{(a+bx{)}^{3/2}(c+dx{)}^{5/2}}{x^7}dx$$ Optimal antiderivative

$$-\frac{\sqrt{a+bx}(c+dx{)}^{3/2}(5ad+7bc)(bc-ad{)}^3}{768a^3{c}^3{x}^2}+\frac{\sqrt{a+bx}(c+dx{)}^{5/2}(5ad+7bc)(bc-ad{)}^2}{960a^2{c}^3{x}^3}+\frac{\sqrt{a+bx}\sqrt{c+dx}(5ad+7bc)(bc-ad{)}^4}{512a^4{c}^{3}x}-\frac{(5ad+7bc)(bc-ad{)}^5{\tanh }^{-1}\left(\frac{\sqrt{c}\sqrt{a+bx}}{\sqrt{a}\sqrt{c+dx}}\right)}{512a^{9/2}{c}^{7/2}}+\frac{\sqrt{a+bx}(c+dx{)}^{7/2}(5ad+7bc)(bc-ad)}{160a{c}^3{x}^4}+\frac{(a+bx{)}^{3/2}(c+dx{)}^{7/2}(5ad+7bc)}{60a{c}^2{x}^5}-\frac{(a+bx{)}^{5/2}(c+dx{)}^{7/2}}{6ac{x}^6}$$ command

integrate((b*x+a)^(3/2)*(d*x+c)^(5/2)/x^7,x, algorithm="fricas")

Fricas 1.3.6 output

$$\text{Exception raised: TypeError}$$ Fricas 1.3.5 output

$$[-\frac{15(7b^6{c}^6-30a{b}^5{c}^{5}d+45a^2{b}^4{c}^4{d}^2-20a^3{b}^3{c}^3{d}^3-15a^4{b}^2{c}^2{d}^4+18a^{5}bc{d}^5-5a^6{d}^6)\sqrt{ac}{x}^6\log \left(\frac{8a^2{c}^2+(b^2{c}^2+6abcd+a^2{d}^2)x^2+4(2ac+(bc+ad)x)\sqrt{ac}\sqrt{bx+a}\sqrt{dx+c}+8(ab{c}^2+a^{2}cd)x}{x^2}\right)+4(1280a^6{c}^6-(105a{b}^5{c}^6-415a^2{b}^4{c}^{5}d+546a^3{b}^3{c}^4{d}^2-150a^4{b}^2{c}^3{d}^3+245a^{5}b{c}^2{d}^4-75a^{6}c{d}^5)x^5+2(35a^2{b}^4{c}^6-136a^3{b}^3{c}^{5}d+174a^4{b}^2{c}^4{d}^2+80a^{5}b{c}^3{d}^3-25a^6{c}^2{d}^4)x^4-8(7a^3{b}^3{c}^6-27a^4{b}^2{c}^{5}d-423a^{5}b{c}^4{d}^2-5a^6{c}^3{d}^3)x^3+16(3a^4{b}^2{c}^6+278a^{5}b{c}^{5}d+135a^6{c}^4{d}^2)x^2+128(13a^{5}b{c}^6+25a^6{c}^{5}d)x)\sqrt{bx+a}\sqrt{dx+c}}{30720a^5{c}^4{x}^6},\frac{15(7b^6{c}^6-30a{b}^5{c}^{5}d+45a^2{b}^4{c}^4{d}^2-20a^3{b}^3{c}^3{d}^3-15a^4{b}^2{c}^2{d}^4+18a^{5}bc{d}^5-5a^6{d}^6)\sqrt{-ac}{x}^6\arctan \left(\frac{(2ac+(bc+ad)x)\sqrt{-ac}\sqrt{bx+a}\sqrt{dx+c}}{2(abcd{x}^2+a^2{c}^2+(ab{c}^2+a^{2}cd)x)}\right)-2(1280a^6{c}^6-(105a{b}^5{c}^6-415a^2{b}^4{c}^{5}d+546a^3{b}^3{c}^4{d}^2-150a^4{b}^2{c}^3{d}^3+245a^{5}b{c}^2{d}^4-75a^{6}c{d}^5)x^5+2(35a^2{b}^4{c}^6-136a^3{b}^3{c}^{5}d+174a^4{b}^2{c}^4{d}^2+80a^{5}b{c}^3{d}^3-25a^6{c}^2{d}^4)x^4-8(7a^3{b}^3{c}^6-27a^4{b}^2{c}^{5}d-423a^{5}b{c}^4{d}^2-5a^6{c}^3{d}^3)x^3+16(3a^4{b}^2{c}^6+278a^{5}b{c}^{5}d+135a^6{c}^4{d}^2)x^2+128(13a^{5}b{c}^6+25a^6{c}^{5}d)x)\sqrt{bx+a}\sqrt{dx+c}}{15360a^5{c}^4{x}^6}]$$

3.2.2 Test number 34

Test folder name

test_cases/1_Algebraic_functions/1.2_Trinomial_products/1.2.1_Quadratic/1.2.1.3-d+e_x-^m-f+g_x-a+b_x+c_x^2-^p

