3.2.92 \(\int \frac {\tan ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) [192]

Optimal. Leaf size=321 \[ -\frac {\left (8 a^2-5 a b-b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^5 d}-\frac {\left (8 a^2+5 a b-b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^5 d}+\frac {a^3 \left (a^4+13 a^2 b^2+10 b^4\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^5 d}-\frac {a^5}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}-\frac {a^4 \left (a^2+5 b^2\right )}{\left (a^2-b^2\right )^4 d (a+b \sin (c+d x))}+\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 \left (a^2-b^2\right )^3 d}-\frac {\sec ^2(c+d x) \left (8 a^3 \left (a^2+5 b^2\right )-b \left (27 a^4+22 a^2 b^2-b^4\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^4 d} \]

[Out]

-1/16*(8*a^2-5*a*b-b^2)*ln(1-sin(d*x+c))/(a+b)^5/d-1/16*(8*a^2+5*a*b-b^2)*ln(1+sin(d*x+c))/(a-b)^5/d+a^3*(a^4+
13*a^2*b^2+10*b^4)*ln(a+b*sin(d*x+c))/(a^2-b^2)^5/d-1/2*a^5/(a^2-b^2)^3/d/(a+b*sin(d*x+c))^2-a^4*(a^2+5*b^2)/(
a^2-b^2)^4/d/(a+b*sin(d*x+c))+1/4*sec(d*x+c)^4*(a*(a^2+3*b^2)-b*(3*a^2+b^2)*sin(d*x+c))/(a^2-b^2)^3/d-1/8*sec(
d*x+c)^2*(8*a^3*(a^2+5*b^2)-b*(27*a^4+22*a^2*b^2-b^4)*sin(d*x+c))/(a^2-b^2)^4/d

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Rubi [A]
time = 0.80, antiderivative size = 321, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 3, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2800, 1661, 1643} \begin {gather*} -\frac {\left (8 a^2-5 a b-b^2\right ) \log (1-\sin (c+d x))}{16 d (a+b)^5}-\frac {\left (8 a^2+5 a b-b^2\right ) \log (\sin (c+d x)+1)}{16 d (a-b)^5}+\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 d \left (a^2-b^2\right )^3}-\frac {a^5}{2 d \left (a^2-b^2\right )^3 (a+b \sin (c+d x))^2}-\frac {a^4 \left (a^2+5 b^2\right )}{d \left (a^2-b^2\right )^4 (a+b \sin (c+d x))}+\frac {a^3 \left (a^4+13 a^2 b^2+10 b^4\right ) \log (a+b \sin (c+d x))}{d \left (a^2-b^2\right )^5}-\frac {\sec ^2(c+d x) \left (8 a^3 \left (a^2+5 b^2\right )-b \left (27 a^4+22 a^2 b^2-b^4\right ) \sin (c+d x)\right )}{8 d \left (a^2-b^2\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Tan[c + d*x]^5/(a + b*Sin[c + d*x])^3,x]

[Out]

-1/16*((8*a^2 - 5*a*b - b^2)*Log[1 - Sin[c + d*x]])/((a + b)^5*d) - ((8*a^2 + 5*a*b - b^2)*Log[1 + Sin[c + d*x
]])/(16*(a - b)^5*d) + (a^3*(a^4 + 13*a^2*b^2 + 10*b^4)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^5*d) - a^5/(2*(a
^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) - (a^4*(a^2 + 5*b^2))/((a^2 - b^2)^4*d*(a + b*Sin[c + d*x])) + (Sec[c +
d*x]^4*(a*(a^2 + 3*b^2) - b*(3*a^2 + b^2)*Sin[c + d*x]))/(4*(a^2 - b^2)^3*d) - (Sec[c + d*x]^2*(8*a^3*(a^2 + 5
*b^2) - b*(27*a^4 + 22*a^2*b^2 - b^4)*Sin[c + d*x]))/(8*(a^2 - b^2)^4*d)

Rule 1643

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1661

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[(a*g - c*f*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rule 2800

Int[((a_) + (b_.)*sin[(e_.) + (f_.)*(x_)])^(m_.)*tan[(e_.) + (f_.)*(x_)]^(p_.), x_Symbol] :> Dist[1/f, Subst[I
nt[(x^p*(a + x)^m)/(b^2 - x^2)^((p + 1)/2), x], x, b*Sin[e + f*x]], x] /; FreeQ[{a, b, e, f, m}, x] && NeQ[a^2
- b^2, 0] && IntegerQ[(p + 1)/2]

