3.3.77 \(\int \frac {(e+f x) \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx\) [277]

Optimal. Leaf size=152 \[ \frac {f \tanh ^{-1}(\sin (c+d x))}{6 a d^2}+\frac {2 f \log (\cos (c+d x))}{3 a d^2}-\frac {f \sec ^2(c+d x)}{6 a d^2}-\frac {(e+f x) \sec ^3(c+d x)}{3 a d}+\frac {2 (e+f x) \tan (c+d x)}{3 a d}+\frac {f \sec (c+d x) \tan (c+d x)}{6 a d^2}+\frac {(e+f x) \sec ^2(c+d x) \tan (c+d x)}{3 a d} \]

[Out]

1/6*f*arctanh(sin(d*x+c))/a/d^2+2/3*f*ln(cos(d*x+c))/a/d^2-1/6*f*sec(d*x+c)^2/a/d^2-1/3*(f*x+e)*sec(d*x+c)^3/a
/d+2/3*(f*x+e)*tan(d*x+c)/a/d+1/6*f*sec(d*x+c)*tan(d*x+c)/a/d^2+1/3*(f*x+e)*sec(d*x+c)^2*tan(d*x+c)/a/d

________________________________________________________________________________________

Rubi [A]
time = 0.10, antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {4627, 4270, 4269, 3556, 4494, 3853, 3855} \begin {gather*} -\frac {f \sec ^2(c+d x)}{6 a d^2}+\frac {f \tanh ^{-1}(\sin (c+d x))}{6 a d^2}+\frac {2 f \log (\cos (c+d x))}{3 a d^2}+\frac {f \tan (c+d x) \sec (c+d x)}{6 a d^2}+\frac {2 (e+f x) \tan (c+d x)}{3 a d}-\frac {(e+f x) \sec ^3(c+d x)}{3 a d}+\frac {(e+f x) \tan (c+d x) \sec ^2(c+d x)}{3 a d} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((e + f*x)*Sec[c + d*x]^2)/(a + a*Sin[c + d*x]),x]

[Out]

(f*ArcTanh[Sin[c + d*x]])/(6*a*d^2) + (2*f*Log[Cos[c + d*x]])/(3*a*d^2) - (f*Sec[c + d*x]^2)/(6*a*d^2) - ((e +
f*x)*Sec[c + d*x]^3)/(3*a*d) + (2*(e + f*x)*Tan[c + d*x])/(3*a*d) + (f*Sec[c + d*x]*Tan[c + d*x])/(6*a*d^2) +
((e + f*x)*Sec[c + d*x]^2*Tan[c + d*x])/(3*a*d)

Rule 3556

Int[tan[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-Log[RemoveContent[Cos[c + d*x], x]]/d, x] /; FreeQ[{c, d}, x]

Rule 3853

Int[(csc[(c_.) + (d_.)*(x_)]*(b_.))^(n_), x_Symbol] :> Simp[(-b)*Cos[c + d*x]*((b*Csc[c + d*x])^(n - 1)/(d*(n
- 1))), x] + Dist[b^2*((n - 2)/(n - 1)), Int[(b*Csc[c + d*x])^(n - 2), x], x] /; FreeQ[{b, c, d}, x] && GtQ[n,
1] && IntegerQ[2*n]

Rule 3855

Int[csc[(c_.) + (d_.)*(x_)], x_Symbol] :> Simp[-ArcTanh[Cos[c + d*x]]/d, x] /; FreeQ[{c, d}, x]

Rule 4269

Int[csc[(e_.) + (f_.)*(x_)]^2*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[(-(c + d*x)^m)*(Cot[e + f*x]/f), x
] + Dist[d*(m/f), Int[(c + d*x)^(m - 1)*Cot[e + f*x], x], x] /; FreeQ[{c, d, e, f}, x] && GtQ[m, 0]