3.2.2.1 Problem number 703

$$\int \frac{(A+Bx){(a^2+2abx+b^2{x}^2)}^{5/2}}{x^{14}}dx$$ Optimal antiderivative

$$-\frac{a^4\sqrt{a^2+2abx+b^2{x}^2}(aB+5Ab)}{12x^{12}(a+bx)}-\frac{5a^{3}b\sqrt{a^2+2abx+b^2{x}^2}(aB+2Ab)}{11x^{11}(a+bx)}-\frac{a^2{b}^2\sqrt{a^2+2abx+b^2{x}^2}(aB+Ab)}{x^{10}(a+bx)}-\frac{5a{b}^3\sqrt{a^2+2abx+b^2{x}^2}(2aB+Ab)}{9x^9(a+bx)}-\frac{b^4\sqrt{a^2+2abx+b^2{x}^2}(5aB+Ab)}{8x^8(a+bx)}-\frac{a^{5}A\sqrt{a^2+2abx+b^2{x}^2}}{13x^{13}(a+bx)}-\frac{b^{5}B\sqrt{a^2+2abx+b^2{x}^2}}{7x^7(a+bx)}$$ command

integrate((B*x+A)*(b^2*x^2+2*a*b*x+a^2)^(5/2)/x^14,x, algorithm="fricas")

Fricas 1.3.6 output

$$\text{Timed out}$$ Fricas 1.3.5 output

$$-\frac{10296B{b}^5{x}^6+5544A{a}^5+9009(5Ba{b}^4+A{b}^5)x^5+40040(2B{a}^2{b}^3+Aa{b}^4)x^4+72072(B{a}^3{b}^2+A{a}^2{b}^3)x^3+32760(B{a}^{4}b+2A{a}^3{b}^2)x^2+6006(B{a}^5+5A{a}^{4}b)x}{72072x^{13}}$$

3.2.3 Test number 94

Test folder name

test_cases/4_Trig_functions/4.2_Cosine/4.2.4.2-a+b_cos-^m-c+d_cos-^n-A+B_cos+C_cos^2-

3.2.3.1 Problem number 597

$$\int \frac{(1-{\cos }^2(c+dx))\sec (c+dx)}{a+b\cos (c+dx)}dx$$ Optimal antiderivative

$$\frac{2\sqrt{a-b}\sqrt{a+b}{\tan }^{-1}\left(\frac{\sqrt{a-b}\tan (\frac{1}{2}(c+dx))}{\sqrt{a+b}}\right)}{abd}+\frac{{\tanh }^{-1}(\sin (c+dx))}{ad}-\frac{x}{b}$$ command

integrate((1-cos(d*x+c)^2)*sec(d*x+c)/(a+b*cos(d*x+c)),x, algorithm="fricas")

Fricas 1.3.6 output

$$\text{Exception raised: TypeError}$$ Fricas 1.3.5 output

$$[-\frac{2adx-b\log (\sin (dx+c)+1)+b\log (-\sin (dx+c)+1)-\sqrt{-a^2+b^2}\log \left(\frac{2ab\cos (dx+c)+(2a^2-b^2)\cos {(dx+c)}^2-2\sqrt{-a^2+b^2}(a\cos (dx+c)+b)\sin (dx+c)-a^2+2b^2}{b^2\cos {(dx+c)}^2+2ab\cos (dx+c)+a^2}\right)}{2abd},-\frac{2adx-b\log (\sin (dx+c)+1)+b\log (-\sin (dx+c)+1)-2\sqrt{a^2-b^2}\arctan (-\frac{a\cos (dx+c)+b}{\sqrt{a^2-b^2}\sin (dx+c)})}{2abd}]$$

3.2.4 Test number 139

Test folder name

test_cases/4_Trig_functions/4.7_Miscellaneous/4.7.5_x^m_trig-a+b_log-c_x^n-^p

3.2.4.1 Problem number 48

$$\int \sin (a+\frac{1}{2}i\log \left(c{x}^2\right))dx$$ Optimal antiderivative

$$\frac{i{e}^{-ia}c{x}^3}{4\sqrt{c{x}^2}}-\frac{i{e}^{ia}x\log (x)}{2\sqrt{c{x}^2}}$$ command

integrate(sin(a+1/2*I*log(c*x^2)),x, algorithm="fricas")

Fricas 1.3.6 output

$$\text{Exception raised: NotImplementedError}$$ Fricas 1.3.5 output

$$\frac{(ic{x}^2-2i{e}^{\left(2ia\right)}\log \left(x\right))e^{(-ia)}}{4\sqrt{c}}$$

3.2.4.2 Problem number 50

$$\int {\sin }^2(a+\frac{1}{4}i\log \left(c{x}^2\right))dx$$ Optimal antiderivative

$$-\frac{e^{-2ia}c{x}^3}{8\sqrt{c{x}^2}}-\frac{e^{2ia}x\log (x)}{4\sqrt{c{x}^2}}+\frac{x}{2}$$ command

integrate(sin(a+1/4*I*log(c*x^2))^2,x, algorithm="fricas")

Fricas 1.3.6 output

$$\text{Exception raised: NotImplementedError}$$ Fricas 1.3.5 output

$$\frac{(4x^2{e}^{\left(2ia\right)}-\frac{x{e}^{\left(4ia\right)}\log \left(\frac{(\sqrt{c{x}^2}(x^2+1)e^{\left(2ia\right)}+\frac{(c{x}^3-cx)e^{\left(2ia\right)}}{\sqrt{c}})e^{(-2ia)}}{8x^2}\right)}{\sqrt{c}}+\frac{x{e}^{\left(4ia\right)}\log \left(\frac{(\sqrt{c{x}^2}(x^2+1)e^{\left(2ia\right)}-\frac{(c{x}^3-cx)e^{\left(2ia\right)}}{\sqrt{c}})e^{(-2ia)}}{8x^2}\right)}{\sqrt{c}}-\sqrt{c{x}^2}(x^2-1))e^{(-2ia)}}{8x}$$

