3.4.84 \(\int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx\) [384]

3.4.84.1 Optimal result

Integrand size = 33, antiderivative size = 630 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx=\frac {\left (16 a^4 A+41 a^2 A b^2-105 A b^4-42 a^3 b B+90 a b^3 B\right ) \cot (c+d x) E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a^4 b \sqrt {a+b} d}+\frac {\left (105 A b^3+5 a b^2 (7 A-18 B)+4 a^3 (4 A+3 B)-6 a^2 b (A+5 B)\right ) \cot (c+d x) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{24 a^4 \sqrt {a+b} d}+\frac {\sqrt {a+b} \left (12 a^2 A b+35 A b^3-8 a^3 B-30 a b^2 B\right ) \cot (c+d x) \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right ) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (1+\sec (c+d x))}{a-b}}}{8 a^5 d}+\frac {\left (16 a^2 A+35 A b^2-30 a b B\right ) \sin (c+d x)}{24 a^3 d \sqrt {a+b \sec (c+d x)}}-\frac {(7 A b-6 a B) \cos (c+d x) \sin (c+d x)}{12 a^2 d \sqrt {a+b \sec (c+d x)}}+\frac {A \cos ^2(c+d x) \sin (c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}+\frac {b \left (16 a^4 A+41 a^2 A b^2-105 A b^4-42 a^3 b B+90 a b^3 B\right ) \tan (c+d x)}{24 a^4 \left (a^2-b^2\right ) d \sqrt {a+b \sec (c+d x)}} \]

output

1/24*(16*A*a^4+41*A*a^2*b^2-105*A*b^4-42*B*a^3*b+90*B*a*b^3)*cot(d*x+c)*El 
lipticE((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*(1-sec( 
d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^4/b/d/(a+b)^(1/2)+1 
/24*(105*A*b^3+5*a*b^2*(7*A-18*B)+4*a^3*(4*A+3*B)-6*a^2*b*(A+5*B))*cot(d*x 
+c)*EllipticF((a+b*sec(d*x+c))^(1/2)/(a+b)^(1/2),((a+b)/(a-b))^(1/2))*(b*( 
1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^4/d/(a+b)^(1/ 
2)+1/8*(12*A*a^2*b+35*A*b^3-8*B*a^3-30*B*a*b^2)*cot(d*x+c)*EllipticPi((a+b 
*sec(d*x+c))^(1/2)/(a+b)^(1/2),(a+b)/a,((a+b)/(a-b))^(1/2))*(a+b)^(1/2)*(b 
*(1-sec(d*x+c))/(a+b))^(1/2)*(-b*(1+sec(d*x+c))/(a-b))^(1/2)/a^5/d+1/24*(1 
6*A*a^2+35*A*b^2-30*B*a*b)*sin(d*x+c)/a^3/d/(a+b*sec(d*x+c))^(1/2)-1/12*(7 
*A*b-6*B*a)*cos(d*x+c)*sin(d*x+c)/a^2/d/(a+b*sec(d*x+c))^(1/2)+1/3*A*cos(d 
*x+c)^2*sin(d*x+c)/a/d/(a+b*sec(d*x+c))^(1/2)+1/24*b*(16*A*a^4+41*A*a^2*b^ 
2-105*A*b^4-42*B*a^3*b+90*B*a*b^3)*tan(d*x+c)/a^4/(a^2-b^2)/d/(a+b*sec(d*x 
+c))^(1/2)
 

3.4.84.2 Mathematica [B] (warning: unable to verify)

Leaf count is larger than twice the leaf count of optimal. \(2319\) vs. \(2(630)=1260\).

Time = 25.40 (sec) , antiderivative size = 2319, normalized size of antiderivative = 3.68 \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx=\text {Result too large to show} \]

input

Integrate[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2) 
,x]
 