Rule 4270

Int[(csc[(e_.) + (f_.)*(x_)]*(b_.))^(n_)*((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(-b^2)*(c + d*x)*Cot[e + f*x]
*((b*Csc[e + f*x])^(n - 2)/(f*(n - 1))), x] + (Dist[b^2*((n - 2)/(n - 1)), Int[(c + d*x)*(b*Csc[e + f*x])^(n -
2), x], x] - Simp[b^2*d*((b*Csc[e + f*x])^(n - 2)/(f^2*(n - 1)*(n - 2))), x]) /; FreeQ[{b, c, d, e, f}, x] &&
GtQ[n, 1] && NeQ[n, 2]

Rule 4494

Int[((c_.) + (d_.)*(x_))^(m_.)*Sec[(a_.) + (b_.)*(x_)]^(n_.)*Tan[(a_.) + (b_.)*(x_)]^(p_.), x_Symbol] :> Simp[
(c + d*x)^m*(Sec[a + b*x]^n/(b*n)), x] - Dist[d*(m/(b*n)), Int[(c + d*x)^(m - 1)*Sec[a + b*x]^n, x], x] /; Fre
eQ[{a, b, c, d, n}, x] && EqQ[p, 1] && GtQ[m, 0]

Rule 4627

Int[(((e_.) + (f_.)*(x_))^(m_.)*Sec[(c_.) + (d_.)*(x_)]^(n_.))/((a_) + (b_.)*Sin[(c_.) + (d_.)*(x_)]), x_Symbo
l] :> Dist[1/a, Int[(e + f*x)^m*Sec[c + d*x]^(n + 2), x], x] - Dist[1/b, Int[(e + f*x)^m*Sec[c + d*x]^(n + 1)*
Tan[c + d*x], x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && IGtQ[m, 0] && EqQ[a^2 - b^2, 0]

Rubi steps

\begin {align*} \int \frac {(e+f x) \sec ^2(c+d x)}{a+a \sin (c+d x)} \, dx &=\frac {\int (e+f x) \sec ^4(c+d x) \, dx}{a}-\frac {\int (e+f x) \sec ^3(c+d x) \tan (c+d x) \, dx}{a}\\ &=-\frac {f \sec ^2(c+d x)}{6 a d^2}-\frac {(e+f x) \sec ^3(c+d x)}{3 a d}+\frac {(e+f x) \sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {2 \int (e+f x) \sec ^2(c+d x) \, dx}{3 a}+\frac {f \int \sec ^3(c+d x) \, dx}{3 a d}\\ &=-\frac {f \sec ^2(c+d x)}{6 a d^2}-\frac {(e+f x) \sec ^3(c+d x)}{3 a d}+\frac {2 (e+f x) \tan (c+d x)}{3 a d}+\frac {f \sec (c+d x) \tan (c+d x)}{6 a d^2}+\frac {(e+f x) \sec ^2(c+d x) \tan (c+d x)}{3 a d}+\frac {f \int \sec (c+d x) \, dx}{6 a d}-\frac {(2 f) \int \tan (c+d x) \, dx}{3 a d}\\ &=\frac {f \tanh ^{-1}(\sin (c+d x))}{6 a d^2}+\frac {2 f \log (\cos (c+d x))}{3 a d^2}-\frac {f \sec ^2(c+d x)}{6 a d^2}-\frac {(e+f x) \sec ^3(c+d x)}{3 a d}+\frac {2 (e+f x) \tan (c+d x)}{3 a d}+\frac {f \sec (c+d x) \tan (c+d x)}{6 a d^2}+\frac {(e+f x) \sec ^2(c+d x) \tan (c+d x)}{3 a d}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.99, size = 231, normalized size = 1.52 \begin {gather*} \frac {-2 d (e+f x) (\cos (2 (c+d x))-2 \sin (c+d x))+\cos (c+d x) \left (d e-f-c f+3 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+5 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )+\left (d e-c f+3 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )+5 f \log \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )\right ) \sin (c+d x)\right )}{6 a d^2 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right ) \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right ) (1+\sin (c+d x))} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((e + f*x)*Sec[c + d*x]^2)/(a + a*Sin[c + d*x]),x]