3.2.4.3 Problem number 52

$$\int {\sin }^3(a+\frac{1}{6}i\log \left(c{x}^2\right))dx$$ Optimal antiderivative

$$-\frac{i{e}^{-3ia}c{x}^3}{16\sqrt{c{x}^2}}+\frac{9}{32}i{e}^{-ia}x\sqrt[6]{c{x}^2}-\frac{9i{e}^{ia}x}{16\sqrt[6]{c{x}^2}}+\frac{i{e}^{3ia}x\log (x)}{8\sqrt{c{x}^2}}$$ command

integrate(sin(a+1/6*I*log(c*x^2))^3,x, algorithm="fricas")

Fricas 1.3.6 output

$$\text{Exception raised: NotImplementedError}$$ Fricas 1.3.5 output

$$-\frac{(2cx\sqrt{-\frac{e^{\left(6ia\right)}}{c}}{e}^{\left(3ia\right)}\log \left(\frac{(4\sqrt{c{x}^2}(x^2+1)e^{\left(3ia\right)}+(4ic{x}^3-4icx)\sqrt{-\frac{e^{\left(6ia\right)}}{c}})e^{(-3ia)}}{32x^2}\right)-2cx\sqrt{-\frac{e^{\left(6ia\right)}}{c}}{e}^{\left(3ia\right)}\log \left(\frac{(4\sqrt{c{x}^2}(x^2+1)e^{\left(3ia\right)}+(-4ic{x}^3+4icx)\sqrt{-\frac{e^{\left(6ia\right)}}{c}})e^{(-3ia)}}{32x^2}\right)-9i{\left(c{x}^2\right)}^{\frac{1}{6}}c{x}^2{e}^{\left(2ia\right)}+18i{\left(c{x}^2\right)}^{\frac{5}{6}}{e}^{\left(4ia\right)}-\sqrt{c{x}^2}(-2ic{x}^2+2ic))e^{(-3ia)}}{32cx}$$

3.2.4.4 Problem number 135

$$\int x^3\tan (a+i\log (x))dx$$ Optimal antiderivative

$$-i{e}^{2ia}{x}^2+i{e}^{4ia}\log (x^2+e^{2ia})+\frac{i{x}^4}{4}$$ command

integrate(x^3*tan(a+I*log(x)),x, algorithm="fricas")

Fricas 1.3.6 output

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(\frac{-i{x}^3{e}^{(2ia-2\log \left(x\right))}+i{x}^3}{e^{(2ia-2\log \left(x\right))}+1},x)$$ Fricas 1.3.5 output

$$\frac{1}{4}i{x}^4-i{x}^2{e}^{\left(2ia\right)}+i{e}^{\left(4ia\right)}\log (x^2+e^{\left(2ia\right)})$$

3.2.4.5 Problem number 136

$$\int x^2\tan (a+i\log (x))dx$$ Optimal antiderivative

$$-2i{e}^{2ia}x+2i{e}^{3ia}{\tan }^{-1}\left(e^{-ia}x\right)+\frac{i{x}^3}{3}$$ command

integrate(x^2*tan(a+I*log(x)),x, algorithm="fricas")

Fricas 1.3.6 output

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(\frac{-i{x}^2{e}^{(2ia-2\log \left(x\right))}+i{x}^2}{e^{(2ia-2\log \left(x\right))}+1},x)$$ Fricas 1.3.5 output

$$\frac{1}{3}i{x}^3-2ix{e}^{\left(2ia\right)}-e^{\left(3ia\right)}\log (x+i{e}^{\left(ia\right)})+e^{\left(3ia\right)}\log (x-i{e}^{\left(ia\right)})$$

3.2.4.6 Problem number 137

$$\int x\tan (a+i\log (x))dx$$ Optimal antiderivative

$$\frac{i{x}^2}{2}-i{e}^{2ia}\log (x^2+e^{2ia})$$ command

integrate(x*tan(a+I*log(x)),x, algorithm="fricas")

Fricas 1.3.6 output

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(\frac{-ix{e}^{(2ia-2\log \left(x\right))}+ix}{e^{(2ia-2\log \left(x\right))}+1},x)$$ Fricas 1.3.5 output

$$\frac{1}{2}i{x}^2-i{e}^{\left(2ia\right)}\log (x^2+e^{\left(2ia\right)})$$

3.2.4.7 Problem number 138

$$\int \tan (a+i\log (x))dx$$ Optimal antiderivative

$$ix-2i{e}^{ia}{\tan }^{-1}\left(e^{-ia}x\right)$$ command

integrate(tan(a+I*log(x)),x, algorithm="fricas")

Fricas 1.3.6 output

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(\frac{-i{e}^{(2ia-2\log \left(x\right))}+i}{e^{(2ia-2\log \left(x\right))}+1},x)$$ Fricas 1.3.5 output

$$e^{\left(ia\right)}\log (x+i{e}^{\left(ia\right)})-e^{\left(ia\right)}\log (x-i{e}^{\left(ia\right)})+ix$$

3.2.4.8 Problem number 140

$$\int \frac{\tan (a+i\log (x))}{x^2}dx$$ Optimal antiderivative

$$2i{e}^{-ia}{\tan }^{-1}\left(e^{-ia}x\right)+\frac{i}{x}$$ command

integrate(tan(a+I*log(x))/x^2,x, algorithm="fricas")

Fricas 1.3.6 output

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(\frac{-i{e}^{(2ia-2\log \left(x\right))}+i}{x^2{e}^{(2ia-2\log \left(x\right))}+x^2},x)$$ Fricas 1.3.5 output

$$-\frac{(x\log (x+i{e}^{\left(ia\right)})-x\log (x-i{e}^{\left(ia\right)})-i{e}^{\left(ia\right)})e^{(-ia)}}{x}$$

3.2.4.9 Problem number 141

$$\int \frac{\tan (a+i\log (x))}{x^3}dx$$ Optimal antiderivative

$$\frac{i}{2x^2}-i{e}^{-2ia}\log (1+\frac{e^{2ia}}{x^2})$$ command

integrate(tan(a+I*log(x))/x^3,x, algorithm="fricas")

