Optimal. Leaf size=229 \[ -\frac {b^3}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac {b^2 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}+\frac {(a-2 b) \log (1-\cos (c+d x))}{4 (a+b)^4 d}-\frac {(a+2 b) \log (1+\cos (c+d x))}{4 (a-b)^4 d}+\frac {b \left (3 a^4+8 a^2 b^2+b^4\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.43, antiderivative size = 229, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {3957, 2800,
1661, 1643} \begin {gather*} \frac {b^2 \left (3 a^2+b^2\right )}{d \left (a^2-b^2\right )^3 (a \cos (c+d x)+b)}+\frac {\csc ^2(c+d x) \left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right )}{2 d \left (a^2-b^2\right )^3}-\frac {b^3}{2 d \left (a^2-b^2\right )^2 (a \cos (c+d x)+b)^2}+\frac {b \left (3 a^4+8 a^2 b^2+b^4\right ) \log (a \cos (c+d x)+b)}{d \left (a^2-b^2\right )^4}+\frac {(a-2 b) \log (1-\cos (c+d x))}{4 d (a+b)^4}-\frac {(a+2 b) \log (\cos (c+d x)+1)}{4 d (a-b)^4} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1643
Rule 1661
Rule 2800
Rule 3957
Rubi steps
\begin {align*} \int \frac {\csc ^3(c+d x)}{(a+b \sec (c+d x))^3} \, dx &=-\int \frac {\cot ^3(c+d x)}{(-b-a \cos (c+d x))^3} \, dx\\ &=\frac {\text {Subst}\left (\int \frac {x^3}{(-b+x)^3 \left (a^2-x^2\right )^2} \, dx,x,-a \cos (c+d x)\right )}{d}\\ &=\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}+\frac {\text {Subst}\left (\int \frac {\frac {a^4 b^3 \left (a^2+3 b^2\right )}{\left (a^2-b^2\right )^3}-\frac {a^2 b^2 \left (3 a^4+3 a^2 b^2-2 b^4\right ) x}{\left (a^2-b^2\right )^3}+\frac {a^4 b \left (3 a^2-7 b^2\right ) x^2}{\left (a^2-b^2\right )^3}+\frac {a^4 \left (a^2+3 b^2\right ) x^3}{\left (a^2-b^2\right )^3}}{(-b+x)^3 \left (a^2-x^2\right )} \, dx,x,-a \cos (c+d x)\right )}{2 a^2 d}\\ &=\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}+\frac {\text {Subst}\left (\int \left (\frac {a^2 (a+2 b)}{2 (a-b)^4 (a-x)}-\frac {2 a^2 b^3}{\left (a^2-b^2\right )^2 (b-x)^3}+\frac {2 a^2 b^2 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 (b-x)^2}-\frac {2 a^2 b \left (3 a^4+8 a^2 b^2+b^4\right )}{\left (a^2-b^2\right )^4 (b-x)}+\frac {a^2 (a-2 b)}{2 (a+b)^4 (a+x)}\right ) \, dx,x,-a \cos (c+d x)\right )}{2 a^2 d}\\ &=-\frac {b^3}{2 \left (a^2-b^2\right )^2 d (b+a \cos (c+d x))^2}+\frac {b^2 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3 d (b+a \cos (c+d x))}+\frac {\left (b \left (3 a^2+b^2\right )-a \left (a^2+3 b^2\right ) \cos (c+d x)\right ) \csc ^2(c+d x)}{2 \left (a^2-b^2\right )^3 d}+\frac {(a-2 b) \log (1-\cos (c+d x))}{4 (a+b)^4 d}-\frac {(a+2 b) \log (1+\cos (c+d x))}{4 (a-b)^4 d}+\frac {b \left (3 a^4+8 a^2 b^2+b^4\right ) \log (b+a \cos (c+d x))}{\left (a^2-b^2\right )^4 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] Result contains complex when optimal does not.
