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
\[ \int \frac{(a+b x)^{3/2} (c+d x)^{5/2}}{x^7} \, dx \] Optimal antiderivative
\[ -\frac{\sqrt{a+b x} (c+d x)^{3/2} (5 a d+7 b c) (b c-a d)^3}{768 a^3 c^3 x^2}+\frac{\sqrt{a+b x} (c+d x)^{5/2} (5 a d+7 b c) (b c-a d)^2}{960 a^2 c^3 x^3}+\frac{\sqrt{a+b x} \sqrt{c+d x} (5 a d+7 b c) (b c-a d)^4}{512 a^4 c^3 x}-\frac{(5 a d+7 b c) (b c-a d)^5 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{a+b x}}{\sqrt{a} \sqrt{c+d x}}\right )}{512 a^{9/2} c^{7/2}}+\frac{\sqrt{a+b x} (c+d x)^{7/2} (5 a d+7 b c) (b c-a d)}{160 a c^3 x^4}+\frac{(a+b x)^{3/2} (c+d x)^{7/2} (5 a d+7 b c)}{60 a c^2 x^5}-\frac{(a+b x)^{5/2} (c+d x)^{7/2}}{6 a c x^6} \] 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
\[ \left [-\frac{15 \,{\left (7 \, b^{6} c^{6} - 30 \, a b^{5} c^{5} d + 45 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 15 \, a^{4} b^{2} c^{2} d^{4} + 18 \, a^{5} b c d^{5} - 5 \, a^{6} d^{6}\right )} \sqrt{a c} x^{6} \log \left (\frac{8 \, a^{2} c^{2} +{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} + 4 \,{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{a c} \sqrt{b x + a} \sqrt{d x + c} + 8 \,{\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ) + 4 \,{\left (1280 \, a^{6} c^{6} -{\left (105 \, a b^{5} c^{6} - 415 \, a^{2} b^{4} c^{5} d + 546 \, a^{3} b^{3} c^{4} d^{2} - 150 \, a^{4} b^{2} c^{3} d^{3} + 245 \, a^{5} b c^{2} d^{4} - 75 \, a^{6} c d^{5}\right )} x^{5} + 2 \,{\left (35 \, a^{2} b^{4} c^{6} - 136 \, a^{3} b^{3} c^{5} d + 174 \, a^{4} b^{2} c^{4} d^{2} + 80 \, a^{5} b c^{3} d^{3} - 25 \, a^{6} c^{2} d^{4}\right )} x^{4} - 8 \,{\left (7 \, a^{3} b^{3} c^{6} - 27 \, a^{4} b^{2} c^{5} d - 423 \, a^{5} b c^{4} d^{2} - 5 \, a^{6} c^{3} d^{3}\right )} x^{3} + 16 \,{\left (3 \, a^{4} b^{2} c^{6} + 278 \, a^{5} b c^{5} d + 135 \, a^{6} c^{4} d^{2}\right )} x^{2} + 128 \,{\left (13 \, a^{5} b c^{6} + 25 \, a^{6} c^{5} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{30720 \, a^{5} c^{4} x^{6}}, \frac{15 \,{\left (7 \, b^{6} c^{6} - 30 \, a b^{5} c^{5} d + 45 \, a^{2} b^{4} c^{4} d^{2} - 20 \, a^{3} b^{3} c^{3} d^{3} - 15 \, a^{4} b^{2} c^{2} d^{4} + 18 \, a^{5} b c d^{5} - 5 \, a^{6} d^{6}\right )} \sqrt{-a c} x^{6} \arctan \left (\frac{{\left (2 \, a c +{\left (b c + a d\right )} x\right )} \sqrt{-a c} \sqrt{b x + a} \sqrt{d x + c}}{2 \,{\left (a b c d x^{2} + a^{2} c^{2} +{\left (a b c^{2} + a^{2} c d\right )} x\right )}}\right ) - 2 \,{\left (1280 \, a^{6} c^{6} -{\left (105 \, a b^{5} c^{6} - 415 \, a^{2} b^{4} c^{5} d + 546 \, a^{3} b^{3} c^{4} d^{2} - 150 \, a^{4} b^{2} c^{3} d^{3} + 245 \, a^{5} b c^{2} d^{4} - 75 \, a^{6} c d^{5}\right )} x^{5} + 2 \,{\left (35 \, a^{2} b^{4} c^{6} - 136 \, a^{3} b^{3} c^{5} d + 174 \, a^{4} b^{2} c^{4} d^{2} + 80 \, a^{5} b c^{3} d^{3} - 25 \, a^{6} c^{2} d^{4}\right )} x^{4} - 8 \,{\left (7 \, a^{3} b^{3} c^{6} - 27 \, a^{4} b^{2} c^{5} d - 423 \, a^{5} b c^{4} d^{2} - 5 \, a^{6} c^{3} d^{3}\right )} x^{3} + 16 \,{\left (3 \, a^{4} b^{2} c^{6} + 278 \, a^{5} b c^{5} d + 135 \, a^{6} c^{4} d^{2}\right )} x^{2} + 128 \,{\left (13 \, a^{5} b c^{6} + 25 \, a^{6} c^{5} d\right )} x\right )} \sqrt{b x + a} \sqrt{d x + c}}{15360 \, a^{5} c^{4} x^{6}}\right ] \]
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
\[ \int \frac{(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{x^{14}} \, dx \] Optimal antiderivative
\[ -\frac{a^4 \sqrt{a^2+2 a b x+b^2 x^2} (a B+5 A b)}{12 x^{12} (a+b x)}-\frac{5 a^3 b \sqrt{a^2+2 a b x+b^2 x^2} (a B+2 A b)}{11 x^{11} (a+b x)}-\frac{a^2 b^2 \sqrt{a^2+2 a b x+b^2 x^2} (a B+A b)}{x^{10} (a+b x)}-\frac{5 a b^3 \sqrt{a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 x^9 (a+b x)}-\frac{b^4 \sqrt{a^2+2 a b x+b^2 x^2} (5 a B+A b)}{8 x^8 (a+b x)}-\frac{a^5 A \sqrt{a^2+2 a b x+b^2 x^2}}{13 x^{13} (a+b x)}-\frac{b^5 B \sqrt{a^2+2 a b x+b^2 x^2}}{7 x^7 (a+b x)} \] 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
\[ -\frac{10296 \, B b^{5} x^{6} + 5544 \, A a^{5} + 9009 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 40040 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 72072 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 32760 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 6006 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x}{72072 \, x^{13}} \]
