Integrand size = 21, antiderivative size = 264 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {5 b^4 \left (6 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{9/2} d}+\frac {b \sec ^3(c+d x)}{2 \left (a^2-b^2\right ) d (a+b \sin (c+d x))^2}+\frac {7 a b \sec ^3(c+d x)}{2 \left (a^2-b^2\right )^2 d (a+b \sin (c+d x))}-\frac {\sec ^3(c+d x) \left (5 b \left (6 a^2+b^2\right )-a \left (2 a^2+33 b^2\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^3 d}+\frac {\sec (c+d x) \left (15 b^3 \left (6 a^2+b^2\right )+a \left (4 a^4-28 a^2 b^2-81 b^4\right ) \sin (c+d x)\right )}{6 \left (a^2-b^2\right )^4 d} \] Output:
5*b^4*(6*a^2+b^2)*arctan((b+a*tan(1/2*d*x+1/2*c))/(a^2-b^2)^(1/2))/(a^2-b^ 2)^(9/2)/d+1/2*b*sec(d*x+c)^3/(a^2-b^2)/d/(a+b*sin(d*x+c))^2+7/2*a*b*sec(d *x+c)^3/(a^2-b^2)^2/d/(a+b*sin(d*x+c))-1/6*sec(d*x+c)^3*(5*b*(6*a^2+b^2)-a *(2*a^2+33*b^2)*sin(d*x+c))/(a^2-b^2)^3/d+1/6*sec(d*x+c)*(15*b^3*(6*a^2+b^ 2)+a*(4*a^4-28*a^2*b^2-81*b^4)*sin(d*x+c))/(a^2-b^2)^4/d
Time = 3.49 (sec) , antiderivative size = 380, normalized size of antiderivative = 1.44 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\frac {\frac {60 b^4 \left (6 a^2+b^2\right ) \arctan \left (\frac {b+a \tan \left (\frac {1}{2} (c+d x)\right )}{\sqrt {a^2-b^2}}\right )}{\left (a^2-b^2\right )^{9/2}}+\frac {1}{(a+b)^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 \sin \left (\frac {1}{2} (c+d x)\right )}{(a+b)^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}+\frac {2 (4 a+13 b) \sin \left (\frac {1}{2} (c+d x)\right )}{(a+b)^4 \left (\cos \left (\frac {1}{2} (c+d x)\right )-\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {2 \sin \left (\frac {1}{2} (c+d x)\right )}{(a-b)^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^3}-\frac {1}{(a-b)^3 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )^2}+\frac {2 (4 a-13 b) \sin \left (\frac {1}{2} (c+d x)\right )}{(a-b)^4 \left (\cos \left (\frac {1}{2} (c+d x)\right )+\sin \left (\frac {1}{2} (c+d x)\right )\right )}+\frac {6 b^5 \cos (c+d x)}{(a-b)^3 (a+b)^3 (a+b \sin (c+d x))^2}+\frac {66 a b^5 \cos (c+d x)}{(a-b)^4 (a+b)^4 (a+b \sin (c+d x))}}{12 d} \] Input:
Integrate[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]
Output:
((60*b^4*(6*a^2 + b^2)*ArcTan[(b + a*Tan[(c + d*x)/2])/Sqrt[a^2 - b^2]])/( a^2 - b^2)^(9/2) + 1/((a + b)^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^2) + (2*Sin[(c + d*x)/2])/((a + b)^3*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])^3) + (2*(4*a + 13*b)*Sin[(c + d*x)/2])/((a + b)^4*(Cos[(c + d*x)/2] - Sin[(c + d*x)/2])) + (2*Sin[(c + d*x)/2])/((a - b)^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^3) - 1/((a - b)^3*(Cos[(c + d*x)/2] + Sin[(c + d*x)/2])^2) + (2* (4*a - 13*b)*Sin[(c + d*x)/2])/((a - b)^4*(Cos[(c + d*x)/2] + Sin[(c + d*x )/2])) + (6*b^5*Cos[c + d*x])/((a - b)^3*(a + b)^3*(a + b*Sin[c + d*x])^2) + (66*a*b^5*Cos[c + d*x])/((a - b)^4*(a + b)^4*(a + b*Sin[c + d*x])))/(12 *d)
Time = 1.40 (sec) , antiderivative size = 310, normalized size of antiderivative = 1.17, number of steps used = 17, number of rules used = 16, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.762, Rules used = {3042, 3173, 25, 3042, 3343, 25, 3042, 3345, 25, 3042, 3345, 27, 3042, 3139, 1083, 217}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \int \frac {1}{\cos (c+d x)^4 (a+b \sin (c+d x))^3}dx\) |
