Optimal. Leaf size=381 \[ \frac {a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {\left (2 a^6 C+8 b^6 (7 A+6 C)+a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)\right ) \tan (c+d x)}{105 b^2 d}+\frac {a \left (4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{420 b d}+\frac {\left (2 a^4 C+8 b^4 (7 A+6 C)+3 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{210 b^2 d}+\frac {a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{210 b^2 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.68, antiderivative size = 381, normalized size of antiderivative = 1.00, number of steps
used = 10, number of rules used = 8, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.242, Rules used = {4178, 4167,
4087, 4082, 3872, 3855, 3852, 8} \begin {gather*} \frac {a b \left (a^2 (8 A+6 C)+b^2 (6 A+5 C)\right ) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^4}{105 b^2 d}+\frac {a \left (2 a^2 C+42 A b^2+31 b^2 C\right ) \tan (c+d x) (a+b \sec (c+d x))^3}{210 b^2 d}+\frac {\left (2 a^4 C+3 a^2 b^2 (14 A+9 C)+8 b^4 (7 A+6 C)\right ) \tan (c+d x) (a+b \sec (c+d x))^2}{210 b^2 d}+\frac {a \left (4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right ) \tan (c+d x) \sec (c+d x)}{420 b d}+\frac {\left (2 a^6 C+a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)+8 b^6 (7 A+6 C)\right ) \tan (c+d x)}{105 b^2 d}-\frac {a C \tan (c+d x) (a+b \sec (c+d x))^5}{21 b^2 d}+\frac {C \tan (c+d x) \sec (c+d x) (a+b \sec (c+d x))^5}{7 b d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 8
Rule 3852
Rule 3855
Rule 3872
Rule 4082
Rule 4087
Rule 4167
Rule 4178
Rubi steps
\begin {align*} \int \sec ^2(c+d x) (a+b \sec (c+d x))^4 \left (A+C \sec ^2(c+d x)\right ) \, dx &=\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^4 \left (a C+b (7 A+6 C) \sec (c+d x)-2 a C \sec ^2(c+d x)\right ) \, dx}{7 b}\\ &=-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^4 \left (-4 a b C+2 \left (a^2 C+3 b^2 (7 A+6 C)\right ) \sec (c+d x)\right ) \, dx}{42 b^2}\\ &=\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^3 \left (12 b \left (14 A b^2-a^2 C+12 b^2 C\right )+4 a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) \sec (c+d x)\right ) \, dx}{210 b^2}\\ &=\frac {a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{210 b^2 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x))^2 \left (12 a b \left (98 A b^2-2 a^2 C+79 b^2 C\right )+12 \left (2 a^4 C+8 b^4 (7 A+6 C)+3 a^2 b^2 (14 A+9 C)\right ) \sec (c+d x)\right ) \, dx}{840 b^2}\\ &=\frac {\left (2 a^4 C+8 b^4 (7 A+6 C)+3 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{210 b^2 d}+\frac {a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{210 b^2 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) (a+b \sec (c+d x)) \left (-12 b \left (2 a^4 C-16 b^4 (7 A+6 C)-3 a^2 b^2 (126 A+97 C)\right )+12 a \left (4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right ) \sec (c+d x)\right ) \, dx}{2520 b^2}\\ &=\frac {a \left (4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{420 b d}+\frac {\left (2 a^4 C+8 b^4 (7 A+6 C)+3 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{210 b^2 d}+\frac {a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{210 b^2 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {\int \sec (c+d x) \left (1260 a b^3 \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right )+48 \left (2 a^6 C+8 b^6 (7 A+6 C)+a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)\right ) \sec (c+d x)\right ) \, dx}{5040 b^2}\\ &=\frac {a \left (4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{420 b d}+\frac {\left (2 a^4 C+8 b^4 (7 A+6 C)+3 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{210 b^2 d}+\frac {a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{210 b^2 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}+\frac {1}{4} \left (a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right )\right ) \int \sec (c+d x) \, dx+\frac {\left (2 a^6 C+8 b^6 (7 A+6 C)+a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)\right ) \int \sec ^2(c+d x) \, dx}{105 b^2}\\ &=\frac {a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {a \left (4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{420 b d}+\frac {\left (2 a^4 C+8 b^4 (7 A+6 C)+3 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{210 b^2 d}+\frac {a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{210 b^2 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}-\frac {\left (2 a^6 C+8 b^6 (7 A+6 C)+a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)\right ) \text {Subst}(\int 1 \, dx,x,-\tan (c+d x))}{105 b^2 d}\\ &=\frac {a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \tanh ^{-1}(\sin (c+d x))}{4 d}+\frac {\left (2 a^6 C+8 b^6 (7 A+6 C)+a^4 b^2 (42 A+23 C)+8 a^2 b^4 (49 A+39 C)\right ) \tan (c+d x)}{105 b^2 d}+\frac {a \left (4 a^4 C+12 a^2 b^2 (7 A+4 C)+b^4 (406 A+333 C)\right ) \sec (c+d x) \tan (c+d x)}{420 b d}+\frac {\left (2 a^4 C+8 b^4 (7 A+6 C)+3 a^2 b^2 (14 A+9 C)\right ) (a+b \sec (c+d x))^2 \tan (c+d x)}{210 b^2 d}+\frac {a \left (42 A b^2+2 a^2 C+31 b^2 C\right ) (a+b \sec (c+d x))^3 \tan (c+d x)}{210 b^2 d}+\frac {\left (a^2 C+3 b^2 (7 A+6 C)\right ) (a+b \sec (c+d x))^4 \tan (c+d x)}{105 b^2 d}-\frac {a C (a+b \sec (c+d x))^5 \tan (c+d x)}{21 b^2 d}+\frac {C \sec (c+d x) (a+b \sec (c+d x))^5 \tan (c+d x)}{7 b d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 2.78, size = 371, normalized size = 0.97 \begin {gather*} -\frac {\left (C+A \cos ^2(c+d x)\right ) \sec ^6(c+d x) \left (105 a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \cos ^6(c+d x) \left (\log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )-\log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right )-70 a b \left (6 A b^2+6 a^2 C+5 b^2 C\right ) \cos ^2(c+d x) \sin (c+d x)-4 \left (35 a^4 C+42 a^2 b^2 (5 A+4 C)+4 b^4 (7 A+6 C)\right ) \cos ^3(c+d x) \sin (c+d x)-105 a b \left (b^2 (6 A+5 C)+a^2 (8 A+6 C)\right ) \cos ^4(c+d x) \sin (c+d x)-4 \left (35 a^4 (3 A+2 C)+84 a^2 b^2 (5 A+4 C)+8 b^4 (7 A+6 C)\right ) \cos ^5(c+d x) \sin (c+d x)-2 b^2 \left (140 a b C \sin (c+d x)+3 \left (7 A b^2+6 \left (7 a^2+b^2\right ) C\right ) \sin (2 (c+d x))+30 b^2 C \tan (c+d x)\right )\right )}{210 d (A+2 C+A \cos (2 (c+d x)))} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.19, size = 384, normalized size = 1.01 Too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 472, normalized size = 1.24 \begin {gather*} \frac {280 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} C a^{4} + 1680 \, {\left (\tan \left (d x + c\right )^{3} + 3 \, \tan \left (d x + c\right )\right )} A a^{2} b^{2} + 336 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} C a^{2} b^{2} + 56 \, {\left (3 \, \tan \left (d x + c\right )^{5} + 10 \, \tan \left (d x + c\right )^{3} + 15 \, \tan \left (d x + c\right )\right )} A b^{4} + 24 \, {\left (5 \, \tan \left (d x + c\right )^{7} + 21 \, \tan \left (d x + c\right )^{5} + 35 \, \tan \left (d x + c\right )^{3} + 35 \, \tan \left (d x + c\right )\right )} C b^{4} - 35 \, C a b^{3} {\left (\frac {2 \, {\left (15 \, \sin \left (d x + c\right )^{5} - 40 \, \sin \left (d x + c\right )^{3} + 33 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{6} - 3 \, \sin \left (d x + c\right )^{4} + 3 \, \sin \left (d x + c\right )^{2} - 1} - 15 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 15 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 210 \, C a^{3} b {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 210 \, A a b^{3} {\left (\frac {2 \, {\left (3 \, \sin \left (d x + c\right )^{3} - 5 \, \sin \left (d x + c\right )\right )}}{\sin \left (d x + c\right )^{4} - 2 \, \sin \left (d x + c\right )^{2} + 1} - 3 \, \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, \log \left (\sin \left (d x + c\right ) - 1\right )\right )} - 840 \, A a^{3} b {\left (\frac {2 \, \sin \left (d x + c\right )}{\sin \left (d x + c\right )^{2} - 1} - \log \left (\sin \left (d x + c\right ) + 1\right ) + \log \left (\sin \left (d x + c\right ) - 1\right )\right )} + 840 \, A a^{4} \tan \left (d x + c\right )}{840 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 3.22, size = 325, normalized size = 0.85 \begin {gather*} \frac {105 \, {\left (2 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + {\left (6 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{7} \log \left (\sin \left (d x + c\right ) + 1\right ) - 105 \, {\left (2 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + {\left (6 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{7} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 2 \, {\left (4 \, {\left (35 \, {\left (3 \, A + 2 \, C\right )} a^{4} + 84 \, {\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 8 \, {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{6} + 280 \, C a b^{3} \cos \left (d x + c\right ) + 105 \, {\left (2 \, {\left (4 \, A + 3 \, C\right )} a^{3} b + {\left (6 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{5} + 60 \, C b^{4} + 4 \, {\left (35 \, C a^{4} + 42 \, {\left (5 \, A + 4 \, C\right )} a^{2} b^{2} + 4 \, {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{4} + 70 \, {\left (6 \, C a^{3} b + {\left (6 \, A + 5 \, C\right )} a b^{3}\right )} \cos \left (d x + c\right )^{3} + 12 \, {\left (42 \, C a^{2} b^{2} + {\left (7 \, A + 6 \, C\right )} b^{4}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{840 \, d \cos \left (d x + c\right )^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (A + C \sec ^{2}{\left (c + d x \right )}\right ) \left (a + b \sec {\left (c + d x \right )}\right )^{4} \sec ^{2}{\left (c + d x \right )}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 1280 vs.
\(2 (366) = 732\).
time = 0.57, size = 1280, normalized size = 3.36 \begin {gather*} \text {Too large to display} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 7.73, size = 755, normalized size = 1.98 \begin {gather*} \frac {a\,b\,\mathrm {atanh}\left (\frac {a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (8\,A\,a^2+6\,A\,b^2+6\,C\,a^2+5\,C\,b^2\right )}{6\,A\,a\,b^3+8\,A\,a^3\,b+5\,C\,a\,b^3+6\,C\,a^3\,b}\right )\,\left (8\,A\,a^2+6\,A\,b^2+6\,C\,a^2+5\,C\,b^2\right )}{2\,d}-\frac {\left (2\,A\,a^4+2\,A\,b^4+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2+12\,C\,a^2\,b^2-5\,A\,a\,b^3-4\,A\,a^3\,b-\frac {11\,C\,a\,b^3}{2}-5\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{13}+\left (12\,A\,a\,b^3-\frac {20\,A\,b^4}{3}-\frac {28\,C\,a^4}{3}-4\,C\,b^4-56\,A\,a^2\,b^2-40\,C\,a^2\,b^2-12\,A\,a^4+16\,A\,a^3\,b+\frac {14\,C\,a\,b^3}{3}+12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+\left (30\,A\,a^4+\frac {226\,A\,b^4}{15}+\frac {58\,C\,a^4}{3}+\frac {86\,C\,b^4}{5}+116\,A\,a^2\,b^2+\frac {452\,C\,a^2\,b^2}{5}-9\,A\,a\,b^3-20\,A\,a^3\,b-\frac {85\,C\,a\,b^3}{6}-9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+\left (-40\,A\,a^4-\frac {104\,A\,b^4}{5}-24\,C\,a^4-\frac {424\,C\,b^4}{35}-144\,A\,a^2\,b^2-\frac {624\,C\,a^2\,b^2}{5}\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7+\left (30\,A\,a^4+\frac {226\,A\,b^4}{15}+\frac {58\,C\,a^4}{3}+\frac {86\,C\,b^4}{5}+116\,A\,a^2\,b^2+\frac {452\,C\,a^2\,b^2}{5}+9\,A\,a\,b^3+20\,A\,a^3\,b+\frac {85\,C\,a\,b^3}{6}+9\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+\left (-12\,A\,a^4-\frac {20\,A\,b^4}{3}-\frac {28\,C\,a^4}{3}-4\,C\,b^4-56\,A\,a^2\,b^2-40\,C\,a^2\,b^2-12\,A\,a\,b^3-16\,A\,a^3\,b-\frac {14\,C\,a\,b^3}{3}-12\,C\,a^3\,b\right )\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+\left (2\,A\,a^4+2\,A\,b^4+2\,C\,a^4+2\,C\,b^4+12\,A\,a^2\,b^2+12\,C\,a^2\,b^2+5\,A\,a\,b^3+4\,A\,a^3\,b+\frac {11\,C\,a\,b^3}{2}+5\,C\,a^3\,b\right )\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{14}-7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}-35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8+35\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6-21\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4+7\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2-1\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________