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-
\[ \int \frac{\left (1-\cos ^2(c+d x)\right ) \sec (c+d x)}{a+b \cos (c+d x)} \, dx \] Optimal antiderivative
\[ \frac{2 \sqrt{a-b} \sqrt{a+b} \tan ^{-1}\left (\frac{\sqrt{a-b} \tan \left (\frac{1}{2} (c+d x)\right )}{\sqrt{a+b}}\right )}{a b d}+\frac{\tanh ^{-1}(\sin (c+d x))}{a d}-\frac{x}{b} \] 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
\[ \left [-\frac{2 \, a d x - b \log \left (\sin \left (d x + c\right ) + 1\right ) + b \log \left (-\sin \left (d x + c\right ) + 1\right ) - \sqrt{-a^{2} + b^{2}} \log \left (\frac{2 \, a b \cos \left (d x + c\right ) +{\left (2 \, a^{2} - b^{2}\right )} \cos \left (d x + c\right )^{2} - 2 \, \sqrt{-a^{2} + b^{2}}{\left (a \cos \left (d x + c\right ) + b\right )} \sin \left (d x + c\right ) - a^{2} + 2 \, b^{2}}{b^{2} \cos \left (d x + c\right )^{2} + 2 \, a b \cos \left (d x + c\right ) + a^{2}}\right )}{2 \, a b d}, -\frac{2 \, a d x - b \log \left (\sin \left (d x + c\right ) + 1\right ) + b \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, \sqrt{a^{2} - b^{2}} \arctan \left (-\frac{a \cos \left (d x + c\right ) + b}{\sqrt{a^{2} - b^{2}} \sin \left (d x + c\right )}\right )}{2 \, a b d}\right ] \]
Test folder name
test_cases/4_Trig_functions/4.7_Miscellaneous/4.7.5_x^m_trig-a+b_log-c_x^n-^p
\[ \int \sin \left (a+\frac{1}{2} i \log \left (c x^2\right )\right ) \, dx \] Optimal antiderivative
\[ \frac{i e^{-i a} c x^3}{4 \sqrt{c x^2}}-\frac{i e^{i a} x \log (x)}{2 \sqrt{c x^2}} \] 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
\[ \frac{{\left (i \, c x^{2} - 2 i \, e^{\left (2 i \, a\right )} \log \left (x\right )\right )} e^{\left (-i \, a\right )}}{4 \, \sqrt{c}} \]
\[ \int \sin ^2\left (a+\frac{1}{4} i \log \left (c x^2\right )\right ) \, dx \] Optimal antiderivative
\[ -\frac{e^{-2 i a} c x^3}{8 \sqrt{c x^2}}-\frac{e^{2 i a} x \log (x)}{4 \sqrt{c x^2}}+\frac{x}{2} \] 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
\[ \frac{{\left (4 \, x^{2} e^{\left (2 i \, a\right )} - \frac{x e^{\left (4 i \, a\right )} \log \left (\frac{{\left (\sqrt{c x^{2}}{\left (x^{2} + 1\right )} e^{\left (2 i \, a\right )} + \frac{{\left (c x^{3} - c x\right )} e^{\left (2 i \, a\right )}}{\sqrt{c}}\right )} e^{\left (-2 i \, a\right )}}{8 \, x^{2}}\right )}{\sqrt{c}} + \frac{x e^{\left (4 i \, a\right )} \log \left (\frac{{\left (\sqrt{c x^{2}}{\left (x^{2} + 1\right )} e^{\left (2 i \, a\right )} - \frac{{\left (c x^{3} - c x\right )} e^{\left (2 i \, a\right )}}{\sqrt{c}}\right )} e^{\left (-2 i \, a\right )}}{8 \, x^{2}}\right )}{\sqrt{c}} - \sqrt{c x^{2}}{\left (x^{2} - 1\right )}\right )} e^{\left (-2 i \, a\right )}}{8 \, x} \]
\[ \int \sin ^3\left (a+\frac{1}{6} i \log \left (c x^2\right )\right ) \, dx \] Optimal antiderivative
\[ -\frac{i e^{-3 i a} c x^3}{16 \sqrt{c x^2}}+\frac{9}{32} i e^{-i a} x \sqrt [6]{c x^2}-\frac{9 i e^{i a} x}{16 \sqrt [6]{c x^2}}+\frac{i e^{3 i a} x \log (x)}{8 \sqrt{c x^2}} \] 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
\[ -\frac{{\left (2 \, c x \sqrt{-\frac{e^{\left (6 i \, a\right )}}{c}} e^{\left (3 i \, a\right )} \log \left (\frac{{\left (4 \, \sqrt{c x^{2}}{\left (x^{2} + 1\right )} e^{\left (3 i \, a\right )} +{\left (4 i \, c x^{3} - 4 i \, c x\right )} \sqrt{-\frac{e^{\left (6 i \, a\right )}}{c}}\right )} e^{\left (-3 i \, a\right )}}{32 \, x^{2}}\right ) - 2 \, c x \sqrt{-\frac{e^{\left (6 i \, a\right )}}{c}} e^{\left (3 i \, a\right )} \log \left (\frac{{\left (4 \, \sqrt{c x^{2}}{\left (x^{2} + 1\right )} e^{\left (3 i \, a\right )} +{\left (-4 i \, c x^{3} + 4 i \, c x\right )} \sqrt{-\frac{e^{\left (6 i \, a\right )}}{c}}\right )} e^{\left (-3 i \, a\right )}}{32 \, x^{2}}\right ) - 9 i \, \left (c x^{2}\right )^{\frac{1}{6}} c x^{2} e^{\left (2 i \, a\right )} + 18 i \, \left (c x^{2}\right )^{\frac{5}{6}} e^{\left (4 i \, a\right )} - \sqrt{c x^{2}}{\left (-2 i \, c x^{2} + 2 i \, c\right )}\right )} e^{\left (-3 i \, a\right )}}{32 \, c x} \]
\[ \int x^3 \tan (a+i \log (x)) \, dx \] Optimal antiderivative