| \(\Big \downarrow \) 3173 |
| \(\displaystyle \frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}-\frac {\int -\frac {\sec ^4(c+d x) (2 a-5 b \sin (c+d x))}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {\sec ^4(c+d x) (2 a-5 b \sin (c+d x))}{(a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\int \frac {2 a-5 b \sin (c+d x)}{\cos (c+d x)^4 (a+b \sin (c+d x))^2}dx}{2 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3343 |
| \(\displaystyle \frac {\frac {7 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}-\frac {\int -\frac {\sec ^4(c+d x) \left (2 a^2-28 b \sin (c+d x) a+5 b^2\right )}{a+b \sin (c+d x)}dx}{a^2-b^2}}{2 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {\sec ^4(c+d x) \left (2 a^2-28 b \sin (c+d x) a+5 b^2\right )}{a+b \sin (c+d x)}dx}{a^2-b^2}+\frac {7 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\int \frac {2 a^2-28 b \sin (c+d x) a+5 b^2}{\cos (c+d x)^4 (a+b \sin (c+d x))}dx}{a^2-b^2}+\frac {7 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\frac {-\frac {\int -\frac {\sec ^2(c+d x) \left (4 a^4-24 b^2 a^2+2 b \left (2 a^2+33 b^2\right ) \sin (c+d x) a-15 b^4\right )}{a+b \sin (c+d x)}dx}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (5 b \left (6 a^2+b^2\right )-a \left (2 a^2+33 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {7 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {\sec ^2(c+d x) \left (4 a^4-24 b^2 a^2+2 b \left (2 a^2+33 b^2\right ) \sin (c+d x) a-15 b^4\right )}{a+b \sin (c+d x)}dx}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (5 b \left (6 a^2+b^2\right )-a \left (2 a^2+33 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {7 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\int \frac {4 a^4-24 b^2 a^2+2 b \left (2 a^2+33 b^2\right ) \sin (c+d x) a-15 b^4}{\cos (c+d x)^2 (a+b \sin (c+d x))}dx}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (5 b \left (6 a^2+b^2\right )-a \left (2 a^2+33 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {7 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3345 |
| \(\displaystyle \frac {\frac {\frac {\frac {\sec (c+d x) \left (15 b^3 \left (6 a^2+b^2\right )+a \left (4 a^4-28 a^2 b^2-81 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {\int -\frac {15 b^4 \left (6 a^2+b^2\right )}{a+b \sin (c+d x)}dx}{a^2-b^2}}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (5 b \left (6 a^2+b^2\right )-a \left (2 a^2+33 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {7 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle \frac {\frac {\frac {\frac {15 b^4 \left (6 a^2+b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a^2-b^2}+\frac {\sec (c+d x) \left (15 b^3 \left (6 a^2+b^2\right )+a \left (4 a^4-28 a^2 b^2-81 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (5 b \left (6 a^2+b^2\right )-a \left (2 a^2+33 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {7 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3042 |