Fricas 1.3.6 output

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(\frac{-i{e}^{(2ia-2\log \left(x\right))}+i}{x^3{e}^{(2ia-2\log \left(x\right))}+x^3},x)$$ Fricas 1.3.5 output

$$\frac{(-2i{x}^2\log (x^2+e^{\left(2ia\right)})+4i{x}^2\log \left(x\right)+i{e}^{\left(2ia\right)})e^{(-2ia)}}{2x^2}$$

3.2.4.10 Problem number 142

$$\int \frac{\tan (a+i\log (x))}{x^4}dx$$ Optimal antiderivative

$$-\frac{2i{e}^{-2ia}}{x}-2i{e}^{-3ia}{\tan }^{-1}\left(e^{-ia}x\right)+\frac{i}{3x^3}$$ command

integrate(tan(a+I*log(x))/x^4,x, algorithm="fricas")

Fricas 1.3.6 output

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(\frac{-i{e}^{(2ia-2\log \left(x\right))}+i}{x^4{e}^{(2ia-2\log \left(x\right))}+x^4},x)$$ Fricas 1.3.5 output

$$\frac{(3x^3\log (x+i{e}^{\left(ia\right)})-3x^3\log (x-i{e}^{\left(ia\right)})-6i{x}^2{e}^{\left(ia\right)}+i{e}^{\left(3ia\right)})e^{(-3ia)}}{3x^3}$$

3.2.4.11 Problem number 143

$$\int x^3{\tan }^2(a+i\log (x))dx$$ Optimal antiderivative

$$2e^{2ia}{x}^2-\frac{2e^{6ia}}{x^2+e^{2ia}}-4e^{4ia}\log (x^2+e^{2ia})-\frac{x^4}{4}$$ command

integrate(x^3*tan(a+I*log(x))^2,x, algorithm="fricas")

Fricas 1.3.6 output

$$\frac{2x^4+(e^{(2ia-2\log \left(x\right))}+1)\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(-\frac{x^3{e}^{(2ia-2\log \left(x\right))}+9x^3}{e^{(2ia-2\log \left(x\right))}+1},x)}{e^{(2ia-2\log \left(x\right))}+1}$$ Fricas 1.3.5 output

$$-\frac{x^6-7x^4{e}^{\left(2ia\right)}-8x^2{e}^{\left(4ia\right)}+16(x^2{e}^{\left(4ia\right)}+e^{\left(6ia\right)})\log (x^2+e^{\left(2ia\right)})+8e^{\left(6ia\right)}}{4(x^2+e^{\left(2ia\right)})}$$

3.2.4.12 Problem number 144

$$\int x^2{\tan }^2(a+i\log (x))dx$$ Optimal antiderivative

$$-\frac{2e^{2ia}{x}^3}{x^2+e^{2ia}}+6e^{2ia}x-6e^{3ia}{\tan }^{-1}\left(e^{-ia}x\right)-\frac{x^3}{3}$$ command

integrate(x^2*tan(a+I*log(x))^2,x, algorithm="fricas")

Fricas 1.3.6 output

$$\frac{2x^3+(e^{(2ia-2\log \left(x\right))}+1)\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(-\frac{x^2{e}^{(2ia-2\log \left(x\right))}+7x^2}{e^{(2ia-2\log \left(x\right))}+1},x)}{e^{(2ia-2\log \left(x\right))}+1}$$ Fricas 1.3.5 output

$$-\frac{x^5-11x^3{e}^{\left(2ia\right)}-18x{e}^{\left(4ia\right)}-(-9i{x}^2{e}^{\left(3ia\right)}-9i{e}^{\left(5ia\right)})\log (x+i{e}^{\left(ia\right)})-(9i{x}^2{e}^{\left(3ia\right)}+9i{e}^{\left(5ia\right)})\log (x-i{e}^{\left(ia\right)})}{3(x^2+e^{\left(2ia\right)})}$$

3.2.4.13 Problem number 145

$$\int x{\tan }^2(a+i\log (x))dx$$ Optimal antiderivative

$$\frac{2e^{4ia}}{x^2+e^{2ia}}+2e^{2ia}\log (x^2+e^{2ia})-\frac{x^2}{2}$$ command

integrate(x*tan(a+I*log(x))^2,x, algorithm="fricas")

Fricas 1.3.6 output

$$\frac{2x^2+(e^{(2ia-2\log \left(x\right))}+1)\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(-\frac{x{e}^{(2ia-2\log \left(x\right))}+5x}{e^{(2ia-2\log \left(x\right))}+1},x)}{e^{(2ia-2\log \left(x\right))}+1}$$ Fricas 1.3.5 output

$$-\frac{x^4+x^2{e}^{\left(2ia\right)}-4(x^2{e}^{\left(2ia\right)}+e^{\left(4ia\right)})\log (x^2+e^{\left(2ia\right)})-4e^{\left(4ia\right)}}{2(x^2+e^{\left(2ia\right)})}$$

3.2.4.14 Problem number 146

$$\int {\tan }^2(a+i\log (x))dx$$ Optimal antiderivative

$$-\frac{2e^{2ia}x}{x^2+e^{2ia}}+2e^{ia}{\tan }^{-1}\left(e^{-ia}x\right)-x$$ command

integrate(tan(a+I*log(x))^2,x, algorithm="fricas")