[Out]

(-2*d*(e + f*x)*(Cos[2*(c + d*x)] - 2*Sin[c + d*x]) + Cos[c + d*x]*(d*e - f - c*f + 3*f*Log[Cos[(c + d*x)/2] -
Sin[(c + d*x)/2]] + 5*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]] + (d*e - c*f + 3*f*Log[Cos[(c + d*x)/2] - Si
n[(c + d*x)/2]] + 5*f*Log[Cos[(c + d*x)/2] + Sin[(c + d*x)/2]])*Sin[c + d*x]))/(6*a*d^2*(Cos[(c + d*x)/2] - Si
n[(c + d*x)/2])*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])*(1 + Sin[c + d*x]))

________________________________________________________________________________________

Maple [A]
time = 0.18, size = 181, normalized size = 1.19

method	result	size
risch	\(-\frac {4 i f x}{3 a d}-\frac {4 i f c}{3 a \,d^{2}}-\frac {i \left (f \,{\mathrm e}^{3 i \left (d x +c \right )}+4 d x f -8 i d f x \,{\mathrm e}^{i \left (d x +c \right )}+4 d e +{\mathrm e}^{i \left (d x +c \right )} f -8 i d e \,{\mathrm e}^{i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )^{3} d^{2} \left ({\mathrm e}^{i \left (d x +c \right )}-i\right ) a}+\frac {5 f \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{6 a \,d^{2}}+\frac {f \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{2 a \,d^{2}}\)	\(166\)
derivativedivides	\(\frac {\frac {f c}{3 \cos \left (d x +c \right )^{3}}-\frac {e d}{3 \cos \left (d x +c \right )^{3}}-f \left (\frac {d x +c}{3 \cos \left (d x +c \right )^{3}}-\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{6}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{6}\right )+f c \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )-e d \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+f \left (\frac {\left (d x +c \right ) \sin \left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {1}{6 \cos \left (d x +c \right )^{2}}+\frac {2 \left (d x +c \right ) \tan \left (d x +c \right )}{3}+\frac {2 \ln \left (\cos \left (d x +c \right )\right )}{3}\right )}{d^{2} a}\)	\(181\)
default	\(\frac {\frac {f c}{3 \cos \left (d x +c \right )^{3}}-\frac {e d}{3 \cos \left (d x +c \right )^{3}}-f \left (\frac {d x +c}{3 \cos \left (d x +c \right )^{3}}-\frac {\sec \left (d x +c \right ) \tan \left (d x +c \right )}{6}-\frac {\ln \left (\sec \left (d x +c \right )+\tan \left (d x +c \right )\right )}{6}\right )+f c \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )-e d \left (-\frac {2}{3}-\frac {\left (\sec ^{2}\left (d x +c \right )\right )}{3}\right ) \tan \left (d x +c \right )+f \left (\frac {\left (d x +c \right ) \sin \left (d x +c \right )}{3 \cos \left (d x +c \right )^{3}}-\frac {1}{6 \cos \left (d x +c \right )^{2}}+\frac {2 \left (d x +c \right ) \tan \left (d x +c \right )}{3}+\frac {2 \ln \left (\cos \left (d x +c \right )\right )}{3}\right )}{d^{2} a}\)	\(181\)
norman	\(\frac {\frac {2 e}{3 d a}-\frac {2 e \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{d a}+\frac {f x}{3 a d}+\frac {\left (-6 d e +f \right ) \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \,d^{2}}-\frac {\left (2 d e +f \right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a \,d^{2}}-\frac {4 f x \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{3 a d}-\frac {2 f x \left (\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{a d}-\frac {4 f x \left (\tan ^{3}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}+\frac {f x \left (\tan ^{4}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a d}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}+\frac {f \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}{2 a \,d^{2}}+\frac {5 f \ln \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}{6 a \,d^{2}}-\frac {2 f \ln \left (1+\tan ^{2}\left (\frac {d x}{2}+\frac {c}{2}\right )\right )}{3 a \,d^{2}}\)	\(264\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*sec(d*x+c)^2/(a+a*sin(d*x+c)),x,method=_RETURNVERBOSE)