output

((b + a*Cos[c + d*x])^2*Sec[c + d*x]^2*(-1/12*((a^4*A - a^2*A*b^2 + 24*A*b 
^4 - 24*a*b^3*B)*Sin[c + d*x])/(a^4*(-a^2 + b^2)) - (2*(A*b^5*Sin[c + d*x] 
 - a*b^4*B*Sin[c + d*x]))/(a^4*(a^2 - b^2)*(b + a*Cos[c + d*x])) + ((-11*A 
*b + 6*a*B)*Sin[2*(c + d*x)])/(24*a^3) + (A*Sin[3*(c + d*x)])/(12*a^2)))/( 
d*(a + b*Sec[c + d*x])^(3/2)) - ((b + a*Cos[c + d*x])^(3/2)*Sec[c + d*x]^( 
3/2)*Sqrt[(1 - Tan[(c + d*x)/2]^2)^(-1)]*Sqrt[(a + b - a*Tan[(c + d*x)/2]^ 
2 + b*Tan[(c + d*x)/2]^2)/(1 + Tan[(c + d*x)/2]^2)]*(16*a^5*A*Tan[(c + d*x 
)/2] + 16*a^4*A*b*Tan[(c + d*x)/2] + 41*a^3*A*b^2*Tan[(c + d*x)/2] + 41*a^ 
2*A*b^3*Tan[(c + d*x)/2] - 105*a*A*b^4*Tan[(c + d*x)/2] - 105*A*b^5*Tan[(c 
 + d*x)/2] - 42*a^4*b*B*Tan[(c + d*x)/2] - 42*a^3*b^2*B*Tan[(c + d*x)/2] + 
 90*a^2*b^3*B*Tan[(c + d*x)/2] + 90*a*b^4*B*Tan[(c + d*x)/2] - 32*a^5*A*Ta 
n[(c + d*x)/2]^3 - 82*a^3*A*b^2*Tan[(c + d*x)/2]^3 + 210*a*A*b^4*Tan[(c + 
d*x)/2]^3 + 84*a^4*b*B*Tan[(c + d*x)/2]^3 - 180*a^2*b^3*B*Tan[(c + d*x)/2] 
^3 + 16*a^5*A*Tan[(c + d*x)/2]^5 - 16*a^4*A*b*Tan[(c + d*x)/2]^5 + 41*a^3* 
A*b^2*Tan[(c + d*x)/2]^5 - 41*a^2*A*b^3*Tan[(c + d*x)/2]^5 - 105*a*A*b^4*T 
an[(c + d*x)/2]^5 + 105*A*b^5*Tan[(c + d*x)/2]^5 - 42*a^4*b*B*Tan[(c + d*x 
)/2]^5 + 42*a^3*b^2*B*Tan[(c + d*x)/2]^5 + 90*a^2*b^3*B*Tan[(c + d*x)/2]^5 
 - 90*a*b^4*B*Tan[(c + d*x)/2]^5 - 72*a^4*A*b*EllipticPi[-1, ArcSin[Tan[(c 
 + d*x)/2]], (a - b)/(a + b)]*Sqrt[1 - Tan[(c + d*x)/2]^2]*Sqrt[(a + b - a 
*Tan[(c + d*x)/2]^2 + b*Tan[(c + d*x)/2]^2)/(a + b)] - 138*a^2*A*b^3*El...
 

3.4.84.3 Rubi [A] (verified)

Time = 3.25 (sec) , antiderivative size = 672, normalized size of antiderivative = 1.07, number of steps used = 20, number of rules used = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.606, Rules used = {3042, 4522, 27, 3042, 4592, 27, 3042, 4592, 27, 3042, 4548, 27, 3042, 4546, 3042, 4409, 3042, 4271, 4319, 4492}

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.

\(\displaystyle \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx\)

\(\Big \downarrow \) 3042

\(\displaystyle \int \frac {A+B \csc \left (c+d x+\frac {\pi }{2}\right )}{\csc \left (c+d x+\frac {\pi }{2}\right )^3 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx\)

\(\Big \downarrow \) 4522

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {\cos ^2(c+d x) \left (-5 A b \sec ^2(c+d x)-4 a A \sec (c+d x)+7 A b-6 a B\right )}{2 (a+b \sec (c+d x))^{3/2}}dx}{3 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {\cos ^2(c+d x) \left (-5 A b \sec ^2(c+d x)-4 a A \sec (c+d x)+7 A b-6 a B\right )}{(a+b \sec (c+d x))^{3/2}}dx}{6 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {-5 A b \csc \left (c+d x+\frac {\pi }{2}\right )^2-4 a A \csc \left (c+d x+\frac {\pi }{2}\right )+7 A b-6 a B}{\csc \left (c+d x+\frac {\pi }{2}\right )^2 \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{6 a}\)