Fricas 1.3.6 output

$$\frac{(e^{(2ia-2\log \left(x\right))}+1)\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(-\frac{e^{(2ia-2\log \left(x\right))}+3}{e^{(2ia-2\log \left(x\right))}+1},x)+2x}{e^{(2ia-2\log \left(x\right))}+1}$$ Fricas 1.3.5 output

$$-\frac{x^3+3x{e}^{\left(2ia\right)}-(i{x}^2{e}^{\left(ia\right)}+i{e}^{\left(3ia\right)})\log (x+i{e}^{\left(ia\right)})-(-i{x}^2{e}^{\left(ia\right)}-i{e}^{\left(3ia\right)})\log (x-i{e}^{\left(ia\right)})}{x^2+e^{\left(2ia\right)}}$$

3.2.4.15 Problem number 148

$$\int \frac{{\tan }^2(a+i\log (x))}{x^2}dx$$ Optimal antiderivative

$$\frac{3x}{x^2+e^{2ia}}+\frac{e^{2ia}}{x(x^2+e^{2ia})}+2e^{-ia}{\tan }^{-1}\left(e^{-ia}x\right)$$ command

integrate(tan(a+I*log(x))^2/x^2,x, algorithm="fricas")

Fricas 1.3.6 output

$$\frac{(x{e}^{(2ia-2\log \left(x\right))}+x)\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(-\frac{e^{(2ia-2\log \left(x\right))}-1}{x^2{e}^{(2ia-2\log \left(x\right))}+x^2},x)+2}{x{e}^{(2ia-2\log \left(x\right))}+x}$$ Fricas 1.3.5 output

$$\frac{3x^2{e}^{\left(ia\right)}+(i{x}^3+ix{e}^{\left(2ia\right)})\log (x+i{e}^{\left(ia\right)})+(-i{x}^3-ix{e}^{\left(2ia\right)})\log (x-i{e}^{\left(ia\right)})+e^{\left(3ia\right)}}{x^3{e}^{\left(ia\right)}+x{e}^{\left(3ia\right)}}$$

3.2.4.16 Problem number 149

$$\int \frac{{\tan }^2(a+i\log (x))}{x^3}dx$$ Optimal antiderivative

$$-\frac{2e^{-2ia}}{1+\frac{e^{2ia}}{x^2}}-2e^{-2ia}\log (1+\frac{e^{2ia}}{x^2})+\frac{1}{2x^2}$$ command

integrate(tan(a+I*log(x))^2/x^3,x, algorithm="fricas")

Fricas 1.3.6 output

$$\frac{(x^2{e}^{(2ia-2\log \left(x\right))}+x^2)\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(-\frac{e^{(2ia-2\log \left(x\right))}-3}{x^3{e}^{(2ia-2\log \left(x\right))}+x^3},x)+2}{x^2{e}^{(2ia-2\log \left(x\right))}+x^2}$$ Fricas 1.3.5 output

$$\frac{5x^2{e}^{\left(2ia\right)}-4(x^4+x^2{e}^{\left(2ia\right)})\log (x^2+e^{\left(2ia\right)})+8(x^4+x^2{e}^{\left(2ia\right)})\log \left(x\right)+e^{\left(4ia\right)}}{2(x^4{e}^{\left(2ia\right)}+x^2{e}^{\left(4ia\right)})}$$

3.2.4.17 Problem number 186

$$\int x^3\cot (a+i\log (x))dx$$ Optimal antiderivative

$$-i{e}^{2ia}{x}^2-i{e}^{4ia}\log (-x^2+e^{2ia})-\frac{i{x}^4}{4}$$ command

integrate(x^3*cot(a+I*log(x)),x, algorithm="fricas")

Fricas 1.3.6 output

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(\frac{i{x}^3{e}^{(2ia-2\log \left(x\right))}+i{x}^3}{e^{(2ia-2\log \left(x\right))}-1},x)$$ Fricas 1.3.5 output

$$-\frac{1}{4}i{x}^4-i{x}^2{e}^{\left(2ia\right)}-i{e}^{\left(4ia\right)}\log (x^2-e^{\left(2ia\right)})$$

3.2.4.18 Problem number 187

$$\int x^2\cot (a+i\log (x))dx$$ Optimal antiderivative

$$-2i{e}^{2ia}x+2i{e}^{3ia}{\tanh }^{-1}\left(e^{-ia}x\right)-\frac{i{x}^3}{3}$$ command

integrate(x^2*cot(a+I*log(x)),x, algorithm="fricas")

Fricas 1.3.6 output

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(\frac{i{x}^2{e}^{(2ia-2\log \left(x\right))}+i{x}^2}{e^{(2ia-2\log \left(x\right))}-1},x)$$ Fricas 1.3.5 output

$$-\frac{1}{3}i{x}^3-2ix{e}^{\left(2ia\right)}+i{e}^{\left(3ia\right)}\log (x+e^{\left(ia\right)})-i{e}^{\left(3ia\right)}\log (x-e^{\left(ia\right)})$$

3.2.4.19 Problem number 188

$$\int x\cot (a+i\log (x))dx$$ Optimal antiderivative

$$-i{e}^{2ia}\log (-x^2+e^{2ia})-\frac{i{x}^2}{2}$$ command

integrate(x*cot(a+I*log(x)),x, algorithm="fricas")

Fricas 1.3.6 output

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(\frac{ix{e}^{(2ia-2\log \left(x\right))}+ix}{e^{(2ia-2\log \left(x\right))}-1},x)$$ Fricas 1.3.5 output

$$-\frac{1}{2}i{x}^2-i{e}^{\left(2ia\right)}\log (x^2-e^{\left(2ia\right)})$$

3.2.4.20 Problem number 189

$$\int \cot (a+i\log (x))dx$$ Optimal antiderivative

$$2i{e}^{ia}{\tanh }^{-1}\left(e^{-ia}x\right)-ix$$ command

integrate(cot(a+I*log(x)),x, algorithm="fricas")