\[ -i e^{2 i a} x^2+i e^{4 i a} \log \left (x^2+e^{2 i a}\right )+\frac{i x^4}{4} \] command
integrate(x^3*tan(a+I*log(x)),x, algorithm="fricas")
Fricas 1.3.6 output
\[{\rm integral}\left (\frac{-i \, x^{3} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i \, x^{3}}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1}, x\right ) \] Fricas 1.3.5 output
\[ \frac{1}{4} i \, x^{4} - i \, x^{2} e^{\left (2 i \, a\right )} + i \, e^{\left (4 i \, a\right )} \log \left (x^{2} + e^{\left (2 i \, a\right )}\right ) \]
\[ \int x^2 \tan (a+i \log (x)) \, dx \] Optimal antiderivative
\[ -2 i e^{2 i a} x+2 i e^{3 i a} \tan ^{-1}\left (e^{-i a} x\right )+\frac{i x^3}{3} \] command
integrate(x^2*tan(a+I*log(x)),x, algorithm="fricas")
Fricas 1.3.6 output
\[{\rm integral}\left (\frac{-i \, x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i \, x^{2}}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1}, x\right ) \] Fricas 1.3.5 output
\[ \frac{1}{3} i \, x^{3} - 2 i \, x e^{\left (2 i \, a\right )} - e^{\left (3 i \, a\right )} \log \left (x + i \, e^{\left (i \, a\right )}\right ) + e^{\left (3 i \, a\right )} \log \left (x - i \, e^{\left (i \, a\right )}\right ) \]
\[ \int x \tan (a+i \log (x)) \, dx \] Optimal antiderivative
\[ \frac{i x^2}{2}-i e^{2 i a} \log \left (x^2+e^{2 i a}\right ) \] command
integrate(x*tan(a+I*log(x)),x, algorithm="fricas")
Fricas 1.3.6 output
\[{\rm integral}\left (\frac{-i \, x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i \, x}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1}, x\right ) \] Fricas 1.3.5 output
\[ \frac{1}{2} i \, x^{2} - i \, e^{\left (2 i \, a\right )} \log \left (x^{2} + e^{\left (2 i \, a\right )}\right ) \]
\[ \int \tan (a+i \log (x)) \, dx \] Optimal antiderivative
\[ i x-2 i e^{i a} \tan ^{-1}\left (e^{-i a} x\right ) \] command
integrate(tan(a+I*log(x)),x, algorithm="fricas")
Fricas 1.3.6 output
\[{\rm integral}\left (\frac{-i \, e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1}, x\right ) \] Fricas 1.3.5 output
\[ e^{\left (i \, a\right )} \log \left (x + i \, e^{\left (i \, a\right )}\right ) - e^{\left (i \, a\right )} \log \left (x - i \, e^{\left (i \, a\right )}\right ) + i \, x \]
\[ \int \frac{\tan (a+i \log (x))}{x^2} \, dx \] Optimal antiderivative
\[ 2 i e^{-i a} \tan ^{-1}\left (e^{-i a} x\right )+\frac{i}{x} \] command
integrate(tan(a+I*log(x))/x^2,x, algorithm="fricas")
Fricas 1.3.6 output
\[{\rm integral}\left (\frac{-i \, e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i}{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + x^{2}}, x\right ) \] Fricas 1.3.5 output
\[ -\frac{{\left (x \log \left (x + i \, e^{\left (i \, a\right )}\right ) - x \log \left (x - i \, e^{\left (i \, a\right )}\right ) - i \, e^{\left (i \, a\right )}\right )} e^{\left (-i \, a\right )}}{x} \]
\[ \int \frac{\tan (a+i \log (x))}{x^3} \, dx \] Optimal antiderivative
\[ \frac{i}{2 x^2}-i e^{-2 i a} \log \left (1+\frac{e^{2 i a}}{x^2}\right ) \] command
integrate(tan(a+I*log(x))/x^3,x, algorithm="fricas")
Fricas 1.3.6 output
\[{\rm integral}\left (\frac{-i \, e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i}{x^{3} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + x^{3}}, x\right ) \] Fricas 1.3.5 output
\[ \frac{{\left (-2 i \, x^{2} \log \left (x^{2} + e^{\left (2 i \, a\right )}\right ) + 4 i \, x^{2} \log \left (x\right ) + i \, e^{\left (2 i \, a\right )}\right )} e^{\left (-2 i \, a\right )}}{2 \, x^{2}} \]
\[ \int \frac{\tan (a+i \log (x))}{x^4} \, dx \] Optimal antiderivative
\[ -\frac{2 i e^{-2 i a}}{x}-2 i e^{-3 i a} \tan ^{-1}\left (e^{-i a} x\right )+\frac{i}{3 x^3} \] command
integrate(tan(a+I*log(x))/x^4,x, algorithm="fricas")
Fricas 1.3.6 output
\[{\rm integral}\left (\frac{-i \, e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i}{x^{4} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + x^{4}}, x\right ) \] Fricas 1.3.5 output
\[ \frac{{\left (3 \, x^{3} \log \left (x + i \, e^{\left (i \, a\right )}\right ) - 3 \, x^{3} \log \left (x - i \, e^{\left (i \, a\right )}\right ) - 6 i \, x^{2} e^{\left (i \, a\right )} + i \, e^{\left (3 i \, a\right )}\right )} e^{\left (-3 i \, a\right )}}{3 \, x^{3}} \]