\(\Big \downarrow \) 4592

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {\cos (c+d x) \left (16 A a^2-30 b B a+6 (A b+2 a B) \sec (c+d x) a+35 A b^2-3 b (7 A b-6 a B) \sec ^2(c+d x)\right )}{2 (a+b \sec (c+d x))^{3/2}}dx}{2 a}}{6 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {\cos (c+d x) \left (16 A a^2-30 b B a+6 (A b+2 a B) \sec (c+d x) a+35 A b^2-3 b (7 A b-6 a B) \sec ^2(c+d x)\right )}{(a+b \sec (c+d x))^{3/2}}dx}{4 a}}{6 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {16 A a^2-30 b B a+6 (A b+2 a B) \csc \left (c+d x+\frac {\pi }{2}\right ) a+35 A b^2-3 b (7 A b-6 a B) \csc \left (c+d x+\frac {\pi }{2}\right )^2}{\csc \left (c+d x+\frac {\pi }{2}\right ) \left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{4 a}}{6 a}\)

\(\Big \downarrow \) 4592

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {-b \left (16 A a^2-30 b B a+35 A b^2\right ) \sec ^2(c+d x)+6 a b (7 A b-6 a B) \sec (c+d x)+3 \left (-8 B a^3+12 A b a^2-30 b^2 B a+35 A b^3\right )}{2 (a+b \sec (c+d x))^{3/2}}dx}{a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {-b \left (16 A a^2-30 b B a+35 A b^2\right ) \sec ^2(c+d x)+6 a b (7 A b-6 a B) \sec (c+d x)+3 \left (-8 B a^3+12 A b a^2-30 b^2 B a+35 A b^3\right )}{(a+b \sec (c+d x))^{3/2}}dx}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\int \frac {-b \left (16 A a^2-30 b B a+35 A b^2\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+6 a b (7 A b-6 a B) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (-8 B a^3+12 A b a^2-30 b^2 B a+35 A b^3\right )}{\left (a+b \csc \left (c+d x+\frac {\pi }{2}\right )\right )^{3/2}}dx}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 4548

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {-\frac {2 \int -\frac {b \left (16 A a^4-42 b B a^3+41 A b^2 a^2+90 b^3 B a-105 A b^4\right ) \sec ^2(c+d x)+2 a b \left (-6 B a^3+11 A b a^2+30 b^2 B a-35 A b^3\right ) \sec (c+d x)+3 \left (a^2-b^2\right ) \left (-8 B a^3+12 A b a^2-30 b^2 B a+35 A b^3\right )}{2 \sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\int \frac {b \left (16 A a^4-42 b B a^3+41 A b^2 a^2+90 b^3 B a-105 A b^4\right ) \sec ^2(c+d x)+2 a b \left (-6 B a^3+11 A b a^2+30 b^2 B a-35 A b^3\right ) \sec (c+d x)+3 \left (a^2-b^2\right ) \left (-8 B a^3+12 A b a^2-30 b^2 B a+35 A b^3\right )}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\int \frac {b \left (16 A a^4-42 b B a^3+41 A b^2 a^2+90 b^3 B a-105 A b^4\right ) \csc \left (c+d x+\frac {\pi }{2}\right )^2+2 a b \left (-6 B a^3+11 A b a^2+30 b^2 B a-35 A b^3\right ) \csc \left (c+d x+\frac {\pi }{2}\right )+3 \left (a^2-b^2\right ) \left (-8 B a^3+12 A b a^2-30 b^2 B a+35 A b^3\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 4546

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \int \frac {\sec (c+d x) (\sec (c+d x)+1)}{\sqrt {a+b \sec (c+d x)}}dx+\int \frac {3 \left (a^2-b^2\right ) \left (-8 B a^3+12 A b a^2-30 b^2 B a+35 A b^3\right )+\left (2 a b \left (-6 B a^3+11 A b a^2+30 b^2 B a-35 A b^3\right )-b \left (16 A a^4-42 b B a^3+41 A b^2 a^2+90 b^3 B a-105 A b^4\right )\right ) \sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+\int \frac {3 \left (a^2-b^2\right ) \left (-8 B a^3+12 A b a^2-30 b^2 B a+35 A b^3\right )+\left (2 a b \left (-6 B a^3+11 A b a^2+30 b^2 B a-35 A b^3\right )-b \left (16 A a^4-42 b B a^3+41 A b^2 a^2+90 b^3 B a-105 A b^4\right )\right ) \csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 4409

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {3 \left (a^2-b^2\right ) \left (-8 a^3 B+12 a^2 A b-30 a b^2 B+35 A b^3\right ) \int \frac {1}{\sqrt {a+b \sec (c+d x)}}dx-b (a-b) \left (4 a^3 (4 A+3 B)-6 a^2 b (A+5 B)+5 a b^2 (7 A-18 B)+105 A b^3\right ) \int \frac {\sec (c+d x)}{\sqrt {a+b \sec (c+d x)}}dx+b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 3042