[Out]

1/d^2/a*(1/3*f*c/cos(d*x+c)^3-1/3*e*d/cos(d*x+c)^3-f*(1/3*(d*x+c)/cos(d*x+c)^3-1/6*sec(d*x+c)*tan(d*x+c)-1/6*l
n(sec(d*x+c)+tan(d*x+c)))+f*c*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)-e*d*(-2/3-1/3*sec(d*x+c)^2)*tan(d*x+c)+f*(1/3
*(d*x+c)*sin(d*x+c)/cos(d*x+c)^3-1/6/cos(d*x+c)^2+2/3*(d*x+c)*tan(d*x+c)+2/3*ln(cos(d*x+c))))

________________________________________________________________________________________

Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 1115 vs. \(2 (138) = 276\).
time = 0.31, size = 1115, normalized size = 7.34 \begin {gather*} -\frac {\frac {8 \, c f {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )}}{a d + \frac {2 \, a d \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a d \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {a d \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}} + \frac {{\left (4 \, {\left (8 \, {\left (d x + c\right )} \cos \left (d x + c\right ) - \sin \left (3 \, d x + 3 \, c\right ) - \sin \left (d x + c\right )\right )} \cos \left (4 \, d x + 4 \, c\right ) + 16 \, {\left (2 \, d x + 4 \, {\left (d x + c\right )} \sin \left (d x + c\right ) + 2 \, c + \cos \left (d x + c\right )\right )} \cos \left (3 \, d x + 3 \, c\right ) + 8 \, \cos \left (3 \, d x + 3 \, c\right )^{2} + 8 \, \cos \left (d x + c\right )^{2} + 5 \, {\left (2 \, {\left (2 \, \sin \left (3 \, d x + 3 \, c\right ) + 2 \, \sin \left (d x + c\right ) + 1\right )} \cos \left (4 \, d x + 4 \, c\right ) - \cos \left (4 \, d x + 4 \, c\right )^{2} - 4 \, \cos \left (3 \, d x + 3 \, c\right )^{2} - 8 \, \cos \left (3 \, d x + 3 \, c\right ) \cos \left (d x + c\right ) - 4 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (3 \, d x + 3 \, c\right ) + \cos \left (d x + c\right )\right )} \sin \left (4 \, d x + 4 \, c\right ) - \sin \left (4 \, d x + 4 \, c\right )^{2} - 4 \, {\left (2 \, \sin \left (d x + c\right ) + 1\right )} \sin \left (3 \, d x + 3 \, c\right ) - 4 \, \sin \left (3 \, d x + 3 \, c\right )^{2} - 4 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) - 1\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} + 2 \, \sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (2 \, {\left (2 \, \sin \left (3 \, d x + 3 \, c\right ) + 2 \, \sin \left (d x + c\right ) + 1\right )} \cos \left (4 \, d x + 4 \, c\right ) - \cos \left (4 \, d x + 4 \, c\right )^{2} - 4 \, \cos \left (3 \, d x + 3 \, c\right )^{2} - 8 \, \cos \left (3 \, d x + 3 \, c\right ) \cos \left (d x + c\right ) - 4 \, \cos \left (d x + c\right )^{2} - 4 \, {\left (\cos \left (3 \, d x + 3 \, c\right ) + \cos \left (d x + c\right )\right )} \sin \left (4 \, d x + 4 \, c\right ) - \sin \left (4 \, d x + 4 \, c\right )^{2} - 4 \, {\left (2 \, \sin \left (d x + c\right ) + 1\right )} \sin \left (3 \, d x + 3 \, c\right ) - 4 \, \sin \left (3 \, d x + 3 \, c\right )^{2} - 4 \, \sin \left (d x + c\right )^{2} - 4 \, \sin \left (d x + c\right ) - 1\right )} \log \left (\cos \left (d x + c\right )^{2} + \sin \left (d x + c\right )^{2} - 2 \, \sin \left (d x + c\right ) + 1\right ) + 4 \, {\left (4 \, d x + 8 \, {\left (d x + c\right )} \sin \left (d x + c\right ) + 4 \, c + \cos \left (3 \, d x + 3 \, c\right ) + \cos \left (d x + c\right )\right )} \sin \left (4 \, d x + 4 \, c\right ) - 4 \, {\left (16 \, {\left (d x + c\right )} \cos \left (d x + c\right ) - 4 \, \sin \left (d x + c\right ) - 1\right )} \sin \left (3 \, d x + 3 \, c\right ) + 8 \, \sin \left (3 \, d x + 3 \, c\right )^{2} + 8 \, \sin \left (d x + c\right )^{2} + 4 \, \sin \left (d x + c\right )\right )} f}{a d \cos \left (4 \, d x + 4 \, c\right )^{2} + 4 \, a d \cos \left (3 \, d x + 3 \, c\right )^{2} + 8 \, a d \cos \left (3 \, d x + 3 \, c\right ) \cos \left (d x + c\right ) + 4 \, a d \cos \left (d x + c\right )^{2} + a d \sin \left (4 \, d x + 4 \, c\right )^{2} + 4 \, a d \sin \left (3 \, d x + 3 \, c\right )^{2} + 4 \, a d \sin \left (d x + c\right )^{2} + 4 \, a d \sin \left (d x + c\right ) + a d - 2 \, {\left (2 \, a d \sin \left (3 \, d x + 3 \, c\right ) + 2 \, a d \sin \left (d x + c\right ) + a d\right )} \cos \left (4 \, d x + 4 \, c\right ) + 4 \, {\left (a d \cos \left (3 \, d x + 3 \, c\right ) + a d \cos \left (d x + c\right )\right )} \sin \left (4 \, d x + 4 \, c\right ) + 4 \, {\left (2 \, a d \sin \left (d x + c\right ) + a d\right )} \sin \left (3 \, d x + 3 \, c\right )} - \frac {8 \, e {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {3 \, \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {3 \, \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - 1\right )}}{a + \frac {2 \, a \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} - \frac {2 \, a \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {a \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}}}}{12 \, d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sec(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="maxima")