\[ \int x^3 \tan ^2(a+i \log (x)) \, dx \] Optimal antiderivative
\[ 2 e^{2 i a} x^2-\frac{2 e^{6 i a}}{x^2+e^{2 i a}}-4 e^{4 i a} \log \left (x^2+e^{2 i a}\right )-\frac{x^4}{4} \] command
integrate(x^3*tan(a+I*log(x))^2,x, algorithm="fricas")
Fricas 1.3.6 output
\[ \frac{2 \, x^{4} +{\left (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1\right )}{\rm integral}\left (-\frac{x^{3} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 9 \, x^{3}}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1}, x\right )}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1} \] Fricas 1.3.5 output
\[ -\frac{x^{6} - 7 \, x^{4} e^{\left (2 i \, a\right )} - 8 \, x^{2} e^{\left (4 i \, a\right )} + 16 \,{\left (x^{2} e^{\left (4 i \, a\right )} + e^{\left (6 i \, a\right )}\right )} \log \left (x^{2} + e^{\left (2 i \, a\right )}\right ) + 8 \, e^{\left (6 i \, a\right )}}{4 \,{\left (x^{2} + e^{\left (2 i \, a\right )}\right )}} \]
\[ \int x^2 \tan ^2(a+i \log (x)) \, dx \] Optimal antiderivative
\[ -\frac{2 e^{2 i a} x^3}{x^2+e^{2 i a}}+6 e^{2 i a} x-6 e^{3 i a} \tan ^{-1}\left (e^{-i a} x\right )-\frac{x^3}{3} \] command
integrate(x^2*tan(a+I*log(x))^2,x, algorithm="fricas")
Fricas 1.3.6 output
\[ \frac{2 \, x^{3} +{\left (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1\right )}{\rm integral}\left (-\frac{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 7 \, x^{2}}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1}, x\right )}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1} \] Fricas 1.3.5 output
\[ -\frac{x^{5} - 11 \, x^{3} e^{\left (2 i \, a\right )} - 18 \, x e^{\left (4 i \, a\right )} -{\left (-9 i \, x^{2} e^{\left (3 i \, a\right )} - 9 i \, e^{\left (5 i \, a\right )}\right )} \log \left (x + i \, e^{\left (i \, a\right )}\right ) -{\left (9 i \, x^{2} e^{\left (3 i \, a\right )} + 9 i \, e^{\left (5 i \, a\right )}\right )} \log \left (x - i \, e^{\left (i \, a\right )}\right )}{3 \,{\left (x^{2} + e^{\left (2 i \, a\right )}\right )}} \]
\[ \int x \tan ^2(a+i \log (x)) \, dx \] Optimal antiderivative
\[ \frac{2 e^{4 i a}}{x^2+e^{2 i a}}+2 e^{2 i a} \log \left (x^2+e^{2 i a}\right )-\frac{x^2}{2} \] command
integrate(x*tan(a+I*log(x))^2,x, algorithm="fricas")
Fricas 1.3.6 output
\[ \frac{2 \, x^{2} +{\left (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1\right )}{\rm integral}\left (-\frac{x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 5 \, x}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1}, x\right )}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1} \] Fricas 1.3.5 output
\[ -\frac{x^{4} + x^{2} e^{\left (2 i \, a\right )} - 4 \,{\left (x^{2} e^{\left (2 i \, a\right )} + e^{\left (4 i \, a\right )}\right )} \log \left (x^{2} + e^{\left (2 i \, a\right )}\right ) - 4 \, e^{\left (4 i \, a\right )}}{2 \,{\left (x^{2} + e^{\left (2 i \, a\right )}\right )}} \]
\[ \int \tan ^2(a+i \log (x)) \, dx \] Optimal antiderivative
\[ -\frac{2 e^{2 i a} x}{x^2+e^{2 i a}}+2 e^{i a} \tan ^{-1}\left (e^{-i a} x\right )-x \] command
integrate(tan(a+I*log(x))^2,x, algorithm="fricas")
Fricas 1.3.6 output
\[ \frac{{\left (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1\right )}{\rm integral}\left (-\frac{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 3}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1}, x\right ) + 2 \, x}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1} \] Fricas 1.3.5 output
\[ -\frac{x^{3} + 3 \, x e^{\left (2 i \, a\right )} -{\left (i \, x^{2} e^{\left (i \, a\right )} + i \, e^{\left (3 i \, a\right )}\right )} \log \left (x + i \, e^{\left (i \, a\right )}\right ) -{\left (-i \, x^{2} e^{\left (i \, a\right )} - i \, e^{\left (3 i \, a\right )}\right )} \log \left (x - i \, e^{\left (i \, a\right )}\right )}{x^{2} + e^{\left (2 i \, a\right )}} \]
\[ \int \frac{\tan ^2(a+i \log (x))}{x^2} \, dx \] Optimal antiderivative
\[ \frac{3 x}{x^2+e^{2 i a}}+\frac{e^{2 i a}}{x \left (x^2+e^{2 i a}\right )}+2 e^{-i a} \tan ^{-1}\left (e^{-i a} x\right ) \] command
integrate(tan(a+I*log(x))^2/x^2,x, algorithm="fricas")
Fricas 1.3.6 output