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {3 \left (a^2-b^2\right ) \left (-8 a^3 B+12 a^2 A b-30 a b^2 B+35 A b^3\right ) \int \frac {1}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-b (a-b) \left (4 a^3 (4 A+3 B)-6 a^2 b (A+5 B)+5 a b^2 (7 A-18 B)+105 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx}{a \left (a^2-b^2\right )}-\frac {2 b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 4271

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {-b (a-b) \left (4 a^3 (4 A+3 B)-6 a^2 b (A+5 B)+5 a b^2 (7 A-18 B)+105 A b^3\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx+b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {6 \sqrt {a+b} \left (a^2-b^2\right ) \left (-8 a^3 B+12 a^2 A b-30 a b^2 B+35 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{a \left (a^2-b^2\right )}-\frac {2 b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 4319

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \int \frac {\csc \left (c+d x+\frac {\pi }{2}\right ) \left (\csc \left (c+d x+\frac {\pi }{2}\right )+1\right )}{\sqrt {a+b \csc \left (c+d x+\frac {\pi }{2}\right )}}dx-\frac {2 (a-b) \sqrt {a+b} \left (4 a^3 (4 A+3 B)-6 a^2 b (A+5 B)+5 a b^2 (7 A-18 B)+105 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {6 \sqrt {a+b} \left (a^2-b^2\right ) \left (-8 a^3 B+12 a^2 A b-30 a b^2 B+35 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}}{a \left (a^2-b^2\right )}-\frac {2 b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{2 a}}{4 a}}{6 a}\)

\(\Big \downarrow \) 4492

\(\displaystyle \frac {A \sin (c+d x) \cos ^2(c+d x)}{3 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {(7 A b-6 a B) \sin (c+d x) \cos (c+d x)}{2 a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {\left (16 a^2 A-30 a b B+35 A b^2\right ) \sin (c+d x)}{a d \sqrt {a+b \sec (c+d x)}}-\frac {\frac {-\frac {2 (a-b) \sqrt {a+b} \left (4 a^3 (4 A+3 B)-6 a^2 b (A+5 B)+5 a b^2 (7 A-18 B)+105 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{d}-\frac {6 \sqrt {a+b} \left (a^2-b^2\right ) \left (-8 a^3 B+12 a^2 A b-30 a b^2 B+35 A b^3\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} \operatorname {EllipticPi}\left (\frac {a+b}{a},\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right ),\frac {a+b}{a-b}\right )}{a d}-\frac {2 (a-b) \sqrt {a+b} \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \cot (c+d x) \sqrt {\frac {b (1-\sec (c+d x))}{a+b}} \sqrt {-\frac {b (\sec (c+d x)+1)}{a-b}} E\left (\arcsin \left (\frac {\sqrt {a+b \sec (c+d x)}}{\sqrt {a+b}}\right )|\frac {a+b}{a-b}\right )}{b d}}{a \left (a^2-b^2\right )}-\frac {2 b \left (16 a^4 A-42 a^3 b B+41 a^2 A b^2+90 a b^3 B-105 A b^4\right ) \tan (c+d x)}{a d \left (a^2-b^2\right ) \sqrt {a+b \sec (c+d x)}}}{2 a}}{4 a}}{6 a}\)

input

Int[(Cos[c + d*x]^3*(A + B*Sec[c + d*x]))/(a + b*Sec[c + d*x])^(3/2),x]
 