Rubi steps

\begin {align*} \int \frac {\tan ^5(c+d x)}{(a+b \sin (c+d x))^3} \, dx &=\frac {\text {Subst}\left (\int \frac {x^5}{(a+x)^3 \left (b^2-x^2\right )^3} \, dx,x,b \sin (c+d x)\right )}{d}\\ &=\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 \left (a^2-b^2\right )^3 d}+\frac {\text {Subst}\left (\int \frac {\frac {a^3 b^6 \left (3 a^2+b^2\right )}{\left (a^2-b^2\right )^3}-\frac {a^2 b^4 \left (4 a^4+3 a^2 b^2-3 b^4\right ) x}{\left (a^2-b^2\right )^3}-\frac {a b^6 \left (23 a^2-3 b^2\right ) x^2}{\left (a^2-b^2\right )^3}-\frac {b^2 \left (4 a^6-12 a^4 b^2+21 a^2 b^4-b^6\right ) x^3}{\left (a^2-b^2\right )^3}}{(a+x)^3 \left (b^2-x^2\right )^2} \, dx,x,b \sin (c+d x)\right )}{4 b^2 d}\\ &=\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 \left (a^2-b^2\right )^3 d}-\frac {\sec ^2(c+d x) \left (8 a^3 \left (a^2+5 b^2\right )-b \left (27 a^4+22 a^2 b^2-b^4\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^4 d}+\frac {\text {Subst}\left (\int \frac {-\frac {a^3 b^6 \left (21 a^4+26 a^2 b^2+b^4\right )}{\left (a^2-b^2\right )^4}+\frac {a^2 b^4 \left (8 a^6+a^4 b^2-54 a^2 b^4-3 b^6\right ) x}{\left (a^2-b^2\right )^4}+\frac {a b^6 \left (65 a^4-14 a^2 b^2-3 b^4\right ) x^2}{\left (a^2-b^2\right )^4}+\frac {b^6 \left (27 a^4+22 a^2 b^2-b^4\right ) x^3}{\left (a^2-b^2\right )^4}}{(a+x)^3 \left (b^2-x^2\right )} \, dx,x,b \sin (c+d x)\right )}{8 b^4 d}\\ &=\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 \left (a^2-b^2\right )^3 d}-\frac {\sec ^2(c+d x) \left (8 a^3 \left (a^2+5 b^2\right )-b \left (27 a^4+22 a^2 b^2-b^4\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^4 d}+\frac {\text {Subst}\left (\int \left (-\frac {b^4 \left (-8 a^2+5 a b+b^2\right )}{2 (a+b)^5 (b-x)}+\frac {8 a^5 b^4}{\left (a^2-b^2\right )^3 (a+x)^3}+\frac {8 a^4 b^4 \left (a^2+5 b^2\right )}{\left (a^2-b^2\right )^4 (a+x)^2}+\frac {8 a^3 b^4 \left (a^4+13 a^2 b^2+10 b^4\right )}{\left (a^2-b^2\right )^5 (a+x)}-\frac {b^4 \left (8 a^2+5 a b-b^2\right )}{2 (a-b)^5 (b+x)}\right ) \, dx,x,b \sin (c+d x)\right )}{8 b^4 d}\\ &=-\frac {\left (8 a^2-5 a b-b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^5 d}-\frac {\left (8 a^2+5 a b-b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^5 d}+\frac {a^3 \left (a^4+13 a^2 b^2+10 b^4\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^5 d}-\frac {a^5}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}-\frac {a^4 \left (a^2+5 b^2\right )}{\left (a^2-b^2\right )^4 d (a+b \sin (c+d x))}+\frac {\sec ^4(c+d x) \left (a \left (a^2+3 b^2\right )-b \left (3 a^2+b^2\right ) \sin (c+d x)\right )}{4 \left (a^2-b^2\right )^3 d}-\frac {\sec ^2(c+d x) \left (8 a^3 \left (a^2+5 b^2\right )-b \left (27 a^4+22 a^2 b^2-b^4\right ) \sin (c+d x)\right )}{8 \left (a^2-b^2\right )^4 d}\\ \end {align*}

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Mathematica [A]
time = 6.24, size = 304, normalized size = 0.95 \begin {gather*} -\frac {\left (8 a^2-5 a b-b^2\right ) \log (1-\sin (c+d x))}{16 (a+b)^5 d}-\frac {\left (8 a^2+5 a b-b^2\right ) \log (1+\sin (c+d x))}{16 (a-b)^5 d}+\frac {a^3 \left (a^4+13 a^2 b^2+10 b^4\right ) \log (a+b \sin (c+d x))}{\left (a^2-b^2\right )^5 d}+\frac {1}{16 (a+b)^3 d (1-\sin (c+d x))^2}-\frac {7 a+b}{16 (a+b)^4 d (1-\sin (c+d x))}+\frac {1}{16 (a-b)^3 d (1+\sin (c+d x))^2}-\frac {7 a-b}{16 (a-b)^4 d (1+\sin (c+d x))}-\frac {a^5}{2 \left (a^2-b^2\right )^3 d (a+b \sin (c+d x))^2}-\frac {a^4 \left (a^2+5 b^2\right )}{\left (a^2-b^2\right )^4 d (a+b \sin (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Tan[c + d*x]^5/(a + b*Sin[c + d*x])^3,x]

[Out]