\[ \frac{{\left (x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + x\right )}{\rm integral}\left (-\frac{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + x^{2}}, x\right ) + 2}{x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + x} \] Fricas 1.3.5 output
\[ \frac{3 \, x^{2} e^{\left (i \, a\right )} +{\left (i \, x^{3} + i \, x e^{\left (2 i \, a\right )}\right )} \log \left (x + i \, e^{\left (i \, a\right )}\right ) +{\left (-i \, x^{3} - i \, x e^{\left (2 i \, a\right )}\right )} \log \left (x - i \, e^{\left (i \, a\right )}\right ) + e^{\left (3 i \, a\right )}}{x^{3} e^{\left (i \, a\right )} + x e^{\left (3 i \, a\right )}} \]
\[ \int \frac{\tan ^2(a+i \log (x))}{x^3} \, dx \] Optimal antiderivative
\[ -\frac{2 e^{-2 i a}}{1+\frac{e^{2 i a}}{x^2}}-2 e^{-2 i a} \log \left (1+\frac{e^{2 i a}}{x^2}\right )+\frac{1}{2 x^2} \] command
integrate(tan(a+I*log(x))^2/x^3,x, algorithm="fricas")
Fricas 1.3.6 output
\[ \frac{{\left (x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + x^{2}\right )}{\rm integral}\left (-\frac{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 3}{x^{3} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + x^{3}}, x\right ) + 2}{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + x^{2}} \] Fricas 1.3.5 output
\[ \frac{5 \, x^{2} e^{\left (2 i \, a\right )} - 4 \,{\left (x^{4} + x^{2} e^{\left (2 i \, a\right )}\right )} \log \left (x^{2} + e^{\left (2 i \, a\right )}\right ) + 8 \,{\left (x^{4} + x^{2} e^{\left (2 i \, a\right )}\right )} \log \left (x\right ) + e^{\left (4 i \, a\right )}}{2 \,{\left (x^{4} e^{\left (2 i \, a\right )} + x^{2} e^{\left (4 i \, a\right )}\right )}} \]
\[ \int x^3 \cot (a+i \log (x)) \, dx \] Optimal antiderivative
\[ -i e^{2 i a} x^2-i e^{4 i a} \log \left (-x^2+e^{2 i a}\right )-\frac{i x^4}{4} \] command
integrate(x^3*cot(a+I*log(x)),x, algorithm="fricas")
Fricas 1.3.6 output
\[{\rm integral}\left (\frac{i \, x^{3} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i \, x^{3}}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right ) \] Fricas 1.3.5 output
\[ -\frac{1}{4} i \, x^{4} - i \, x^{2} e^{\left (2 i \, a\right )} - i \, e^{\left (4 i \, a\right )} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right ) \]
\[ \int x^2 \cot (a+i \log (x)) \, dx \] Optimal antiderivative
\[ -2 i e^{2 i a} x+2 i e^{3 i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac{i x^3}{3} \] command
integrate(x^2*cot(a+I*log(x)),x, algorithm="fricas")
Fricas 1.3.6 output
\[{\rm integral}\left (\frac{i \, x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i \, x^{2}}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right ) \] Fricas 1.3.5 output
\[ -\frac{1}{3} i \, x^{3} - 2 i \, x e^{\left (2 i \, a\right )} + i \, e^{\left (3 i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - i \, e^{\left (3 i \, a\right )} \log \left (x - e^{\left (i \, a\right )}\right ) \]
\[ \int x \cot (a+i \log (x)) \, dx \] Optimal antiderivative
\[ -i e^{2 i a} \log \left (-x^2+e^{2 i a}\right )-\frac{i x^2}{2} \] command
integrate(x*cot(a+I*log(x)),x, algorithm="fricas")
Fricas 1.3.6 output
\[{\rm integral}\left (\frac{i \, x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i \, x}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right ) \] Fricas 1.3.5 output
\[ -\frac{1}{2} i \, x^{2} - i \, e^{\left (2 i \, a\right )} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right ) \]
\[ \int \cot (a+i \log (x)) \, dx \] Optimal antiderivative
\[ 2 i e^{i a} \tanh ^{-1}\left (e^{-i a} x\right )-i x \] command
integrate(cot(a+I*log(x)),x, algorithm="fricas")
Fricas 1.3.6 output
\[{\rm integral}\left (\frac{i \, e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right ) \] Fricas 1.3.5 output
\[ i \, e^{\left (i \, a\right )} \log \left (x + e^{\left (i \, a\right )}\right ) - i \, e^{\left (i \, a\right )} \log \left (x - e^{\left (i \, a\right )}\right ) - i \, x \]
\[ \int \frac{\cot (a+i \log (x))}{x^2} \, dx \] Optimal antiderivative
\[ 2 i e^{-i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac{i}{x} \] command
integrate(cot(a+I*log(x))/x^2,x, algorithm="fricas")
Fricas 1.3.6 output