[Out]

-1/12*(8*c*f*(sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 + 3*sin(d*x + c)^3/(cos(
d*x + c) + 1)^3 - 1)/(a*d + 2*a*d*sin(d*x + c)/(cos(d*x + c) + 1) - 2*a*d*sin(d*x + c)^3/(cos(d*x + c) + 1)^3
- a*d*sin(d*x + c)^4/(cos(d*x + c) + 1)^4) + (4*(8*(d*x + c)*cos(d*x + c) - sin(3*d*x + 3*c) - sin(d*x + c))*c
os(4*d*x + 4*c) + 16*(2*d*x + 4*(d*x + c)*sin(d*x + c) + 2*c + cos(d*x + c))*cos(3*d*x + 3*c) + 8*cos(3*d*x +
3*c)^2 + 8*cos(d*x + c)^2 + 5*(2*(2*sin(3*d*x + 3*c) + 2*sin(d*x + c) + 1)*cos(4*d*x + 4*c) - cos(4*d*x + 4*c)
^2 - 4*cos(3*d*x + 3*c)^2 - 8*cos(3*d*x + 3*c)*cos(d*x + c) - 4*cos(d*x + c)^2 - 4*(cos(3*d*x + 3*c) + cos(d*x
+ c))*sin(4*d*x + 4*c) - sin(4*d*x + 4*c)^2 - 4*(2*sin(d*x + c) + 1)*sin(3*d*x + 3*c) - 4*sin(3*d*x + 3*c)^2
- 4*sin(d*x + c)^2 - 4*sin(d*x + c) - 1)*log(cos(d*x + c)^2 + sin(d*x + c)^2 + 2*sin(d*x + c) + 1) + 3*(2*(2*s
in(3*d*x + 3*c) + 2*sin(d*x + c) + 1)*cos(4*d*x + 4*c) - cos(4*d*x + 4*c)^2 - 4*cos(3*d*x + 3*c)^2 - 8*cos(3*d
*x + 3*c)*cos(d*x + c) - 4*cos(d*x + c)^2 - 4*(cos(3*d*x + 3*c) + cos(d*x + c))*sin(4*d*x + 4*c) - sin(4*d*x +
4*c)^2 - 4*(2*sin(d*x + c) + 1)*sin(3*d*x + 3*c) - 4*sin(3*d*x + 3*c)^2 - 4*sin(d*x + c)^2 - 4*sin(d*x + c) -
1)*log(cos(d*x + c)^2 + sin(d*x + c)^2 - 2*sin(d*x + c) + 1) + 4*(4*d*x + 8*(d*x + c)*sin(d*x + c) + 4*c + co
s(3*d*x + 3*c) + cos(d*x + c))*sin(4*d*x + 4*c) - 4*(16*(d*x + c)*cos(d*x + c) - 4*sin(d*x + c) - 1)*sin(3*d*x
+ 3*c) + 8*sin(3*d*x + 3*c)^2 + 8*sin(d*x + c)^2 + 4*sin(d*x + c))*f/(a*d*cos(4*d*x + 4*c)^2 + 4*a*d*cos(3*d*
x + 3*c)^2 + 8*a*d*cos(3*d*x + 3*c)*cos(d*x + c) + 4*a*d*cos(d*x + c)^2 + a*d*sin(4*d*x + 4*c)^2 + 4*a*d*sin(3
*d*x + 3*c)^2 + 4*a*d*sin(d*x + c)^2 + 4*a*d*sin(d*x + c) + a*d - 2*(2*a*d*sin(3*d*x + 3*c) + 2*a*d*sin(d*x +
c) + a*d)*cos(4*d*x + 4*c) + 4*(a*d*cos(3*d*x + 3*c) + a*d*cos(d*x + c))*sin(4*d*x + 4*c) + 4*(2*a*d*sin(d*x +
c) + a*d)*sin(3*d*x + 3*c)) - 8*e*(sin(d*x + c)/(cos(d*x + c) + 1) + 3*sin(d*x + c)^2/(cos(d*x + c) + 1)^2 +
3*sin(d*x + c)^3/(cos(d*x + c) + 1)^3 - 1)/(a + 2*a*sin(d*x + c)/(cos(d*x + c) + 1) - 2*a*sin(d*x + c)^3/(cos(
d*x + c) + 1)^3 - a*sin(d*x + c)^4/(cos(d*x + c) + 1)^4))/d

________________________________________________________________________________________

Fricas [A]
time = 0.38, size = 159, normalized size = 1.05 \begin {gather*} \frac {4 \, d f x - 8 \, {\left (d f x + d e\right )} \cos \left (d x + c\right )^{2} - 2 \, f \cos \left (d x + c\right ) + 4 \, d e + 5 \, {\left (f \cos \left (d x + c\right ) \sin \left (d x + c\right ) + f \cos \left (d x + c\right )\right )} \log \left (\sin \left (d x + c\right ) + 1\right ) + 3 \, {\left (f \cos \left (d x + c\right ) \sin \left (d x + c\right ) + f \cos \left (d x + c\right )\right )} \log \left (-\sin \left (d x + c\right ) + 1\right ) + 8 \, {\left (d f x + d e\right )} \sin \left (d x + c\right )}{12 \, {\left (a d^{2} \cos \left (d x + c\right ) \sin \left (d x + c\right ) + a d^{2} \cos \left (d x + c\right )\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sec(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="fricas")

[Out]

1/12*(4*d*f*x - 8*(d*f*x + d*e)*cos(d*x + c)^2 - 2*f*cos(d*x + c) + 4*d*e + 5*(f*cos(d*x + c)*sin(d*x + c) + f
*cos(d*x + c))*log(sin(d*x + c) + 1) + 3*(f*cos(d*x + c)*sin(d*x + c) + f*cos(d*x + c))*log(-sin(d*x + c) + 1)
+ 8*(d*f*x + d*e)*sin(d*x + c))/(a*d^2*cos(d*x + c)*sin(d*x + c) + a*d^2*cos(d*x + c))