-1/16*((8*a^2 - 5*a*b - b^2)*Log[1 - Sin[c + d*x]])/((a + b)^5*d) - ((8*a^2 + 5*a*b - b^2)*Log[1 + Sin[c + d*x
]])/(16*(a - b)^5*d) + (a^3*(a^4 + 13*a^2*b^2 + 10*b^4)*Log[a + b*Sin[c + d*x]])/((a^2 - b^2)^5*d) + 1/(16*(a
+ b)^3*d*(1 - Sin[c + d*x])^2) - (7*a + b)/(16*(a + b)^4*d*(1 - Sin[c + d*x])) + 1/(16*(a - b)^3*d*(1 + Sin[c
+ d*x])^2) - (7*a - b)/(16*(a - b)^4*d*(1 + Sin[c + d*x])) - a^5/(2*(a^2 - b^2)^3*d*(a + b*Sin[c + d*x])^2) -
(a^4*(a^2 + 5*b^2))/((a^2 - b^2)^4*d*(a + b*Sin[c + d*x]))

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Maple [A]
time = 0.98, size = 263, normalized size = 0.82

method	result	size
derivativedivides	\(\frac {\frac {1}{16 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {-b -7 a}{16 \left (a +b \right )^{4} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-8 a^{2}+5 a b +b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{5}}-\frac {a^{5}}{2 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {a^{3} \left (a^{4}+13 a^{2} b^{2}+10 b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{5} \left (a -b \right )^{5}}-\frac {a^{4} \left (a^{2}+5 b^{2}\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )}+\frac {1}{16 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {-b +7 a}{16 \left (a -b \right )^{4} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (-8 a^{2}-5 a b +b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{5}}}{d}\)	\(263\)
default	\(\frac {\frac {1}{16 \left (a +b \right )^{3} \left (\sin \left (d x +c \right )-1\right )^{2}}-\frac {-b -7 a}{16 \left (a +b \right )^{4} \left (\sin \left (d x +c \right )-1\right )}+\frac {\left (-8 a^{2}+5 a b +b^{2}\right ) \ln \left (\sin \left (d x +c \right )-1\right )}{16 \left (a +b \right )^{5}}-\frac {a^{5}}{2 \left (a +b \right )^{3} \left (a -b \right )^{3} \left (a +b \sin \left (d x +c \right )\right )^{2}}+\frac {a^{3} \left (a^{4}+13 a^{2} b^{2}+10 b^{4}\right ) \ln \left (a +b \sin \left (d x +c \right )\right )}{\left (a +b \right )^{5} \left (a -b \right )^{5}}-\frac {a^{4} \left (a^{2}+5 b^{2}\right )}{\left (a +b \right )^{4} \left (a -b \right )^{4} \left (a +b \sin \left (d x +c \right )\right )}+\frac {1}{16 \left (a -b \right )^{3} \left (1+\sin \left (d x +c \right )\right )^{2}}-\frac {-b +7 a}{16 \left (a -b \right )^{4} \left (1+\sin \left (d x +c \right )\right )}+\frac {\left (-8 a^{2}-5 a b +b^{2}\right ) \ln \left (1+\sin \left (d x +c \right )\right )}{16 \left (a -b \right )^{5}}}{d}\)	\(263\)
risch	\(\text {Expression too large to display}\)	\(2267\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(d*x+c)^5/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)

[Out]

1/d*(1/16/(a+b)^3/(sin(d*x+c)-1)^2-1/16*(-b-7*a)/(a+b)^4/(sin(d*x+c)-1)+1/16/(a+b)^5*(-8*a^2+5*a*b+b^2)*ln(sin
(d*x+c)-1)-1/2*a^5/(a+b)^3/(a-b)^3/(a+b*sin(d*x+c))^2+a^3*(a^4+13*a^2*b^2+10*b^4)/(a+b)^5/(a-b)^5*ln(a+b*sin(d
*x+c))-a^4*(a^2+5*b^2)/(a+b)^4/(a-b)^4/(a+b*sin(d*x+c))+1/16/(a-b)^3/(1+sin(d*x+c))^2-1/16*(-b+7*a)/(a-b)^4/(1
+sin(d*x+c))+1/16/(a-b)^5*(-8*a^2-5*a*b+b^2)*ln(1+sin(d*x+c)))