| \(\displaystyle \frac {\frac {\frac {\frac {15 b^4 \left (6 a^2+b^2\right ) \int \frac {1}{a+b \sin (c+d x)}dx}{a^2-b^2}+\frac {\sec (c+d x) \left (15 b^3 \left (6 a^2+b^2\right )+a \left (4 a^4-28 a^2 b^2-81 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (5 b \left (6 a^2+b^2\right )-a \left (2 a^2+33 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {7 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 3139 |
| \(\displaystyle \frac {\frac {\frac {\frac {30 b^4 \left (6 a^2+b^2\right ) \int \frac {1}{a \tan ^2\left (\frac {1}{2} (c+d x)\right )+2 b \tan \left (\frac {1}{2} (c+d x)\right )+a}d\tan \left (\frac {1}{2} (c+d x)\right )}{d \left (a^2-b^2\right )}+\frac {\sec (c+d x) \left (15 b^3 \left (6 a^2+b^2\right )+a \left (4 a^4-28 a^2 b^2-81 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (5 b \left (6 a^2+b^2\right )-a \left (2 a^2+33 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {7 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 1083 |
| \(\displaystyle \frac {\frac {\frac {\frac {\sec (c+d x) \left (15 b^3 \left (6 a^2+b^2\right )+a \left (4 a^4-28 a^2 b^2-81 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}-\frac {60 b^4 \left (6 a^2+b^2\right ) \int \frac {1}{-\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )^2-4 \left (a^2-b^2\right )}d\left (2 b+2 a \tan \left (\frac {1}{2} (c+d x)\right )\right )}{d \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (5 b \left (6 a^2+b^2\right )-a \left (2 a^2+33 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}+\frac {7 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}}{2 \left (a^2-b^2\right )}+\frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}\) |
| \(\Big \downarrow \) 217 |
| \(\displaystyle \frac {b \sec ^3(c+d x)}{2 d \left (a^2-b^2\right ) (a+b \sin (c+d x))^2}+\frac {\frac {7 a b \sec ^3(c+d x)}{d \left (a^2-b^2\right ) (a+b \sin (c+d x))}+\frac {\frac {\frac {30 b^4 \left (6 a^2+b^2\right ) \arctan \left (\frac {2 a \tan \left (\frac {1}{2} (c+d x)\right )+2 b}{2 \sqrt {a^2-b^2}}\right )}{d \left (a^2-b^2\right )^{3/2}}+\frac {\sec (c+d x) \left (15 b^3 \left (6 a^2+b^2\right )+a \left (4 a^4-28 a^2 b^2-81 b^4\right ) \sin (c+d x)\right )}{d \left (a^2-b^2\right )}}{3 \left (a^2-b^2\right )}-\frac {\sec ^3(c+d x) \left (5 b \left (6 a^2+b^2\right )-a \left (2 a^2+33 b^2\right ) \sin (c+d x)\right )}{3 d \left (a^2-b^2\right )}}{a^2-b^2}}{2 \left (a^2-b^2\right )}\) |
Input:
Int[Sec[c + d*x]^4/(a + b*Sin[c + d*x])^3,x]
Output:
(b*Sec[c + d*x]^3)/(2*(a^2 - b^2)*d*(a + b*Sin[c + d*x])^2) + ((7*a*b*Sec[ c + d*x]^3)/((a^2 - b^2)*d*(a + b*Sin[c + d*x])) + (-1/3*(Sec[c + d*x]^3*( 5*b*(6*a^2 + b^2) - a*(2*a^2 + 33*b^2)*Sin[c + d*x]))/((a^2 - b^2)*d) + (( 30*b^4*(6*a^2 + b^2)*ArcTan[(2*b + 2*a*Tan[(c + d*x)/2])/(2*Sqrt[a^2 - b^2 ])])/((a^2 - b^2)^(3/2)*d) + (Sec[c + d*x]*(15*b^3*(6*a^2 + b^2) + a*(4*a^ 4 - 28*a^2*b^2 - 81*b^4)*Sin[c + d*x]))/((a^2 - b^2)*d))/(3*(a^2 - b^2)))/ (a^2 - b^2))/(2*(a^2 - b^2))
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^( -1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] & & (LtQ[a, 0] || LtQ[b, 0])