\[{\rm integral}\left (\frac{i \, e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i}{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{2}}, x\right ) \] Fricas 1.3.5 output
\[ \frac{{\left (i \, x \log \left (x + e^{\left (i \, a\right )}\right ) - i \, x \log \left (x - e^{\left (i \, a\right )}\right ) - i \, e^{\left (i \, a\right )}\right )} e^{\left (-i \, a\right )}}{x} \]
\[ \int \frac{\cot (a+i \log (x))}{x^3} \, dx \] Optimal antiderivative
\[ -i e^{-2 i a} \log \left (1-\frac{e^{2 i a}}{x^2}\right )-\frac{i}{2 x^2} \] command
integrate(cot(a+I*log(x))/x^3,x, algorithm="fricas")
Fricas 1.3.6 output
\[{\rm integral}\left (\frac{i \, e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i}{x^{3} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{3}}, x\right ) \] Fricas 1.3.5 output
\[ \frac{{\left (-2 i \, x^{2} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right ) + 4 i \, x^{2} \log \left (x\right ) - i \, e^{\left (2 i \, a\right )}\right )} e^{\left (-2 i \, a\right )}}{2 \, x^{2}} \]
\[ \int \frac{\cot (a+i \log (x))}{x^4} \, dx \] Optimal antiderivative
\[ -\frac{2 i e^{-2 i a}}{x}+2 i e^{-3 i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac{i}{3 x^3} \] command
integrate(cot(a+I*log(x))/x^4,x, algorithm="fricas")
Fricas 1.3.6 output
\[{\rm integral}\left (\frac{i \, e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + i}{x^{4} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{4}}, x\right ) \] Fricas 1.3.5 output
\[ \frac{{\left (3 i \, x^{3} \log \left (x + e^{\left (i \, a\right )}\right ) - 3 i \, x^{3} \log \left (x - e^{\left (i \, a\right )}\right ) - 6 i \, x^{2} e^{\left (i \, a\right )} - i \, e^{\left (3 i \, a\right )}\right )} e^{\left (-3 i \, a\right )}}{3 \, x^{3}} \]
\[ \int x^3 \cot ^2(a+i \log (x)) \, dx \] Optimal antiderivative
\[ -2 e^{2 i a} x^2-\frac{2 e^{6 i a}}{-x^2+e^{2 i a}}-4 e^{4 i a} \log \left (-x^2+e^{2 i a}\right )-\frac{x^4}{4} \] command
integrate(x^3*cot(a+I*log(x))^2,x, algorithm="fricas")
Fricas 1.3.6 output
\[ -\frac{2 \, x^{4} -{\left (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1\right )}{\rm integral}\left (-\frac{x^{3} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 9 \, x^{3}}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right )}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1} \] Fricas 1.3.5 output
\[ -\frac{x^{6} + 7 \, x^{4} e^{\left (2 i \, a\right )} - 8 \, x^{2} e^{\left (4 i \, a\right )} + 16 \,{\left (x^{2} e^{\left (4 i \, a\right )} - e^{\left (6 i \, a\right )}\right )} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right ) - 8 \, e^{\left (6 i \, a\right )}}{4 \,{\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} \]
\[ \int x^2 \cot ^2(a+i \log (x)) \, dx \] Optimal antiderivative
\[ -\frac{2 e^{2 i a} x^3}{-x^2+e^{2 i a}}-6 e^{2 i a} x+6 e^{3 i a} \tanh ^{-1}\left (e^{-i a} x\right )-\frac{x^3}{3} \] command
integrate(x^2*cot(a+I*log(x))^2,x, algorithm="fricas")
Fricas 1.3.6 output
\[ -\frac{2 \, x^{3} -{\left (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1\right )}{\rm integral}\left (-\frac{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 7 \, x^{2}}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right )}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1} \] Fricas 1.3.5 output
\[ -\frac{x^{5} + 11 \, x^{3} e^{\left (2 i \, a\right )} - 18 \, x e^{\left (4 i \, a\right )} - 9 \,{\left (x^{2} e^{\left (3 i \, a\right )} - e^{\left (5 i \, a\right )}\right )} \log \left (x + e^{\left (i \, a\right )}\right ) + 9 \,{\left (x^{2} e^{\left (3 i \, a\right )} - e^{\left (5 i \, a\right )}\right )} \log \left (x - e^{\left (i \, a\right )}\right )}{3 \,{\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} \]
\[ \int x \cot ^2(a+i \log (x)) \, dx \] Optimal antiderivative
\[ -\frac{2 e^{4 i a}}{-x^2+e^{2 i a}}-2 e^{2 i a} \log \left (-x^2+e^{2 i a}\right )-\frac{x^2}{2} \] command
integrate(x*cot(a+I*log(x))^2,x, algorithm="fricas")
Fricas 1.3.6 output
\[ -\frac{2 \, x^{2} -{\left (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1\right )}{\rm integral}\left (-\frac{x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 5 \, x}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right )}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1} \] Fricas 1.3.5 output