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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 730 vs. \(2 (311) = 622\).
time = 0.63, size = 730, normalized size = 2.27 \begin {gather*} \frac {\frac {16 \, {\left (a^{7} + 13 \, a^{5} b^{2} + 10 \, a^{3} b^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right )}{a^{10} - 5 \, a^{8} b^{2} + 10 \, a^{6} b^{4} - 10 \, a^{4} b^{6} + 5 \, a^{2} b^{8} - b^{10}} - \frac {{\left (8 \, a^{2} + 5 \, a b - b^{2}\right )} \log \left (\sin \left (d x + c\right ) + 1\right )}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}} - \frac {{\left (8 \, a^{2} - 5 \, a b - b^{2}\right )} \log \left (\sin \left (d x + c\right ) - 1\right )}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}} - \frac {2 \, {\left (18 \, a^{7} + 72 \, a^{5} b^{2} + 6 \, a^{3} b^{4} + {\left (8 \, a^{6} b + 67 \, a^{4} b^{3} + 22 \, a^{2} b^{5} - b^{7}\right )} \sin \left (d x + c\right )^{5} + 2 \, {\left (6 \, a^{7} + 41 \, a^{5} b^{2} + 2 \, a^{3} b^{4} - a b^{6}\right )} \sin \left (d x + c\right )^{4} - {\left (5 \, a^{6} b + 159 \, a^{4} b^{3} + 27 \, a^{2} b^{5} + b^{7}\right )} \sin \left (d x + c\right )^{3} - 4 \, {\left (8 \, a^{7} + 37 \, a^{5} b^{2} + 4 \, a^{3} b^{4} - a b^{6}\right )} \sin \left (d x + c\right )^{2} - {\left (a^{6} b - 86 \, a^{4} b^{3} - 11 \, a^{2} b^{5}\right )} \sin \left (d x + c\right )\right )}}{a^{10} - 4 \, a^{8} b^{2} + 6 \, a^{6} b^{4} - 4 \, a^{4} b^{6} + a^{2} b^{8} + {\left (a^{8} b^{2} - 4 \, a^{6} b^{4} + 6 \, a^{4} b^{6} - 4 \, a^{2} b^{8} + b^{10}\right )} \sin \left (d x + c\right )^{6} + 2 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \sin \left (d x + c\right )^{5} + {\left (a^{10} - 6 \, a^{8} b^{2} + 14 \, a^{6} b^{4} - 16 \, a^{4} b^{6} + 9 \, a^{2} b^{8} - 2 \, b^{10}\right )} \sin \left (d x + c\right )^{4} - 4 \, {\left (a^{9} b - 4 \, a^{7} b^{3} + 6 \, a^{5} b^{5} - 4 \, a^{3} b^{7} + a b^{9}\right )} \sin \left (d x + c\right )^{3} - {\left (2 \, a^{10} - 9 \, a^{8} b^{2} + 16 \, a^{6} b^{4} - 14 \, a^{4} b^{6} + 6 \, a^{2} b^{8} - b^{10}\right )} \sin \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 )} \sin \left (d x + c\right )}}{16 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5/(a+b*sin(d*x+c))^3,x, algorithm="maxima")

[Out]

1/16*(16*(a^7 + 13*a^5*b^2 + 10*a^3*b^4)*log(b*sin(d*x + c) + a)/(a^10 - 5*a^8*b^2 + 10*a^6*b^4 - 10*a^4*b^6 +
5*a^2*b^8 - b^10) - (8*a^2 + 5*a*b - b^2)*log(sin(d*x + c) + 1)/(a^5 - 5*a^4*b + 10*a^3*b^2 - 10*a^2*b^3 + 5*
a*b^4 - b^5) - (8*a^2 - 5*a*b - b^2)*log(sin(d*x + c) - 1)/(a^5 + 5*a^4*b + 10*a^3*b^2 + 10*a^2*b^3 + 5*a*b^4
+ b^5) - 2*(18*a^7 + 72*a^5*b^2 + 6*a^3*b^4 + (8*a^6*b + 67*a^4*b^3 + 22*a^2*b^5 - b^7)*sin(d*x + c)^5 + 2*(6*
a^7 + 41*a^5*b^2 + 2*a^3*b^4 - a*b^6)*sin(d*x + c)^4 - (5*a^6*b + 159*a^4*b^3 + 27*a^2*b^5 + b^7)*sin(d*x + c)
^3 - 4*(8*a^7 + 37*a^5*b^2 + 4*a^3*b^4 - a*b^6)*sin(d*x + c)^2 - (a^6*b - 86*a^4*b^3 - 11*a^2*b^5)*sin(d*x + c
))/(a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8 + (a^8*b^2 - 4*a^6*b^4 + 6*a^4*b^6 - 4*a^2*b^8 + b^10)*
sin(d*x + c)^6 + 2*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*sin(d*x + c)^5 + (a^10 - 6*a^8*b^2 + 14
*a^6*b^4 - 16*a^4*b^6 + 9*a^2*b^8 - 2*b^10)*sin(d*x + c)^4 - 4*(a^9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*
b^9)*sin(d*x + c)^3 - (2*a^10 - 9*a^8*b^2 + 16*a^6*b^4 - 14*a^4*b^6 + 6*a^2*b^8 - b^10)*sin(d*x + c)^2 + 2*(a^
9*b - 4*a^7*b^3 + 6*a^5*b^5 - 4*a^3*b^7 + a*b^9)*sin(d*x + c)))/d