Int[((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Simp[-2 Subst[I nt[1/Simp[b^2 - 4*a*c - x^2, x], x], x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*sin[(c_.) + (d_.)*(x_)])^(-1), x_Symbol] :> With[{e = Fre eFactors[Tan[(c + d*x)/2], x]}, Simp[2*(e/d) Subst[Int[1/(a + 2*b*e*x + a *e^2*x^2), x], x, Tan[(c + d*x)/2]/e], x]] /; FreeQ[{a, b, c, d}, x] && NeQ [a^2 - b^2, 0]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_), x_Symbol] :> Simp[(-b)*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) Int[(g*Cos[e + f*x])^p*(a + b*Sin[e + f*x])^(m + 1)*(a*(m + 1) - b*(m + p + 2)*Sin[e + f*x]), x], x] /; FreeQ[{a, b, e, f, g, p}, x] && NeQ[a^2 - b ^2, 0] && LtQ[m, -1] && IntegersQ[2*m, 2*p]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(-(b*c - a*d))*(g*Cos[e + f*x])^(p + 1)*((a + b*Sin[e + f*x])^(m + 1)/(f*g*(a^2 - b^2)*(m + 1))), x] + Simp[1/((a^2 - b^2)*(m + 1)) Int[(g*Cos[e + f*x])^p *(a + b*Sin[e + f*x])^(m + 1)*Simp[(a*c - b*d)*(m + 1) - (b*c - a*d)*(m + p + 2)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ [a^2 - b^2, 0] && LtQ[m, -1] && IntegerQ[2*m]
Int[(cos[(e_.) + (f_.)*(x_)]*(g_.))^(p_)*((a_) + (b_.)*sin[(e_.) + (f_.)*(x _)])^(m_.)*((c_.) + (d_.)*sin[(e_.) + (f_.)*(x_)]), x_Symbol] :> Simp[(g*Co s[e + f*x])^(p + 1)*(a + b*Sin[e + f*x])^(m + 1)*((b*c - a*d - (a*c - b*d)* Sin[e + f*x])/(f*g*(a^2 - b^2)*(p + 1))), x] + Simp[1/(g^2*(a^2 - b^2)*(p + 1)) Int[(g*Cos[e + f*x])^(p + 2)*(a + b*Sin[e + f*x])^m*Simp[c*(a^2*(p + 2) - b^2*(m + p + 2)) + a*b*d*m + b*(a*c - b*d)*(m + p + 3)*Sin[e + f*x], x], x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && NeQ[a^2 - b^2, 0] && Lt Q[p, -1] && IntegerQ[2*m]
Time = 2.24 (sec) , antiderivative size = 352, normalized size of antiderivative = 1.33
| method | result | size |
| derivativedivides | \(\frac {-\frac {1}{3 \left (a -b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (a -b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 a -5 b}{2 \left (a -b \right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 b^{4} \left (\frac {\frac {b^{2} \left (13 a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}+\frac {b \left (12 a^{4}+23 b^{2} a^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a^{2}}+\frac {b^{2} \left (35 a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+6 a^{2} b -\frac {b^{3}}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {5 \left (6 a^{2}+b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{4} \left (a +b \right )^{4}}-\frac {1}{3 \left (a +b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 a +5 b}{2 \left (a +b \right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(352\) |
| default | \(\frac {-\frac {1}{3 \left (a -b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{3}}+\frac {1}{2 \left (a -b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )^{2}}-\frac {2 a -5 b}{2 \left (a -b \right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )+1\right )}+\frac {2 b^{4} \left (\frac {\frac {b^{2} \left (13 a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{3}}{2 a}+\frac {b \left (12 a^{4}+23 b^{2} a^{2}-2 b^{4}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2}}{2 a^{2}}+\frac {b^{2} \left (35 a^{2}-2 b^{2}\right ) \tan \left (\frac {d x}{2}+\frac {c}{2}\right )}{2 a}+6 a^{2} b -\frac {b^{3}}{2}}{\left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )^{2} a +2 b \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+a \right )^{2}}+\frac {5 \left (6 a^{2}+b^{2}\right ) \arctan \left (\frac {2 a \tan \left (\frac {d x}{2}+\frac {c}{2}\right )+2 b}{2 \sqrt {a^{2}-b^{2}}}\right )}{2 \sqrt {a^{2}-b^{2}}}\right )}{\left (a -b \right )^{4} \left (a +b \right )^{4}}-\frac {1}{3 \left (a +b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{3}}-\frac {1}{2 \left (a +b \right )^{3} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )^{2}}-\frac {2 a +5 b}{2 \left (a +b \right )^{4} \left (\tan \left (\frac {d x}{2}+\frac {c}{2}\right )-1\right )}}{d}\) | \(352\) |
| risch | \(\frac {i \left (-4 a^{5} b^{2}+28 a^{3} b^{4}+81 a \,b^{6}-548 i a^{2} b^{5} {\mathrm e}^{3 i \left (d x +c \right )}+32 i a^{6} b \,{\mathrm e}^{3 i \left (d x +c \right )}-448 i a^{4} b^{3} {\mathrm e}^{5 i \left (d x +c \right )}+48 i a^{6} b \,{\mathrm e}^{5 i \left (d x +c \right )}+100 i a^{2} b^{5} {\mathrm e}^{7 i \left (d x +c \right )}-208 i a^{2} b^{5} {\mathrm e}^{5 i \left (d x +c \right )}-344 i a^{4} b^{3} {\mathrm e}^{3 i \left (d x +c \right )}-120 i a^{4} b^{3} {\mathrm e}^{7 i \left (d x +c \right )}+90 i a^{2} b^{5} {\mathrm e}^{9 i \left (d x +c \right )}-234 i a^{2} b^{5} {\mathrm e}^{i \left (d x +c \right )}+16 i a^{6} b \,{\mathrm e}^{i \left (d x +c \right )}-112 i a^{4} b^{3} {\mathrm e}^{i \left (d x +c \right )}-45 a \,b^{6} {\mathrm e}^{8 i \left (d x +c \right )}-358 a^{3} b^{4} {\mathrm e}^{4 i \left (d x +c \right )}+20 i b^{7} {\mathrm e}^{3 i \left (d x +c \right )}-110 a \,b^{6} {\mathrm e}^{6 i \left (d x +c \right )}-268 a^{5} b^{2} {\mathrm e}^{4 i \left (d x +c \right )}-670 a^{3} b^{4} {\mathrm e}^{6 i \left (d x +c \right )}-60 a^{5} b^{2} {\mathrm e}^{6 i \left (d x +c \right )}+20 i b^{7} {\mathrm e}^{7 i \left (d x +c \right )}-22 i b^{7} {\mathrm e}^{5 i \left (d x +c \right )}+15 i b^{7} {\mathrm e}^{9 i \left (d x +c \right )}+15 i b^{7} {\mathrm e}^{i \left (d x +c \right )}-52 a \,b^{6} {\mathrm e}^{4 i \left (d x +c \right )}-116 a^{5} b^{2} {\mathrm e}^{2 i \left (d x +c \right )}-26 a^{3} b^{4} {\mathrm e}^{2 i \left (d x +c \right )}+126 a \,b^{6} {\mathrm e}^{2 i \left (d x +c \right )}-270 a^{3} b^{4} {\mathrm e}^{8 i \left (d x +c \right )}+48 a^{7} {\mathrm e}^{4 i \left (d x +c \right )}+16 a^{7} {\mathrm e}^{2 i \left (d x +c \right )}\right )}{3 \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} \left (-i {\mathrm e}^{2 i \left (d x +c \right )} b +i b +2 a \,{\mathrm e}^{i \left (d x +c \right )}\right )^{2} \left (a^{2}-b^{2}\right )^{4} d}-\frac {15 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{2}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{4} \left (a -b \right )^{4} d}-\frac {5 b^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}-a^{2}+b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{4} \left (a -b \right )^{4} d}+\frac {15 b^{4} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right ) a^{2}}{\sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{4} \left (a -b \right )^{4} d}+\frac {5 b^{6} \ln \left ({\mathrm e}^{i \left (d x +c \right )}+\frac {i a \sqrt {-a^{2}+b^{2}}+a^{2}-b^{2}}{\sqrt {-a^{2}+b^{2}}\, b}\right )}{2 \sqrt {-a^{2}+b^{2}}\, \left (a +b \right )^{4} \left (a -b \right )^{4} d}\) | \(909\) |