________________________________________________________________________________________

Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \frac {\int \frac {e \sec ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx + \int \frac {f x \sec ^{2}{\left (c + d x \right )}}{\sin {\left (c + d x \right )} + 1}\, dx}{a} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sec(d*x+c)**2/(a+a*sin(d*x+c)),x)

[Out]

(Integral(e*sec(c + d*x)**2/(sin(c + d*x) + 1), x) + Integral(f*x*sec(c + d*x)**2/(sin(c + d*x) + 1), x))/a

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 6656 vs. \(2 (141) = 282\).
time = 6.41, size = 6656, normalized size = 43.79 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*sec(d*x+c)^2/(a+a*sin(d*x+c)),x, algorithm="giac")

[Out]

-1/12*(4*d*f*x*tan(1/2*d*x)^4*tan(1/2*c)^4 + 16*d*f*x*tan(1/2*d*x)^4*tan(1/2*c)^3 + 16*d*f*x*tan(1/2*d*x)^3*ta
n(1/2*c)^4 + 4*d*e*tan(1/2*d*x)^4*tan(1/2*c)^4 - 3*f*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan
(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 +
2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*c)
^2 + 1))*tan(1/2*d*x)^4*tan(1/2*c)^4 - 5*f*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) -
2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*
d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*t
an(1/2*d*x)^4*tan(1/2*c)^4 - 24*d*f*x*tan(1/2*d*x)^4*tan(1/2*c)^2 - 64*d*f*x*tan(1/2*d*x)^3*tan(1/2*c)^3 + 16*
d*e*tan(1/2*d*x)^4*tan(1/2*c)^3 + 6*f*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c) + 2*tan
(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1/2*d*x)*
tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(1/
2*d*x)^4*tan(1/2*c)^3 + 10*f*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*d*x)
^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*tan(1/2*c
)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(1/2*d*x)^4*
tan(1/2*c)^3 - 24*d*f*x*tan(1/2*d*x)^2*tan(1/2*c)^4 + 16*d*e*tan(1/2*d*x)^3*tan(1/2*c)^4 + 6*f*log(2*(tan(1/2*
d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2
*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*ta
n(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(1/2*d*x)^3*tan(1/2*c)^4 + 10*f*log(2*(tan(1/2*d*x)^4*ta
n(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*t
an(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x
) + 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(1/2*d*x)^3*tan(1/2*c)^4 + 2*f*tan(1/2*d*x)^4*tan(1/2*c)^4 + 16*d
*f*x*tan(1/2*d*x)^4*tan(1/2*c) - 24*d*e*tan(1/2*d*x)^4*tan(1/2*c)^2 - 64*d*e*tan(1/2*d*x)^3*tan(1/2*c)^3 + 12*
f*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d
*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + ta
n(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(1/2*d*x)^3*tan(1/2*c)^3 + 20*f*log(2*(
tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2
*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^
2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(1/2*d*x)^3*tan(1/2*c)^3 + 16*d*f*x*tan(1/2*d*x)
*tan(1/2*c)^4 - 24*d*e*tan(1/2*d*x)^2*tan(1/2*c)^4 + 4*d*f*x*tan(1/2*d*x)^4 + 64*d*f*x*tan(1/2*d*x)^3*tan(1/2*
c) + 16*d*e*tan(1/2*d*x)^4*tan(1/2*c) - 6*f*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c) +
2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1/2
*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*
tan(1/2*d*x)^4*tan(1/2*c) - 10*f*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*
d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 - 2*tan(1/2*d*x)*tan(1
/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(1/2*d*x
)^4*tan(1/2*c) + 144*d*f*x*tan(1/2*d*x)^2*tan(1/2*c)^2 - 36*f*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 + 2*tan(1/2*d
*x)^4*tan(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 - 2*tan(1/2*
d*x)^3 + 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2*tan(1/2*c) + 1)/(t
an(1/2*c)^2 + 1))*tan(1/2*d*x)^3*tan(1/2*c)^2 - 60*f*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2*tan(1/2*d*x)^4*tan
(1/2*c) - 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/2*c)^2 + 2*tan(1/2*d*x)^3 -
2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 + 2*tan(1/2*d*x) + 2*tan(1/2*c) + 1)/(tan(1/2*c)
^2 + 1))*tan(1/2*d*x)^3*tan(1/2*c)^2 + 64*d*f*x*tan(1/2*d*x)*tan(1/2*c)^3 - 36*f*log(2*(tan(1/2*d*x)^4*tan(1/2
*c)^2 + 2*tan(1/2*d*x)^4*tan(1/2*c) + 2*tan(1/2*d*x)^3*tan(1/2*c)^2 + tan(1/2*d*x)^4 + 2*tan(1/2*d*x)^2*tan(1/
2*c)^2 - 2*tan(1/2*d*x)^3 + 2*tan(1/2*d*x)*tan(1/2*c)^2 + 2*tan(1/2*d*x)^2 + tan(1/2*c)^2 - 2*tan(1/2*d*x) - 2
*tan(1/2*c) + 1)/(tan(1/2*c)^2 + 1))*tan(1/2*d*x)^2*tan(1/2*c)^3 - 60*f*log(2*(tan(1/2*d*x)^4*tan(1/2*c)^2 - 2
*tan(1/2*d*x)^4*tan(1/2*c) - 2*tan(1/2*d*x)^3*t...