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 981 vs. \(2 (311) = 622\).
time = 0.75, size = 981, normalized size = 3.06 \begin {gather*} -\frac {4 \, a^{9} - 16 \, a^{7} b^{2} + 24 \, a^{5} b^{4} - 16 \, a^{3} b^{6} + 4 \, a b^{8} - 4 \, {\left (6 \, a^{9} + 35 \, a^{7} b^{2} - 39 \, a^{5} b^{4} - 3 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (d x + c\right )^{4} - 16 \, {\left (a^{9} - 3 \, a^{7} b^{2} + 3 \, a^{5} b^{4} - a^{3} b^{6}\right )} \cos \left (d x + c\right )^{2} - 16 \, {\left ({\left (a^{7} b^{2} + 13 \, a^{5} b^{4} + 10 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (a^{8} b + 13 \, a^{6} b^{3} + 10 \, a^{4} b^{5}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (a^{9} + 14 \, a^{7} b^{2} + 23 \, a^{5} b^{4} + 10 \, a^{3} b^{6}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (b \sin \left (d x + c\right ) + a\right ) + {\left ({\left (8 \, a^{7} b^{2} + 45 \, a^{6} b^{3} + 104 \, a^{5} b^{4} + 125 \, a^{4} b^{5} + 80 \, a^{3} b^{6} + 23 \, a^{2} b^{7} - b^{9}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (8 \, a^{8} b + 45 \, a^{7} b^{2} + 104 \, a^{6} b^{3} + 125 \, a^{5} b^{4} + 80 \, a^{4} b^{5} + 23 \, a^{3} b^{6} - a b^{8}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (8 \, a^{9} + 45 \, a^{8} b + 112 \, a^{7} b^{2} + 170 \, a^{6} b^{3} + 184 \, a^{5} b^{4} + 148 \, a^{4} b^{5} + 80 \, a^{3} b^{6} + 22 \, a^{2} b^{7} - b^{9}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + {\left ({\left (8 \, a^{7} b^{2} - 45 \, a^{6} b^{3} + 104 \, a^{5} b^{4} - 125 \, a^{4} b^{5} + 80 \, a^{3} b^{6} - 23 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{6} - 2 \, {\left (8 \, a^{8} b - 45 \, a^{7} b^{2} + 104 \, a^{6} b^{3} - 125 \, a^{5} b^{4} + 80 \, a^{4} b^{5} - 23 \, a^{3} b^{6} + a b^{8}\right )} \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (8 \, a^{9} - 45 \, a^{8} b + 112 \, a^{7} b^{2} - 170 \, a^{6} b^{3} + 184 \, a^{5} b^{4} - 148 \, a^{4} b^{5} + 80 \, a^{3} b^{6} - 22 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{4}\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) - 2 \, {\left (2 \, a^{8} b - 8 \, a^{6} b^{3} + 12 \, a^{4} b^{5} - 8 \, a^{2} b^{7} + 2 \, b^{9} + {\left (8 \, a^{8} b + 59 \, a^{6} b^{3} - 45 \, a^{4} b^{5} - 23 \, a^{2} b^{7} + b^{9}\right )} \cos \left (d x + c\right )^{4} - {\left (11 \, a^{8} b - 36 \, a^{6} b^{3} + 42 \, a^{4} b^{5} - 20 \, a^{2} b^{7} + 3 \, b^{9}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )}{16 \, {\left ({\left (a^{10} b^{2} - 5 \, a^{8} b^{4} + 10 \, a^{6} b^{6} - 10 \, a^{4} b^{8} + 5 \, a^{2} b^{10} - b^{12}\right )} d \cos \left (d x + c\right )^{6} - 2 \, {\left (a^{11} b - 5 \, a^{9} b^{3} + 10 \, a^{7} b^{5} - 10 \, a^{5} b^{7} + 5 \, a^{3} b^{9} - a b^{11}\right )} d \cos \left (d x + c\right )^{4} \sin \left (d x + c\right ) - {\left (a^{12} - 4 \, a^{10} b^{2} + 5 \, a^{8} b^{4} - 5 \, a^{4} b^{8} + 4 \, a^{2} b^{10} - b^{12}\right )} d \cos \left (d x + c\right )^{4}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/16*(4*a^9 - 16*a^7*b^2 + 24*a^5*b^4 - 16*a^3*b^6 + 4*a*b^8 - 4*(6*a^9 + 35*a^7*b^2 - 39*a^5*b^4 - 3*a^3*b^6
+ a*b^8)*cos(d*x + c)^4 - 16*(a^9 - 3*a^7*b^2 + 3*a^5*b^4 - a^3*b^6)*cos(d*x + c)^2 - 16*((a^7*b^2 + 13*a^5*b
^4 + 10*a^3*b^6)*cos(d*x + c)^6 - 2*(a^8*b + 13*a^6*b^3 + 10*a^4*b^5)*cos(d*x + c)^4*sin(d*x + c) - (a^9 + 14*
a^7*b^2 + 23*a^5*b^4 + 10*a^3*b^6)*cos(d*x + c)^4)*log(b*sin(d*x + c) + a) + ((8*a^7*b^2 + 45*a^6*b^3 + 104*a^
5*b^4 + 125*a^4*b^5 + 80*a^3*b^6 + 23*a^2*b^7 - b^9)*cos(d*x + c)^6 - 2*(8*a^8*b + 45*a^7*b^2 + 104*a^6*b^3 +
125*a^5*b^4 + 80*a^4*b^5 + 23*a^3*b^6 - a*b^8)*cos(d*x + c)^4*sin(d*x + c) - (8*a^9 + 45*a^8*b + 112*a^7*b^2 +
170*a^6*b^3 + 184*a^5*b^4 + 148*a^4*b^5 + 80*a^3*b^6 + 22*a^2*b^7 - b^9)*cos(d*x + c)^4)*log(sin(d*x + c) + 1
) + ((8*a^7*b^2 - 45*a^6*b^3 + 104*a^5*b^4 - 125*a^4*b^5 + 80*a^3*b^6 - 23*a^2*b^7 + b^9)*cos(d*x + c)^6 - 2*(
8*a^8*b - 45*a^7*b^2 + 104*a^6*b^3 - 125*a^5*b^4 + 80*a^4*b^5 - 23*a^3*b^6 + a*b^8)*cos(d*x + c)^4*sin(d*x + c
) - (8*a^9 - 45*a^8*b + 112*a^7*b^2 - 170*a^6*b^3 + 184*a^5*b^4 - 148*a^4*b^5 + 80*a^3*b^6 - 22*a^2*b^7 + b^9)
*cos(d*x + c)^4)*log(-sin(d*x + c) + 1) - 2*(2*a^8*b - 8*a^6*b^3 + 12*a^4*b^5 - 8*a^2*b^7 + 2*b^9 + (8*a^8*b +
59*a^6*b^3 - 45*a^4*b^5 - 23*a^2*b^7 + b^9)*cos(d*x + c)^4 - (11*a^8*b - 36*a^6*b^3 + 42*a^4*b^5 - 20*a^2*b^7
+ 3*b^9)*cos(d*x + c)^2)*sin(d*x + c))/((a^10*b^2 - 5*a^8*b^4 + 10*a^6*b^6 - 10*a^4*b^8 + 5*a^2*b^10 - b^12)*
d*cos(d*x + c)^6 - 2*(a^11*b - 5*a^9*b^3 + 10*a^7*b^5 - 10*a^5*b^7 + 5*a^3*b^9 - a*b^11)*d*cos(d*x + c)^4*sin(
d*x + c) - (a^12 - 4*a^10*b^2 + 5*a^8*b^4 - 5*a^4*b^8 + 4*a^2*b^10 - b^12)*d*cos(d*x + c)^4)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\tan ^{5}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)**5/(a+b*sin(d*x+c))**3,x)