\[ -\frac{x^{4} - x^{2} e^{\left (2 i \, a\right )} + 4 \,{\left (x^{2} e^{\left (2 i \, a\right )} - e^{\left (4 i \, a\right )}\right )} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right ) - 4 \, e^{\left (4 i \, a\right )}}{2 \,{\left (x^{2} - e^{\left (2 i \, a\right )}\right )}} \]
\[ \int \cot ^2(a+i \log (x)) \, dx \] Optimal antiderivative
\[ -\frac{2 e^{2 i a} x}{-x^2+e^{2 i a}}+2 e^{i a} \tanh ^{-1}\left (e^{-i a} x\right )-x \] command
integrate(cot(a+I*log(x))^2,x, algorithm="fricas")
Fricas 1.3.6 output
\[ \frac{{\left (e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1\right )}{\rm integral}\left (-\frac{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 3}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1}, x\right ) - 2 \, x}{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - 1} \] Fricas 1.3.5 output
\[ -\frac{x^{3} - 3 \, x e^{\left (2 i \, a\right )} -{\left (x^{2} e^{\left (i \, a\right )} - e^{\left (3 i \, a\right )}\right )} \log \left (x + e^{\left (i \, a\right )}\right ) +{\left (x^{2} e^{\left (i \, a\right )} - e^{\left (3 i \, a\right )}\right )} \log \left (x - e^{\left (i \, a\right )}\right )}{x^{2} - e^{\left (2 i \, a\right )}} \]
\[ \int \frac{\cot ^2(a+i \log (x))}{x^2} \, dx \] Optimal antiderivative
\[ -\frac{3 x}{-x^2+e^{2 i a}}+\frac{e^{2 i a}}{x \left (-x^2+e^{2 i a}\right )}-2 e^{-i a} \tanh ^{-1}\left (e^{-i a} x\right ) \] command
integrate(cot(a+I*log(x))^2/x^2,x, algorithm="fricas")
Fricas 1.3.6 output
\[ \frac{{\left (x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x\right )}{\rm integral}\left (-\frac{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 1}{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{2}}, x\right ) - 2}{x e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x} \] Fricas 1.3.5 output
\[ \frac{3 \, x^{2} e^{\left (i \, a\right )} -{\left (x^{3} - x e^{\left (2 i \, a\right )}\right )} \log \left (x + e^{\left (i \, a\right )}\right ) +{\left (x^{3} - x e^{\left (2 i \, a\right )}\right )} \log \left (x - e^{\left (i \, a\right )}\right ) - e^{\left (3 i \, a\right )}}{x^{3} e^{\left (i \, a\right )} - x e^{\left (3 i \, a\right )}} \]
\[ \int \frac{\cot ^2(a+i \log (x))}{x^3} \, dx \] Optimal antiderivative
\[ \frac{2 e^{-2 i a}}{1-\frac{e^{2 i a}}{x^2}}+2 e^{-2 i a} \log \left (1-\frac{e^{2 i a}}{x^2}\right )+\frac{1}{2 x^2} \] command
integrate(cot(a+I*log(x))^2/x^3,x, algorithm="fricas")
Fricas 1.3.6 output
\[ \frac{{\left (x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{2}\right )}{\rm integral}\left (-\frac{e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} + 3}{x^{3} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{3}}, x\right ) - 2}{x^{2} e^{\left (2 i \, a - 2 \, \log \left (x\right )\right )} - x^{2}} \] Fricas 1.3.5 output
\[ \frac{5 \, x^{2} e^{\left (2 i \, a\right )} + 4 \,{\left (x^{4} - x^{2} e^{\left (2 i \, a\right )}\right )} \log \left (x^{2} - e^{\left (2 i \, a\right )}\right ) - 8 \,{\left (x^{4} - x^{2} e^{\left (2 i \, a\right )}\right )} \log \left (x\right ) - e^{\left (4 i \, a\right )}}{2 \,{\left (x^{4} e^{\left (2 i \, a\right )} - x^{2} e^{\left (4 i \, a\right )}\right )}} \]
Test folder name
test_cases/6_Hyperbolic_functions/6.6_Hyperbolic_cosecant/6.6.3_Hyperbolic_cosecant_functions
\[ \int \frac{1}{\sqrt{a+i a \text{csch}(c+d x)}} \, dx \] Optimal antiderivative
\[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a+i a \text{csch}(c+d x)}}\right )}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{2} \sqrt{a+i a \text{csch}(c+d x)}}\right )}{\sqrt{a} d} \] 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 \,{\left (\sqrt{2}{\left (a d e^{\left (2 \, d x + 2 \, c\right )} - a d\right )} \sqrt{\frac{a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt{\frac{1}{a d^{2}}} + a e^{\left (d x + c\right )} - i \, a\right )} e^{\left (-d x - c\right )}\right ) + \frac{1}{2} \, \sqrt{2} \sqrt{\frac{1}{a d^{2}}} \log \left (-2 \,{\left (\sqrt{2}{\left (a d e^{\left (2 \, d x + 2 \, c\right )} - a d\right )} \sqrt{\frac{a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt{\frac{1}{a