time = 6.21, size = 332, normalized size = 1.45 \begin {gather*} -\frac {2 i \left (3 a^4 b+8 a^2 b^3+b^5\right ) (c+d x)}{(a-b)^4 (a+b)^4 d}-\frac {i (-a-2 b) \text {ArcTan}(\tan (c+d x))}{2 (-a+b)^4 d}-\frac {i (a-2 b) \text {ArcTan}(\tan (c+d x))}{2 (a+b)^4 d}-\frac {b^3}{2 (-a+b)^2 (a+b)^2 d (b+a \cos (c+d x))^2}-\frac {b^2 \left (3 a^2+b^2\right )}{(-a+b)^3 (a+b)^3 d (b+a \cos (c+d x))}-\frac {\csc ^2\left (\frac {1}{2} (c+d x)\right )}{8 (a+b)^3 d}+\frac {(-a-2 b) \log \left (\cos ^2\left (\frac {1}{2} (c+d x)\right )\right )}{4 (-a+b)^4 d}+\frac {\left (3 a^4 b+8 a^2 b^3+b^5\right ) \log (b+a \cos (c+d x))}{\left (-a^2+b^2\right )^4 d}+\frac {(a-2 b) \log \left (\sin ^2\left (\frac {1}{2} (c+d x)\right )\right )}{4 (a+b)^4 d}-\frac {\sec ^2\left (\frac {1}{2} (c+d x)\right )}{8 (-a+b)^3 d} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.31, size = 196, normalized size = 0.86
method | result | size |
derivativedivides | \(\frac {\frac {1}{4 \left (a -b \right )^{3} \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (-a -2 b \right ) \ln \left (1+\cos \left (d x +c \right )\right )}{4 \left (a -b \right )^{4}}+\frac {1}{4 \left (a +b \right )^{3} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\left (a -2 b \right ) \ln \left (-1+\cos \left (d x +c \right )\right )}{4 \left (a +b \right )^{4}}-\frac {b^{3}}{2 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{4}+8 b^{2} a^{2}+b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {b^{2} \left (3 a^{2}+b^{2}\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}}{d}\) | \(196\) |
default | \(\frac {\frac {1}{4 \left (a -b \right )^{3} \left (1+\cos \left (d x +c \right )\right )}+\frac {\left (-a -2 b \right ) \ln \left (1+\cos \left (d x +c \right )\right )}{4 \left (a -b \right )^{4}}+\frac {1}{4 \left (a +b \right )^{3} \left (-1+\cos \left (d x +c \right )\right )}+\frac {\left (a -2 b \right ) \ln \left (-1+\cos \left (d x +c \right )\right )}{4 \left (a +b \right )^{4}}-\frac {b^{3}}{2 \left (a +b \right )^{2} \left (a -b \right )^{2} \left (b +a \cos \left (d x +c \right )\right )^{2}}+\frac {b \left (3 a^{4}+8 b^{2} a^{2}+b^{4}\right ) \ln \left (b +a \cos \left (d x +c \right )\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4}}+\frac {b^{2} \left (3 a^{2}+b^{2}\right )}{\left (a +b \right )^{3} \left (a -b \right )^{3} \left (b +a \cos \left (d x +c \right )\right )}}{d}\) | \(196\) |
norman | \(\frac {-\frac {1}{8 d \left (a +b \right )}+\frac {\tan ^{8}\left (\frac {d x}{2}+\frac {c}{2}\right )}{8 d \left (a -b \right )}+\frac {\left (a^{5}+22 a^{3} b^{2}+13 a \,b^{4}\right ) \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \left (a^{6}-2 a^{5} b -a^{4} b^{2}+4 a^{3} b^{3}-a^{2} b^{4}-2 a \,b^{5}+b^{6}\right )}-\frac {\left (2 a^{5}+5 a^{4} b +44 a^{3} b^{2}+18 a^{2} b^{3}+26 a \,b^{4}+b^{5}\right ) \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{4 d \left (a^{6}-3 a^{4} b^{2}+3 a^{2} b^{4}-b^{6}\right )}}{\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )^{2}}+\frac {b \left (3 a^{4}+8 b^{2} a^{2}+b^{4}\right ) \ln \left (a \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-b \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )-a -b \right )}{d \left (a^{8}-4 a^{6} b^{2}+6 a^{4} b^{4}-4 b^{6} a^{2}+b^{8}\right )}+\frac {\left (a -2 b \right ) \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{2 d \left (a^{4}+4 b \,a^{3}+6 b^{2} a^{2}+4 b^{3} a +b^{4}\right )}\) | \(382\) |
risch | \(\text {Expression too large to display}\) | \(1332\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [A]
time = 0.29, size = 435, normalized size = 1.90 \begin {gather*} \frac {\frac {4 \, {\left (3 \, a^{4} b + 8 \, a^{2} b^{3} + b^{5}\right )} \log \left (a \cos \left (d x + c\right ) + b\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} - \frac {{\left (a + 2 \, b\right )} \log \left (\cos \left (d x + c\right ) + 1\right )}{a^{4} - 4 \, a^{3} b + 6 \, a^{2} b^{2} - 4 \, a b^{3} + b^{4}} + \frac {{\left (a - 2 \, b\right )} \log \left (\cos \left (d x + c\right ) - 1\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {2 \, {\left (8 \, a^{2} b^{3} + 4 \, b^{5} - {\left (a^{5} + 9 \, a^{3} b^{2} + 2 \, a b^{4}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{4} b - 10 \, a^{2} b^{3} - 3 \, b^{5}\right )} \cos \left (d x + c\right )^{2} + {\left (11 \, a^{3} b^{2} + a b^{4}\right )} \cos \left (d x + c\right )\right )}}{a^{6} b^{2} - 3 \, a^{4} b^{4} + 3 \, a^{2} b^{6} - b^{8} - {\left (a^{8} - 3 \, a^{6} b^{2} + 3 \, a^{4} b^{4} - a^{2} b^{6}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{7} b - 3 \, a^{5} b^{3} + 3 \, a^{3} b^{5} - a b^{7}\right )} \cos \left (d x + c\right )}}{4 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1071 vs.