[Out]

Integral(tan(c + d*x)**5/(a + b*sin(c + d*x))**3, x)

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Giac [A]
time = 14.52, size = 585, normalized size = 1.82 \begin {gather*} \frac {\frac {16 \, {\left (a^{7} b + 13 \, a^{5} b^{3} + 10 \, a^{3} b^{5}\right )} \log \left ({\left | b \sin \left (d x + c\right ) + a \right |}\right )}{a^{10} b - 5 \, a^{8} b^{3} + 10 \, a^{6} b^{5} - 10 \, a^{4} b^{7} + 5 \, a^{2} b^{9} - b^{11}} - \frac {{\left (8 \, a^{2} + 5 \, a b - b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) + 1 \right |}\right )}{a^{5} - 5 \, a^{4} b + 10 \, a^{3} b^{2} - 10 \, a^{2} b^{3} + 5 \, a b^{4} - b^{5}} - \frac {{\left (8 \, a^{2} - 5 \, a b - b^{2}\right )} \log \left ({\left | \sin \left (d x + c\right ) - 1 \right |}\right )}{a^{5} + 5 \, a^{4} b + 10 \, a^{3} b^{2} + 10 \, a^{2} b^{3} + 5 \, a b^{4} + b^{5}} - \frac {2 \, {\left (8 \, a^{6} b \sin \left (d x + c\right )^{5} + 67 \, a^{4} b^{3} \sin \left (d x + c\right )^{5} + 22 \, a^{2} b^{5} \sin \left (d x + c\right )^{5} - b^{7} \sin \left (d x + c\right )^{5} + 12 \, a^{7} \sin \left (d x + c\right )^{4} + 82 \, a^{5} b^{2} \sin \left (d x + c\right )^{4} + 4 \, a^{3} b^{4} \sin \left (d x + c\right )^{4} - 2 \, a b^{6} \sin \left (d x + c\right )^{4} - 5 \, a^{6} b \sin \left (d x + c\right )^{3} - 159 \, a^{4} b^{3} \sin \left (d x + c\right )^{3} - 27 \, a^{2} b^{5} \sin \left (d x + c\right )^{3} - b^{7} \sin \left (d x + c\right )^{3} - 32 \, a^{7} \sin \left (d x + c\right )^{2} - 148 \, a^{5} b^{2} \sin \left (d x + c\right )^{2} - 16 \, a^{3} b^{4} \sin \left (d x + c\right )^{2} + 4 \, a b^{6} \sin \left (d x + c\right )^{2} - a^{6} b \sin \left (d x + c\right ) + 86 \, a^{4} b^{3} \sin \left (d x + c\right ) + 11 \, a^{2} b^{5} \sin \left (d x + c\right ) + 18 \, a^{7} + 72 \, a^{5} b^{2} + 6 \, a^{3} b^{4}\right )}}{{\left (a^{8} - 4 \, a^{6} b^{2} + 6 \, a^{4} b^{4} - 4 \, a^{2} b^{6} + b^{8}\right )} {\left (b \sin \left (d x + c\right )^{3} + a \sin \left (d x + c\right )^{2} - b \sin \left (d x + c\right ) - a\right )}^{2}}}{16 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(tan(d*x+c)^5/(a+b*sin(d*x+c))^3,x, algorithm="giac")