Fricas 1.3.6 output

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(\frac{i{e}^{(2ia-2\log \left(x\right))}+i}{e^{(2ia-2\log \left(x\right))}-1},x)$$ Fricas 1.3.5 output

$$i{e}^{\left(ia\right)}\log (x+e^{\left(ia\right)})-i{e}^{\left(ia\right)}\log (x-e^{\left(ia\right)})-ix$$

3.2.4.21 Problem number 191

$$\int \frac{\cot (a+i\log (x))}{x^2}dx$$ Optimal antiderivative

$$2i{e}^{-ia}{\tanh }^{-1}\left(e^{-ia}x\right)-\frac{i}{x}$$ command

integrate(cot(a+I*log(x))/x^2,x, algorithm="fricas")

Fricas 1.3.6 output

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(\frac{i{e}^{(2ia-2\log \left(x\right))}+i}{x^2{e}^{(2ia-2\log \left(x\right))}-x^2},x)$$ Fricas 1.3.5 output

$$\frac{(ix\log (x+e^{\left(ia\right)})-ix\log (x-e^{\left(ia\right)})-i{e}^{\left(ia\right)})e^{(-ia)}}{x}$$

3.2.4.22 Problem number 192

$$\int \frac{\cot (a+i\log (x))}{x^3}dx$$ Optimal antiderivative

$$-i{e}^{-2ia}\log (1-\frac{e^{2ia}}{x^2})-\frac{i}{2x^2}$$ command

integrate(cot(a+I*log(x))/x^3,x, algorithm="fricas")

Fricas 1.3.6 output

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(\frac{i{e}^{(2ia-2\log \left(x\right))}+i}{x^3{e}^{(2ia-2\log \left(x\right))}-x^3},x)$$ Fricas 1.3.5 output

$$\frac{(-2i{x}^2\log (x^2-e^{\left(2ia\right)})+4i{x}^2\log \left(x\right)-i{e}^{\left(2ia\right)})e^{(-2ia)}}{2x^2}$$

3.2.4.23 Problem number 193

$$\int \frac{\cot (a+i\log (x))}{x^4}dx$$ Optimal antiderivative

$$-\frac{2i{e}^{-2ia}}{x}+2i{e}^{-3ia}{\tanh }^{-1}\left(e^{-ia}x\right)-\frac{i}{3x^3}$$ command

integrate(cot(a+I*log(x))/x^4,x, algorithm="fricas")

Fricas 1.3.6 output

$$\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(\frac{i{e}^{(2ia-2\log \left(x\right))}+i}{x^4{e}^{(2ia-2\log \left(x\right))}-x^4},x)$$ Fricas 1.3.5 output

$$\frac{(3i{x}^3\log (x+e^{\left(ia\right)})-3i{x}^3\log (x-e^{\left(ia\right)})-6i{x}^2{e}^{\left(ia\right)}-i{e}^{\left(3ia\right)})e^{(-3ia)}}{3x^3}$$

3.2.4.24 Problem number 194

$$\int x^3{\cot }^2(a+i\log (x))dx$$ Optimal antiderivative

$$-2e^{2ia}{x}^2-\frac{2e^{6ia}}{-x^2+e^{2ia}}-4e^{4ia}\log (-x^2+e^{2ia})-\frac{x^4}{4}$$ command

integrate(x^3*cot(a+I*log(x))^2,x, algorithm="fricas")

Fricas 1.3.6 output

$$-\frac{2x^4-(e^{(2ia-2\log \left(x\right))}-1)\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(-\frac{x^3{e}^{(2ia-2\log \left(x\right))}-9x^3}{e^{(2ia-2\log \left(x\right))}-1},x)}{e^{(2ia-2\log \left(x\right))}-1}$$ Fricas 1.3.5 output

$$-\frac{x^6+7x^4{e}^{\left(2ia\right)}-8x^2{e}^{\left(4ia\right)}+16(x^2{e}^{\left(4ia\right)}-e^{\left(6ia\right)})\log (x^2-e^{\left(2ia\right)})-8e^{\left(6ia\right)}}{4(x^2-e^{\left(2ia\right)})}$$

3.2.4.25 Problem number 195

$$\int x^2{\cot }^2(a+i\log (x))dx$$ Optimal antiderivative

$$-\frac{2e^{2ia}{x}^3}{-x^2+e^{2ia}}-6e^{2ia}x+6e^{3ia}{\tanh }^{-1}\left(e^{-ia}x\right)-\frac{x^3}{3}$$ command

integrate(x^2*cot(a+I*log(x))^2,x, algorithm="fricas")

Fricas 1.3.6 output

$$-\frac{2x^3-(e^{(2ia-2\log \left(x\right))}-1)\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(-\frac{x^2{e}^{(2ia-2\log \left(x\right))}-7x^2}{e^{(2ia-2\log \left(x\right))}-1},x)}{e^{(2ia-2\log \left(x\right))}-1}$$ Fricas 1.3.5 output

$$-\frac{x^5+11x^3{e}^{\left(2ia\right)}-18x{e}^{\left(4ia\right)}-9(x^2{e}^{\left(3ia\right)}-e^{\left(5ia\right)})\log (x+e^{\left(ia\right)})+9(x^2{e}^{\left(3ia\right)}-e^{\left(5ia\right)})\log (x-e^{\left(ia\right)})}{3(x^2-e^{\left(2ia\right)})}$$

3.2.4.26 Problem number 196

$$\int x{\cot }^2(a+i\log (x))dx$$ Optimal antiderivative

$$-\frac{2e^{4ia}}{-x^2+e^{2ia}}-2e^{2ia}\log (-x^2+e^{2ia})-\frac{x^2}{2}$$ command

integrate(x*cot(a+I*log(x))^2,x, algorithm="fricas")