\(2 (221) = 442\).
time = 4.10, size = 1071, normalized size = 4.68 \begin {gather*} -\frac {16 \, a^{4} b^{3} - 8 \, a^{2} b^{5} - 8 \, b^{7} - 2 \, {\left (a^{7} + 8 \, a^{5} b^{2} - 7 \, a^{3} b^{4} - 2 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + 2 \, {\left (a^{6} b - 11 \, a^{4} b^{3} + 7 \, a^{2} b^{5} + 3 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (11 \, a^{5} b^{2} - 10 \, a^{3} b^{4} - a b^{6}\right )} \cos \left (d x + c\right ) + 4 \, {\left (3 \, a^{4} b^{3} + 8 \, a^{2} b^{5} + b^{7} - {\left (3 \, a^{6} b + 8 \, a^{4} b^{3} + a^{2} b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (3 \, a^{5} b^{2} + 8 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (3 \, a^{6} b + 5 \, a^{4} b^{3} - 7 \, a^{2} b^{5} - b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (3 \, a^{5} b^{2} + 8 \, a^{3} b^{4} + a b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (a \cos \left (d x + c\right ) + b\right ) - {\left (a^{5} b^{2} + 6 \, a^{4} b^{3} + 14 \, a^{3} b^{4} + 16 \, a^{2} b^{5} + 9 \, a b^{6} + 2 \, b^{7} - {\left (a^{7} + 6 \, a^{6} b + 14 \, a^{5} b^{2} + 16 \, a^{4} b^{3} + 9 \, a^{3} b^{4} + 2 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b + 6 \, a^{5} b^{2} + 14 \, a^{4} b^{3} + 16 \, a^{3} b^{4} + 9 \, a^{2} b^{5} + 2 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{7} + 6 \, a^{6} b + 13 \, a^{5} b^{2} + 10 \, a^{4} b^{3} - 5 \, a^{3} b^{4} - 14 \, a^{2} b^{5} - 9 \, a b^{6} - 2 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b + 6 \, a^{5} b^{2} + 14 \, a^{4} b^{3} + 16 \, a^{3} b^{4} + 9 \, a^{2} b^{5} + 2 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right ) + {\left (a^{5} b^{2} - 6 \, a^{4} b^{3} + 14 \, a^{3} b^{4} - 16 \, a^{2} b^{5} + 9 \, a b^{6} - 2 \, b^{7} - {\left (a^{7} - 6 \, a^{6} b + 14 \, a^{5} b^{2} - 16 \, a^{4} b^{3} + 9 \, a^{3} b^{4} - 2 \, a^{2} b^{5}\right )} \cos \left (d x + c\right )^{4} - 2 \, {\left (a^{6} b - 6 \, a^{5} b^{2} + 14 \, a^{4} b^{3} - 16 \, a^{3} b^{4} + 9 \, a^{2} b^{5} - 2 \, a b^{6}\right )} \cos \left (d x + c\right )^{3} + {\left (a^{7} - 6 \, a^{6} b + 13 \, a^{5} b^{2} - 10 \, a^{4} b^{3} - 5 \, a^{3} b^{4} + 14 \, a^{2} b^{5} - 9 \, a b^{6} + 2 \, b^{7}\right )} \cos \left (d x + c\right )^{2} + 2 \, {\left (a^{6} b - 6 \, a^{5} b^{2} + 14 \, a^{4} b^{3} - 16 \, a^{3} b^{4} + 9 \, a^{2} b^{5} - 2 \, a b^{6}\right )} \cos \left (d x + c\right )\right )} \log \left (-\frac {1}{2} \, \cos \left (d x + c\right ) + \frac {1}{2}\right )}{4 \, {\left ({\left (a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8}\right )} d \cos \left (d x + c\right )^{4} + 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right )^{3} - {\left (a^{10} - 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} - 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} - b^{10}\right )} d \cos \left (d x + c\right )^{2} - 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} d \cos \left (d x + c\right ) - {\left (a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right )} d\right )}} \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 \frac {\csc ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{3}}\, 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. 800 vs.