[Out]

1/16*(16*(a^7*b + 13*a^5*b^3 + 10*a^3*b^5)*log(abs(b*sin(d*x + c) + a))/(a^10*b - 5*a^8*b^3 + 10*a^6*b^5 - 10*
a^4*b^7 + 5*a^2*b^9 - b^11) - (8*a^2 + 5*a*b - b^2)*log(abs(sin(d*x + c) + 1))/(a^5 - 5*a^4*b + 10*a^3*b^2 - 1
0*a^2*b^3 + 5*a*b^4 - b^5) - (8*a^2 - 5*a*b - b^2)*log(abs(sin(d*x + c) - 1))/(a^5 + 5*a^4*b + 10*a^3*b^2 + 10
*a^2*b^3 + 5*a*b^4 + b^5) - 2*(8*a^6*b*sin(d*x + c)^5 + 67*a^4*b^3*sin(d*x + c)^5 + 22*a^2*b^5*sin(d*x + c)^5
- b^7*sin(d*x + c)^5 + 12*a^7*sin(d*x + c)^4 + 82*a^5*b^2*sin(d*x + c)^4 + 4*a^3*b^4*sin(d*x + c)^4 - 2*a*b^6*
sin(d*x + c)^4 - 5*a^6*b*sin(d*x + c)^3 - 159*a^4*b^3*sin(d*x + c)^3 - 27*a^2*b^5*sin(d*x + c)^3 - b^7*sin(d*x
+ c)^3 - 32*a^7*sin(d*x + c)^2 - 148*a^5*b^2*sin(d*x + c)^2 - 16*a^3*b^4*sin(d*x + c)^2 + 4*a*b^6*sin(d*x + c
)^2 - a^6*b*sin(d*x + c) + 86*a^4*b^3*sin(d*x + c) + 11*a^2*b^5*sin(d*x + c) + 18*a^7 + 72*a^5*b^2 + 6*a^3*b^4
)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(b*sin(d*x + c)^3 + a*sin(d*x + c)^2 - b*sin(d*x + c) - a)^
2))/d

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Mupad [B]
time = 10.74, size = 1229, normalized size = 3.83 \begin {gather*} \frac {\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (-2\,a^7+11\,a^5\,b^2+38\,a^3\,b^4+a\,b^6\right )}{a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (-a^7+12\,a^5\,b^2+33\,a^3\,b^4+4\,a\,b^6\right )}{a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8}-\frac {4\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (-a^7+12\,a^5\,b^2+33\,a^3\,b^4+4\,a\,b^6\right )}{a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8}+\frac {{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (-2\,a^7+11\,a^5\,b^2+38\,a^3\,b^4+a\,b^6\right )}{a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8}+\frac {2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (6\,a^7+13\,a^5\,b^2+118\,a^3\,b^4+7\,a\,b^6\right )}{a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}\,\left (37\,a^6+58\,a^4\,b^2+a^2\,b^4\right )}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5\,\left (7\,a^6+132\,a^4\,b^2-57\,a^2\,b^4+14\,b^6\right )}{2\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7\,\left (7\,a^6+132\,a^4\,b^2-57\,a^2\,b^4+14\,b^6\right )}{2\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3\,\left (83\,a^6+226\,a^4\,b^2-17\,a^2\,b^4-4\,b^6\right )}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}-\frac {b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9\,\left (83\,a^6+226\,a^4\,b^2-17\,a^2\,b^4-4\,b^6\right )}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}+\frac {b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\,\left (37\,a^6+58\,a^4\,b^2+a^2\,b^4\right )}{4\,\left (a^8-4\,a^6\,b^2+6\,a^4\,b^4-4\,a^2\,b^6+b^8\right )}}{d\,\left ({\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^6\,\left (4\,a^2+24\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{10}\,\left (2\,a^2-4\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2\,\left (2\,a^2-4\,b^2\right )+a^2\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{12}+a^2-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^4\,\left (a^2+16\,b^2\right )-{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^8\,\left (a^2+16\,b^2\right )-12\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^3+8\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^5+8\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^7-12\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^9+4\,a\,b\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^{11}+4\,a\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )\right )}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+1\right )\,\left (\frac {3\,b^2}{2\,{\left (a-b\right )}^5}+\frac {21\,b}{8\,{\left (a-b\right )}^4}+\frac {1}{{\left (a-b\right )}^3}\right )}{d}-\frac {\ln \left (\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )-1\right )\,\left (\frac {1}{{\left (a+b\right )}^3}-\frac {21\,b}{8\,{\left (a+b\right )}^4}+\frac {3\,b^2}{2\,{\left (a+b\right )}^5}\right )}{d}+\frac {\ln \left (a\,{\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )}^2+2\,b\,\mathrm {tan}\left (\frac {c}{2}+\frac {d\,x}{2}\right )+a\right )\,\left (a^7+13\,a^5\,b^2+10\,a^3\,b^4\right )}{d\,\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 )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(tan(c + d*x)^5/(a + b*sin(c + d*x))^3,x)