________________________________________________________________________________________

Mupad [B]
time = 7.67, size = 240, normalized size = 1.58 \begin {gather*} \frac {2\,\left (d\,e+d\,f\,x\right )}{3\,a\,d^2\,\left (3\,{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}-{\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}\,3{}\mathrm {i}-{\mathrm {e}}^{c\,3{}\mathrm {i}+d\,x\,3{}\mathrm {i}}+1{}\mathrm {i}\right )}-\frac {3\,d\,e+3\,d\,f\,x+f\,2{}\mathrm {i}}{6\,a\,d^2\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}+\frac {e+f\,x}{2\,a\,d\,\left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}-\mathrm {i}\right )}-\frac {\left (24\,d\,e+24\,d\,f\,x-f\,8{}\mathrm {i}\right )\,1{}\mathrm {i}}{24\,a\,d^2\,\left ({\mathrm {e}}^{c\,2{}\mathrm {i}+d\,x\,2{}\mathrm {i}}-1+{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}\,2{}\mathrm {i}\right )}-\frac {f\,x\,4{}\mathrm {i}}{3\,a\,d}+\frac {f\,\ln \left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}-\mathrm {i}\right )}{2\,a\,d^2}+\frac {5\,f\,\ln \left ({\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}+1{}\mathrm {i}\right )}{6\,a\,d^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e + f*x)/(cos(c + d*x)^2*(a + a*sin(c + d*x))),x)

[Out]

(2*(d*e + d*f*x))/(3*a*d^2*(3*exp(c*1i + d*x*1i) - exp(c*2i + d*x*2i)*3i - exp(c*3i + d*x*3i) + 1i)) - (f*2i +
3*d*e + 3*d*f*x)/(6*a*d^2*(exp(c*1i + d*x*1i) + 1i)) + (e + f*x)/(2*a*d*(exp(c*1i + d*x*1i) - 1i)) - ((24*d*e
- f*8i + 24*d*f*x)*1i)/(24*a*d^2*(exp(c*1i + d*x*1i)*2i + exp(c*2i + d*x*2i) - 1)) - (f*x*4i)/(3*a*d) + (f*lo
g(exp(c*1i + d*x*1i) - 1i))/(2*a*d^2) + (5*f*log(exp(c*1i + d*x*1i) + 1i))/(6*a*d^2)

________________________________________________________________________________________