Input:
int(sec(d*x+c)^4/(a+b*sin(d*x+c))^3,x,method=_RETURNVERBOSE)
Output:
1/d*(-1/3/(a-b)^3/(tan(1/2*d*x+1/2*c)+1)^3+1/2/(a-b)^3/(tan(1/2*d*x+1/2*c) +1)^2-1/2*(2*a-5*b)/(a-b)^4/(tan(1/2*d*x+1/2*c)+1)+2*b^4/(a-b)^4/(a+b)^4*( (1/2*b^2*(13*a^2-2*b^2)/a*tan(1/2*d*x+1/2*c)^3+1/2*b*(12*a^4+23*a^2*b^2-2* b^4)/a^2*tan(1/2*d*x+1/2*c)^2+1/2*b^2*(35*a^2-2*b^2)/a*tan(1/2*d*x+1/2*c)+ 6*a^2*b-1/2*b^3)/(tan(1/2*d*x+1/2*c)^2*a+2*b*tan(1/2*d*x+1/2*c)+a)^2+5/2*( 6*a^2+b^2)/(a^2-b^2)^(1/2)*arctan(1/2*(2*a*tan(1/2*d*x+1/2*c)+2*b)/(a^2-b^ 2)^(1/2)))-1/3/(a+b)^3/(tan(1/2*d*x+1/2*c)-1)^3-1/2/(a+b)^3/(tan(1/2*d*x+1 /2*c)-1)^2-1/2*(2*a+5*b)/(a+b)^4/(tan(1/2*d*x+1/2*c)-1))
Leaf count of result is larger than twice the leaf count of optimal. 558 vs. \(2 (250) = 500\).
Time = 0.17 (sec) , antiderivative size = 1200, normalized size of antiderivative = 4.55 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
integrate(sec(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm="fricas")
Output:
[1/12*(4*a^8*b - 16*a^6*b^3 + 24*a^4*b^5 - 16*a^2*b^7 + 4*b^9 + 2*(8*a^8*b - 64*a^6*b^3 - 16*a^4*b^5 + 87*a^2*b^7 - 15*b^9)*cos(d*x + c)^4 - 4*(2*a^ 8*b - a^6*b^3 - 9*a^4*b^5 + 13*a^2*b^7 - 5*b^9)*cos(d*x + c)^2 - 15*((6*a^ 2*b^6 + b^8)*cos(d*x + c)^5 - 2*(6*a^3*b^5 + a*b^7)*cos(d*x + c)^3*sin(d*x + c) - (6*a^4*b^4 + 7*a^2*b^6 + b^8)*cos(d*x + c)^3)*sqrt(-a^2 + b^2)*log (((2*a^2 - b^2)*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2 + 2*(a*cos (d*x + c)*sin(d*x + c) + b*cos(d*x + c))*sqrt(-a^2 + b^2))/(b^2*cos(d*x + c)^2 - 2*a*b*sin(d*x + c) - a^2 - b^2)) - 2*(2*a^9 - 8*a^7*b^2 + 12*a^5*b^ 4 - 8*a^3*b^6 + 2*a*b^8 - (4*a^7*b^2 - 32*a^5*b^4 - 53*a^3*b^6 + 81*a*b^8) *cos(d*x + c)^4 + 2*(2*a^9 - 15*a^7*b^2 + 33*a^5*b^4 - 29*a^3*b^6 + 9*a*b^ 8)*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)^5 - 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)^3*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)^3), 1/6*(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 - 64*a^6*b^3 - 16*a^4*b^5 + 87*a^2*b^7 - 15*b^9)*cos(d*x + c)^4 - 2* (2*a^8*b - a^6*b^3 - 9*a^4*b^5 + 13*a^2*b^7 - 5*b^9)*cos(d*x + c)^2 - 15*( (6*a^2*b^6 + b^8)*cos(d*x + c)^5 - 2*(6*a^3*b^5 + a*b^7)*cos(d*x + c)^3*si n(d*x + c) - (6*a^4*b^4 + 7*a^2*b^6 + b^8)*cos(d*x + c)^3)*sqrt(a^2 - b^2) *arctan(-(a*sin(d*x + c) + b)/(sqrt(a^2 - b^2)*cos(d*x + c))) - (2*a^9 ...
\[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\int \frac {\sec ^{4}{\left (c + d x \right )}}{\left (a + b \sin {\left (c + d x \right )}\right )^{3}}\, dx \] Input:
integrate(sec(d*x+c)**4/(a+b*sin(d*x+c))**3,x)
Output:
Integral(sec(c + d*x)**4/(a + b*sin(c + d*x))**3, x)
Exception generated. \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Exception raised: ValueError} \] Input:
integrate(sec(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm="maxima")
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(4*b^2-4*a^2>0)', see `assume?` f or more de
Leaf count of result is larger than twice the leaf count of optimal. 622 vs. \(2 (250) = 500\).