[Out]

((tan(c/2 + (d*x)/2)^2*(a*b^6 - 2*a^7 + 38*a^3*b^4 + 11*a^5*b^2))/(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b
^2) - (4*tan(c/2 + (d*x)/2)^4*(4*a*b^6 - a^7 + 33*a^3*b^4 + 12*a^5*b^2))/(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 -
4*a^6*b^2) - (4*tan(c/2 + (d*x)/2)^8*(4*a*b^6 - a^7 + 33*a^3*b^4 + 12*a^5*b^2))/(a^8 + b^8 - 4*a^2*b^6 + 6*a^4
*b^4 - 4*a^6*b^2) + (tan(c/2 + (d*x)/2)^10*(a*b^6 - 2*a^7 + 38*a^3*b^4 + 11*a^5*b^2))/(a^8 + b^8 - 4*a^2*b^6 +
6*a^4*b^4 - 4*a^6*b^2) + (2*tan(c/2 + (d*x)/2)^6*(7*a*b^6 + 6*a^7 + 118*a^3*b^4 + 13*a^5*b^2))/(a^8 + b^8 - 4
*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2) + (b*tan(c/2 + (d*x)/2)^11*(37*a^6 + a^2*b^4 + 58*a^4*b^2))/(4*(a^8 + b^8 -
4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (b*tan(c/2 + (d*x)/2)^5*(7*a^6 + 14*b^6 - 57*a^2*b^4 + 132*a^4*b^2))/(2*
(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (b*tan(c/2 + (d*x)/2)^7*(7*a^6 + 14*b^6 - 57*a^2*b^4 + 132*
a^4*b^2))/(2*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (b*tan(c/2 + (d*x)/2)^3*(83*a^6 - 4*b^6 - 17*a
^2*b^4 + 226*a^4*b^2))/(4*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (b*tan(c/2 + (d*x)/2)^9*(83*a^6 -
4*b^6 - 17*a^2*b^4 + 226*a^4*b^2))/(4*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) + (b*tan(c/2 + (d*x)/2
)*(37*a^6 + a^2*b^4 + 58*a^4*b^2))/(4*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)))/(d*(tan(c/2 + (d*x)/2)
^6*(4*a^2 + 24*b^2) - tan(c/2 + (d*x)/2)^10*(2*a^2 - 4*b^2) - tan(c/2 + (d*x)/2)^2*(2*a^2 - 4*b^2) + a^2*tan(c
/2 + (d*x)/2)^12 + a^2 - tan(c/2 + (d*x)/2)^4*(a^2 + 16*b^2) - tan(c/2 + (d*x)/2)^8*(a^2 + 16*b^2) - 12*a*b*ta
n(c/2 + (d*x)/2)^3 + 8*a*b*tan(c/2 + (d*x)/2)^5 + 8*a*b*tan(c/2 + (d*x)/2)^7 - 12*a*b*tan(c/2 + (d*x)/2)^9 + 4
*a*b*tan(c/2 + (d*x)/2)^11 + 4*a*b*tan(c/2 + (d*x)/2))) - (log(tan(c/2 + (d*x)/2) + 1)*((3*b^2)/(2*(a - b)^5)
+ (21*b)/(8*(a - b)^4) + 1/(a - b)^3))/d - (log(tan(c/2 + (d*x)/2) - 1)*(1/(a + b)^3 - (21*b)/(8*(a + b)^4) +
(3*b^2)/(2*(a + b)^5)))/d + (log(a + 2*b*tan(c/2 + (d*x)/2) + a*tan(c/2 + (d*x)/2)^2)*(a^7 + 10*a^3*b^4 + 13*a
^5*b^2))/(d*(a^10 - b^10 + 5*a^2*b^8 - 10*a^4*b^6 + 10*a^6*b^4 - 5*a^8*b^2))

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