Fricas 1.3.6 output

$$-\frac{2x^2-(e^{(2ia-2\log \left(x\right))}-1)\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(-\frac{x{e}^{(2ia-2\log \left(x\right))}-5x}{e^{(2ia-2\log \left(x\right))}-1},x)}{e^{(2ia-2\log \left(x\right))}-1}$$ Fricas 1.3.5 output

$$-\frac{x^4-x^2{e}^{\left(2ia\right)}+4(x^2{e}^{\left(2ia\right)}-e^{\left(4ia\right)})\log (x^2-e^{\left(2ia\right)})-4e^{\left(4ia\right)}}{2(x^2-e^{\left(2ia\right)})}$$

3.2.4.27 Problem number 197

$$\int {\cot }^2(a+i\log (x))dx$$ Optimal antiderivative

$$-\frac{2e^{2ia}x}{-x^2+e^{2ia}}+2e^{ia}{\tanh }^{-1}\left(e^{-ia}x\right)-x$$ command

integrate(cot(a+I*log(x))^2,x, algorithm="fricas")

Fricas 1.3.6 output

$$\frac{(e^{(2ia-2\log \left(x\right))}-1)\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(-\frac{e^{(2ia-2\log \left(x\right))}-3}{e^{(2ia-2\log \left(x\right))}-1},x)-2x}{e^{(2ia-2\log \left(x\right))}-1}$$ Fricas 1.3.5 output

$$-\frac{x^3-3x{e}^{\left(2ia\right)}-(x^2{e}^{\left(ia\right)}-e^{\left(3ia\right)})\log (x+e^{\left(ia\right)})+(x^2{e}^{\left(ia\right)}-e^{\left(3ia\right)})\log (x-e^{\left(ia\right)})}{x^2-e^{\left(2ia\right)}}$$

3.2.4.28 Problem number 199

$$\int \frac{{\cot }^2(a+i\log (x))}{x^2}dx$$ Optimal antiderivative

$$-\frac{3x}{-x^2+e^{2ia}}+\frac{e^{2ia}}{x(-x^2+e^{2ia})}-2e^{-ia}{\tanh }^{-1}\left(e^{-ia}x\right)$$ command

integrate(cot(a+I*log(x))^2/x^2,x, algorithm="fricas")

Fricas 1.3.6 output

$$\frac{(x{e}^{(2ia-2\log \left(x\right))}-x)\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(-\frac{e^{(2ia-2\log \left(x\right))}+1}{x^2{e}^{(2ia-2\log \left(x\right))}-x^2},x)-2}{x{e}^{(2ia-2\log \left(x\right))}-x}$$ Fricas 1.3.5 output

$$\frac{3x^2{e}^{\left(ia\right)}-(x^3-x{e}^{\left(2ia\right)})\log (x+e^{\left(ia\right)})+(x^3-x{e}^{\left(2ia\right)})\log (x-e^{\left(ia\right)})-e^{\left(3ia\right)}}{x^3{e}^{\left(ia\right)}-x{e}^{\left(3ia\right)}}$$

3.2.4.29 Problem number 200

$$\int \frac{{\cot }^2(a+i\log (x))}{x^3}dx$$ Optimal antiderivative

$$\frac{2e^{-2ia}}{1-\frac{e^{2ia}}{x^2}}+2e^{-2ia}\log (1-\frac{e^{2ia}}{x^2})+\frac{1}{2x^2}$$ command

integrate(cot(a+I*log(x))^2/x^3,x, algorithm="fricas")

Fricas 1.3.6 output

$$\frac{(x^2{e}^{(2ia-2\log \left(x\right))}-x^2)\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{g}\mathrm{r}\mathrm{a}\mathrm{l}(-\frac{e^{(2ia-2\log \left(x\right))}+3}{x^3{e}^{(2ia-2\log \left(x\right))}-x^3},x)-2}{x^2{e}^{(2ia-2\log \left(x\right))}-x^2}$$ Fricas 1.3.5 output

$$\frac{5x^2{e}^{\left(2ia\right)}+4(x^4-x^2{e}^{\left(2ia\right)})\log (x^2-e^{\left(2ia\right)})-8(x^4-x^2{e}^{\left(2ia\right)})\log \left(x\right)-e^{\left(4ia\right)}}{2(x^4{e}^{\left(2ia\right)}-x^2{e}^{\left(4ia\right)})}$$

3.2.5 Test number 183

Test folder name

test_cases/6_Hyperbolic_functions/6.6_Hyperbolic_cosecant/6.6.3_Hyperbolic_cosecant_functions

3.2.5.1 Problem number 54

$$\int \frac{1}{\sqrt{a+ia\text{csch}(c+dx)}}dx$$ Optimal antiderivative

$$\frac{2{\tanh }^{-1}\left(\frac{\sqrt{a}\coth (c+dx)}{\sqrt{a+ia\text{csch}(c+dx)}}\right)}{\sqrt{a}d}-\frac{\sqrt{2}{\tanh }^{-1}\left(\frac{\sqrt{a}\coth (c+dx)}{\sqrt{2}\sqrt{a+ia\text{csch}(c+dx)}}\right)}{\sqrt{a}d}$$ command

integrate(1/(a+I*a*csch(d*x+c))^(1/2),x, algorithm="fricas")