\(2 (221) = 442\).
time = 0.61, size = 800, normalized size = 3.49 \begin {gather*} \frac {\frac {2 \, {\left (a - 2 \, b\right )} \log \left (\frac {{\left | -\cos \left (d x + c\right ) + 1 \right |}}{{\left | \cos \left (d x + c\right ) + 1 \right |}}\right )}{a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}} + \frac {8 \, {\left (3 \, a^{4} b + 8 \, a^{2} b^{3} + b^{5}\right )} \log \left ({\left | -a - b - \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} \right |}\right )}{a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}} + \frac {{\left (a + b - \frac {2 \, a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {4 \, b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}}{{\left (a^{4} + 4 \, a^{3} b + 6 \, a^{2} b^{2} + 4 \, a b^{3} + b^{4}\right )} {\left (\cos \left (d x + c\right ) - 1\right )}} - \frac {\cos \left (d x + c\right ) - 1}{{\left (a^{3} - 3 \, a^{2} b + 3 \, a b^{2} - b^{3}\right )} {\left (\cos \left (d x + c\right ) + 1\right )}} - \frac {4 \, {\left (9 \, a^{6} b + 6 \, a^{5} b^{2} + 9 \, a^{4} b^{3} + 28 \, a^{3} b^{4} + 11 \, a^{2} b^{5} - 2 \, a b^{6} + 3 \, b^{7} + \frac {18 \, a^{6} b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {12 \, a^{5} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {26 \, a^{4} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {4 \, a^{3} b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {38 \, a^{2} b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {8 \, a b^{6} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {6 \, b^{7} {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} + \frac {9 \, a^{6} b {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {18 \, a^{5} b^{2} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {33 \, a^{4} b^{3} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {48 \, a^{3} b^{4} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {27 \, a^{2} b^{5} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {6 \, a b^{6} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, b^{7} {\left (\cos \left (d x + c\right ) - 1\right )}^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (a + b + \frac {a {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1} - \frac {b {\left (\cos \left (d x + c\right ) - 1\right )}}{\cos \left (d x + c\right ) + 1}\right )}^{2}}}{8 \, d} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 1.70, size = 378, normalized size = 1.65 \begin {gather*} \frac {\ln \left (b+a\,\cos \left (c+d\,x\right )\right )\,\left (\frac {3\,b}{4\,{\left (a+b\right )}^4}-\frac {1}{4\,{\left (a+b\right )}^3}+\frac {3\,b}{4\,{\left (a-b\right )}^4}+\frac {1}{4\,{\left (a-b\right )}^3}\right )}{d}-\frac {\ln \left (\cos \left (c+d\,x\right )-1\right )\,\left (\frac {3\,b}{4\,{\left (a+b\right )}^4}-\frac {1}{4\,{\left (a+b\right )}^3}\right )}{d}-\frac {\frac {{\cos \left (c+d\,x\right )}^3\,\left (a^5+9\,a^3\,b^2+2\,a\,b^4\right )}{2\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}+\frac {{\cos \left (c+d\,x\right )}^2\,\left (-a^4\,b+10\,a^2\,b^3+3\,b^5\right )}{2\,\left (a^6-3\,a^4\,b^2+3\,a^2\,b^4-b^6\right )}-\frac {2\,b\,\left (2\,a^2\,b^2+b^4\right )}{\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}-\frac {a\,\cos \left (c+d\,x\right )\,\left (11\,a^2\,b^2+b^4\right )}{2\,\left (a^2-b^2\right )\,\left (a^4-2\,a^2\,b^2+b^4\right )}}{d\,\left ({\cos \left (c+d\,x\right )}^2\,\left (a^2-b^2\right )+b^2-a^2\,{\cos \left (c+d\,x\right )}^4+2\,a\,b\,\cos \left (c+d\,x\right )-2\,a\,b\,{\cos \left (c+d\,x\right )}^3\right )}-\frac {\ln \left (\cos \left (c+d\,x\right )+1\right )\,\left (a+2\,b\right )}{4\,d\,{\left (a-b\right )}^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________