d^{2}}} - a e^{\left (d x + c\right )} + i \, a\right )} e^{\left (-d x - c\right )}\right ) + \frac{1}{2} \, \sqrt{\frac{1}{a d^{2}}} \log \left (\frac{{\left (2 \,{\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt{\frac{a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt{\frac{1}{a d^{2}}} + 2 \, e^{\left (d x + c\right )} + 2 i\right )} e^{\left (-d x - c\right )}}{d}\right ) - \frac{1}{2} \, \sqrt{\frac{1}{a d^{2}}} \log \left (-\frac{{\left (2 \,{\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt{\frac{a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt{\frac{1}{a d^{2}}} - 2 \, e^{\left (d x + c\right )} - 2 i\right )} e^{\left (-d x - c\right )}}{d}\right ) + \frac{1}{2} \, \sqrt{\frac{1}{a d^{2}}} \log \left (\frac{{\left ({\left (2 \, a d e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, a d e^{\left (d x + c\right )} - 4 \, a d\right )} \sqrt{\frac{1}{a d^{2}}} + \sqrt{\frac{a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}{\left (2 \, e^{\left (3 \, d x + 3 \, c\right )} - 4 i \, e^{\left (2 \, d x + 2 \, c\right )} - 2 \, e^{\left (d x + c\right )} + 4 i\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) - \frac{1}{2} \, \sqrt{\frac{1}{a d^{2}}} \log \left (-\frac{{\left ({\left (2 \, a d e^{\left (2 \, d x + 2 \, c\right )} - 2 i \, a d e^{\left (d x + c\right )} - 4 \, a d\right )} \sqrt{\frac{1}{a d^{2}}} - \sqrt{\frac{a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}{\left (2 \, e^{\left (3 \, d x + 3 \, c\right )} - 4 i \, e^{\left (2 \, d x + 2 \, c\right )} - 2 \, e^{\left (d x + c\right )} + 4 i\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) \]
\[ \int \frac{1}{\sqrt{a-i a \text{csch}(c+d x)}} \, dx \] Optimal antiderivative
\[ \frac{2 \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{a-i a \text{csch}(c+d x)}}\right )}{\sqrt{a} d}-\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{a} \coth (c+d x)}{\sqrt{2} \sqrt{a-i a \text{csch}(c+d x)}}\right )}{\sqrt{a} d} \] 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 \,{\left (\sqrt{2}{\left (a d e^{\left (2 \, d x + 2 \, c\right )} - a d\right )} \sqrt{\frac{a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt{\frac{1}{a d^{2}}} + a e^{\left (d x + c\right )} + i \, a\right )} e^{\left (-d x - c\right )}\right ) + \frac{1}{2} \, \sqrt{2} \sqrt{\frac{1}{a d^{2}}} \log \left (-2 \,{\left (\sqrt{2}{\left (a d e^{\left (2 \, d x + 2 \, c\right )} - a d\right )} \sqrt{\frac{a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt{\frac{1}{a d^{2}}} - a e^{\left (d x + c\right )} - i \, a\right )} e^{\left (-d x - c\right )}\right ) + \frac{1}{2} \, \sqrt{\frac{1}{a d^{2}}} \log \left (\frac{{\left (2 \,{\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt{\frac{a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt{\frac{1}{a d^{2}}} + 2 \, e^{\left (d x + c\right )} - 2 i\right )} e^{\left (-d x - c\right )}}{d}\right ) - \frac{1}{2} \, \sqrt{\frac{1}{a d^{2}}} \log \left (-\frac{{\left (2 \,{\left (d e^{\left (2 \, d x + 2 \, c\right )} - d\right )} \sqrt{\frac{a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}} \sqrt{\frac{1}{a d^{2}}} - 2 \, e^{\left (d x + c\right )} + 2 i\right )} e^{\left (-d x - c\right )}}{d}\right ) + \frac{1}{2} \, \sqrt{\frac{1}{a d^{2}}} \log \left (\frac{{\left ({\left (2 \, a d e^{\left (2 \, d x + 2 \, c\right )} + 2 i \, a d e^{\left (d x + c\right )} - 4 \, a d\right )} \sqrt{\frac{1}{a d^{2}}} + \sqrt{\frac{a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}{\left (2 \, e^{\left (3 \, d x + 3 \, c\right )} + 4 i \, e^{\left (2 \, d x + 2 \, c\right )} - 2 \, e^{\left (d x + c\right )} - 4 i\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) - \frac{1}{2} \, \sqrt{\frac{1}{a d^{2}}} \log \left (-\frac{{\left ({\left (2 \, a d e^{\left (2 \, d x + 2 \, c\right )} + 2 i \, a d e^{\left (d x + c\right )} - 4 \, a d\right )} \sqrt{\frac{1}{a d^{2}}} - \sqrt{\frac{a}{e^{\left (2 \, d x + 2 \, c\right )} - 1}}{\left (2 \, e^{\left (3 \, d x + 3 \, c\right )} + 4 i \, e^{\left (2 \, d x + 2 \, c\right )} - 2 \, e^{\left (d x + c\right )} - 4 i\right )}\right )} e^{\left (-2 \, d x - 2 \, c\right )}}{d}\right ) \]