Fricas 1.3.6 output

$$\text{Exception raised: TypeError}$$ Fricas 1.3.5 output

$$-\frac{1}{2}\sqrt{2}\sqrt{\frac{1}{a{d}^2}}\log \left(2(\sqrt{2}(ad{e}^{(2dx+2c)}-ad)\sqrt{\frac{a}{e^{(2dx+2c)}-1}}\sqrt{\frac{1}{a{d}^2}}+a{e}^{(dx+c)}-ia)e^{(-dx-c)}\right)+\frac{1}{2}\sqrt{2}\sqrt{\frac{1}{a{d}^2}}\log (-2(\sqrt{2}(ad{e}^{(2dx+2c)}-ad)\sqrt{\frac{a}{e^{(2dx+2c)}-1}}\sqrt{\frac{1}{a{d}^2}}-a{e}^{(dx+c)}+ia)e^{(-dx-c)})+\frac{1}{2}\sqrt{\frac{1}{a{d}^2}}\log \left(\frac{(2(d{e}^{(2dx+2c)}-d)\sqrt{\frac{a}{e^{(2dx+2c)}-1}}\sqrt{\frac{1}{a{d}^2}}+2e^{(dx+c)}+2i)e^{(-dx-c)}}{d}\right)-\frac{1}{2}\sqrt{\frac{1}{a{d}^2}}\log (-\frac{(2(d{e}^{(2dx+2c)}-d)\sqrt{\frac{a}{e^{(2dx+2c)}-1}}\sqrt{\frac{1}{a{d}^2}}-2e^{(dx+c)}-2i)e^{(-dx-c)}}{d})+\frac{1}{2}\sqrt{\frac{1}{a{d}^2}}\log \left(\frac{((2ad{e}^{(2dx+2c)}-2iad{e}^{(dx+c)}-4ad)\sqrt{\frac{1}{a{d}^2}}+\sqrt{\frac{a}{e^{(2dx+2c)}-1}}(2e^{(3dx+3c)}-4i{e}^{(2dx+2c)}-2e^{(dx+c)}+4i))e^{(-2dx-2c)}}{d}\right)-\frac{1}{2}\sqrt{\frac{1}{a{d}^2}}\log (-\frac{((2ad{e}^{(2dx+2c)}-2iad{e}^{(dx+c)}-4ad)\sqrt{\frac{1}{a{d}^2}}-\sqrt{\frac{a}{e^{(2dx+2c)}-1}}(2e^{(3dx+3c)}-4i{e}^{(2dx+2c)}-2e^{(dx+c)}+4i))e^{(-2dx-2c)}}{d})$$

3.2.5.2 Problem number 57

$$\int \frac{1}{\sqrt{a-ia\text{csch}(c+dx)}}dx$$ Optimal antiderivative

$$\frac{2{\tanh }^{-1}\left(\frac{\sqrt{a}\coth (c+dx)}{\sqrt{a-ia\text{csch}(c+dx)}}\right)}{\sqrt{a}d}-\frac{\sqrt{2}{\tanh }^{-1}\left(\frac{\sqrt{a}\coth (c+dx)}{\sqrt{2}\sqrt{a-ia\text{csch}(c+dx)}}\right)}{\sqrt{a}d}$$ command

integrate(1/(a-I*a*csch(d*x+c))^(1/2),x, algorithm="fricas")

Fricas 1.3.6 output

$$\text{Exception raised: TypeError}$$ Fricas 1.3.5 output

$$-\frac{1}{2}\sqrt{2}\sqrt{\frac{1}{a{d}^2}}\log \left(2(\sqrt{2}(ad{e}^{(2dx+2c)}-ad)\sqrt{\frac{a}{e^{(2dx+2c)}-1}}\sqrt{\frac{1}{a{d}^2}}+a{e}^{(dx+c)}+ia)e^{(-dx-c)}\right)+\frac{1}{2}\sqrt{2}\sqrt{\frac{1}{a{d}^2}}\log (-2(\sqrt{2}(ad{e}^{(2dx+2c)}-ad)\sqrt{\frac{a}{e^{(2dx+2c)}-1}}\sqrt{\frac{1}{a{d}^2}}-a{e}^{(dx+c)}-ia)e^{(-dx-c)})+\frac{1}{2}\sqrt{\frac{1}{a{d}^2}}\log \left(\frac{(2(d{e}^{(2dx+2c)}-d)\sqrt{\frac{a}{e^{(2dx+2c)}-1}}\sqrt{\frac{1}{a{d}^2}}+2e^{(dx+c)}-2i)e^{(-dx-c)}}{d}\right)-\frac{1}{2}\sqrt{\frac{1}{a{d}^2}}\log (-\frac{(2(d{e}^{(2dx+2c)}-d)\sqrt{\frac{a}{e^{(2dx+2c)}-1}}\sqrt{\frac{1}{a{d}^2}}-2e^{(dx+c)}+2i)e^{(-dx-c)}}{d})+\frac{1}{2}\sqrt{\frac{1}{a{d}^2}}\log \left(\frac{((2ad{e}^{(2dx+2c)}+2iad{e}^{(dx+c)}-4ad)\sqrt{\frac{1}{a{d}^2}}+\sqrt{\frac{a}{e^{(2dx+2c)}-1}}(2e^{(3dx+3c)}+4i{e}^{(2dx+2c)}-2e^{(dx+c)}-4i))e^{(-2dx-2c)}}{d}\right)-\frac{1}{2}\sqrt{\frac{1}{a{d}^2}}\log (-\frac{((2ad{e}^{(2dx+2c)}+2iad{e}^{(dx+c)}-4ad)\sqrt{\frac{1}{a{d}^2}}-\sqrt{\frac{a}{e^{(2dx+2c)}-1}}(2e^{(3dx+3c)}+4i{e}^{(2dx+2c)}-2e^{(dx+c)}-4i))e^{(-2dx-2c)}}{d})$$