Time = 0.21 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.36 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
integrate(sec(d*x+c)^4/(a+b*sin(d*x+c))^3,x, algorithm="giac")
Output:
1/3*(15*(6*a^2*b^4 + b^6)*(pi*floor(1/2*(d*x + c)/pi + 1/2)*sgn(a) + arcta n((a*tan(1/2*d*x + 1/2*c) + b)/sqrt(a^2 - b^2)))/((a^8 - 4*a^6*b^2 + 6*a^4 *b^4 - 4*a^2*b^6 + b^8)*sqrt(a^2 - b^2)) + 3*(13*a^3*b^6*tan(1/2*d*x + 1/2 *c)^3 - 2*a*b^8*tan(1/2*d*x + 1/2*c)^3 + 12*a^4*b^5*tan(1/2*d*x + 1/2*c)^2 + 23*a^2*b^7*tan(1/2*d*x + 1/2*c)^2 - 2*b^9*tan(1/2*d*x + 1/2*c)^2 + 35*a ^3*b^6*tan(1/2*d*x + 1/2*c) - 2*a*b^8*tan(1/2*d*x + 1/2*c) + 12*a^4*b^5 - a^2*b^7)/((a^10 - 4*a^8*b^2 + 6*a^6*b^4 - 4*a^4*b^6 + a^2*b^8)*(a*tan(1/2* d*x + 1/2*c)^2 + 2*b*tan(1/2*d*x + 1/2*c) + a)^2) - 2*(3*a^5*tan(1/2*d*x + 1/2*c)^5 - 12*a^3*b^2*tan(1/2*d*x + 1/2*c)^5 - 27*a*b^4*tan(1/2*d*x + 1/2 *c)^5 - 9*a^4*b*tan(1/2*d*x + 1/2*c)^4 + 36*a^2*b^3*tan(1/2*d*x + 1/2*c)^4 + 9*b^5*tan(1/2*d*x + 1/2*c)^4 - 2*a^5*tan(1/2*d*x + 1/2*c)^3 + 32*a^3*b^ 2*tan(1/2*d*x + 1/2*c)^3 + 42*a*b^4*tan(1/2*d*x + 1/2*c)^3 - 60*a^2*b^3*ta n(1/2*d*x + 1/2*c)^2 - 12*b^5*tan(1/2*d*x + 1/2*c)^2 + 3*a^5*tan(1/2*d*x + 1/2*c) - 12*a^3*b^2*tan(1/2*d*x + 1/2*c) - 27*a*b^4*tan(1/2*d*x + 1/2*c) - 3*a^4*b + 32*a^2*b^3 + 7*b^5)/((a^8 - 4*a^6*b^2 + 6*a^4*b^4 - 4*a^2*b^6 + b^8)*(tan(1/2*d*x + 1/2*c)^2 - 1)^3))/d
Time = 19.49 (sec) , antiderivative size = 1167, normalized size of antiderivative = 4.42 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx=\text {Too large to display} \] Input:
int(1/(cos(c + d*x)^4*(a + b*sin(c + d*x))^3),x)
Output:
(5*b^4*atan(((5*b^4*(6*a^2 + b^2)*(2*a^8*b + 2*b^9 - 8*a^2*b^7 + 12*a^4*b^ 5 - 8*a^6*b^3))/(2*(a + b)^(9/2)*(a - b)^(9/2)) + (5*a*b^4*tan(c/2 + (d*x) /2)*(6*a^2 + b^2)*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2))/((a + b )^(9/2)*(a - b)^(9/2)))/(5*b^6 + 30*a^2*b^4))*(6*a^2 + b^2))/(d*(a + b)^(9 /2)*(a - b)^(9/2)) - ((2*tan(c/2 + (d*x)/2)^5*(255*a*b^6 + 2*a^7 + 62*a^3* b^4 - 4*a^5*b^2))/(3*(a^8 + b^8 - 4*a^2*b^6 + 6*a^4*b^4 - 4*a^6*b^2)) - (6 *a^6*b + 3*b^7 - 50*a^2*b^5 - 64*a^4*b^3)/(3*(a^2 - b^2)*(a^6 - b^6 + 3*a^ 2*b^4 - 3*a^4*b^2)) + (4*tan(c/2 + (d*x)/2)^7*(2*a^6 + 3*b^6 + 36*a^2*b^4 - 6*a^4*b^2))/(3*a*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (2*tan(c/2 + (d* x)/2)^2*(6*a^6*b + 3*b^7 - 64*a^2*b^5 - 50*a^4*b^3))/(3*a^2*(a^6 - b^6 + 3 *a^2*b^4 - 3*a^4*b^2)) - (tan(c/2 + (d*x)/2)^9*(13*a^2*b^6 - 2*b^8 - 2*a^8 + 18*a^4*b^4 + 8*a^6*b^2))/(a*(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)^8*(23*a^2*b^7 - 2*b^9 - 2*a^8*b + 78*a^4*b^5 + 8*a^6*b^3))/(a^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)*(6*a^8 - 6*b^8 + 161*a^2*b^6 + 202*a^4*b^4 - 48*a^6*b^2))/( 3*a*(a^2 - b^2)*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (4*tan(c/2 + (d*x)/ 2)^3*(2*a^8 + 3*b^8 - 133*a^2*b^6 - 86*a^4*b^4 + 4*a^6*b^2))/(3*a*(a^2 - b ^2)*(a^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) - (2*tan(c/2 + (d*x)/2)^4*(8*a^8* b - 9*b^9 + 156*a^2*b^7 + 188*a^4*b^5 - 28*a^6*b^3))/(3*a^2*(a^2 - b^2)*(a ^6 - b^6 + 3*a^2*b^4 - 3*a^4*b^2)) + (2*tan(c/2 + (d*x)/2)^6*(141*a^2*b...
Time = 0.26 (sec) , antiderivative size = 1948, normalized size of antiderivative = 7.38 \[ \int \frac {\sec ^4(c+d x)}{(a+b \sin (c+d x))^3} \, dx =\text {Too large to display} \] Input:
int(sec(d*x+c)^4/(a+b*sin(d*x+c))^3,x)
Output:
(360*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*co s(c + d*x)*sin(c + d*x)**4*a**2*b**7 + 60*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**4*b**9 + 720 *sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**3*a**3*b**6 + 120*sqrt(a**2 - b**2)*atan((tan((c + d* x)/2)*a + b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**3*a*b**8 + 360* sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**4*b**5 - 300*sqrt(a**2 - b**2)*atan((tan((c + d*x )/2)*a + b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*a**2*b**7 - 60 *sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)**2*b**9 - 720*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2) *a + b)/sqrt(a**2 - b**2))*cos(c + d*x)*sin(c + d*x)*a**3*b**6 - 120*sqrt( a**2 - b**2)*atan((tan((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*cos(c + d*x) *sin(c + d*x)*a*b**8 - 360*sqrt(a**2 - b**2)*atan((tan((c + d*x)/2)*a + b) /sqrt(a**2 - b**2))*cos(c + d*x)*a**4*b**5 - 60*sqrt(a**2 - b**2)*atan((ta n((c + d*x)/2)*a + b)/sqrt(a**2 - b**2))*cos(c + d*x)*a**2*b**7 + 6*cos(c + d*x)*sin(c + d*x)**4*a**8*b**2 - 42*cos(c + d*x)*sin(c + d*x)**4*a**6*b* *4 + 110*cos(c + d*x)*sin(c + d*x)**4*a**4*b**6 - 13*cos(c + d*x)*sin(c + d*x)**4*a**2*b**8 - 61*cos(c + d*x)*sin(c + d*x)**4*b**10 + 12*cos(c + d*x )*sin(c + d*x)**3*a**9*b - 84*cos(c + d*x)*sin(c + d*x)**3*a**7*b**3 + ...