output

(A*Cos[c + d*x]^2*Sin[c + d*x])/(3*a*d*Sqrt[a + b*Sec[c + d*x]]) - (((7*A* 
b - 6*a*B)*Cos[c + d*x]*Sin[c + d*x])/(2*a*d*Sqrt[a + b*Sec[c + d*x]]) - ( 
((16*a^2*A + 35*A*b^2 - 30*a*b*B)*Sin[c + d*x])/(a*d*Sqrt[a + b*Sec[c + d* 
x]]) - (((-2*(a - b)*Sqrt[a + b]*(16*a^4*A + 41*a^2*A*b^2 - 105*A*b^4 - 42 
*a^3*b*B + 90*a*b^3*B)*Cot[c + d*x]*EllipticE[ArcSin[Sqrt[a + b*Sec[c + d* 
x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sq 
rt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(b*d) - (2*(a - b)*Sqrt[a + b]*(105 
*A*b^3 + 5*a*b^2*(7*A - 18*B) + 4*a^3*(4*A + 3*B) - 6*a^2*b*(A + 5*B))*Cot 
[c + d*x]*EllipticF[ArcSin[Sqrt[a + b*Sec[c + d*x]]/Sqrt[a + b]], (a + b)/ 
(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + b)]*Sqrt[-((b*(1 + Sec[c + d*x]) 
)/(a - b))])/d - (6*Sqrt[a + b]*(a^2 - b^2)*(12*a^2*A*b + 35*A*b^3 - 8*a^3 
*B - 30*a*b^2*B)*Cot[c + d*x]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Sec[ 
c + d*x]]/Sqrt[a + b]], (a + b)/(a - b)]*Sqrt[(b*(1 - Sec[c + d*x]))/(a + 
b)]*Sqrt[-((b*(1 + Sec[c + d*x]))/(a - b))])/(a*d))/(a*(a^2 - b^2)) - (2*b 
*(16*a^4*A + 41*a^2*A*b^2 - 105*A*b^4 - 42*a^3*b*B + 90*a*b^3*B)*Tan[c + d 
*x])/(a*(a^2 - b^2)*d*Sqrt[a + b*Sec[c + d*x]]))/(2*a))/(4*a))/(6*a)
 

3.4.84.3.1 Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 3042

Int[u_, x_Symbol] :> Int[DeactivateTrig[u, x], x] /; FunctionOfTrigOfLinear 
Q[u, x]
 

rule 4271

Int[1/Sqrt[csc[(c_.) + (d_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[2*(Rt[a 
 + b, 2]/(a*d*Cot[c + d*x]))*Sqrt[b*((1 - Csc[c + d*x])/(a + b))]*Sqrt[(-b) 
*((1 + Csc[c + d*x])/(a - b))]*EllipticPi[(a + b)/a, ArcSin[Sqrt[a + b*Csc[ 
c + d*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, c, d}, x] && 
NeQ[a^2 - b^2, 0]
 

rule 4319

Int[csc[(e_.) + (f_.)*(x_)]/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_S 
ymbol] :> Simp[-2*(Rt[a + b, 2]/(b*f*Cot[e + f*x]))*Sqrt[(b*(1 - Csc[e + f* 
x]))/(a + b)]*Sqrt[(-b)*((1 + Csc[e + f*x])/(a - b))]*EllipticF[ArcSin[Sqrt 
[a + b*Csc[e + f*x]]/Rt[a + b, 2]], (a + b)/(a - b)], x] /; FreeQ[{a, b, e, 
 f}, x] && NeQ[a^2 - b^2, 0]
 

rule 4409

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.) + (c_))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_ 
.) + (a_)], x_Symbol] :> Simp[c   Int[1/Sqrt[a + b*Csc[e + f*x]], x], x] + 
Simp[d   Int[Csc[e + f*x]/Sqrt[a + b*Csc[e + f*x]], x], x] /; FreeQ[{a, b, 
c, d, e, f}, x] && NeQ[b*c - a*d, 0] && NeQ[a^2 - b^2, 0]
 

rule 4492

Int[(csc[(e_.) + (f_.)*(x_)]*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)))/Sqrt[c 
sc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Simp[-2*(A*b - a*B)*Rt[a 
 + b*(B/A), 2]*Sqrt[b*((1 - Csc[e + f*x])/(a + b))]*(Sqrt[(-b)*((1 + Csc[e 
+ f*x])/(a - b))]/(b^2*f*Cot[e + f*x]))*EllipticE[ArcSin[Sqrt[a + b*Csc[e + 
 f*x]]/Rt[a + b*(B/A), 2]], (a*A + b*B)/(a*A - b*B)], x] /; FreeQ[{a, b, e, 
 f, A, B}, x] && NeQ[a^2 - b^2, 0] && EqQ[A^2 - B^2, 0]
 

rule 4522

Int[(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + ( 
a_))^(m_)*(csc[(e_.) + (f_.)*(x_)]*(B_.) + (A_)), x_Symbol] :> Simp[A*Cot[e 
 + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d*Csc[e + f*x])^n/(a*f*n)), x] + Sim 
p[1/(a*d*n)   Int[(a + b*Csc[e + f*x])^m*(d*Csc[e + f*x])^(n + 1)*Simp[a*B* 
n - A*b*(m + n + 1) + A*a*(n + 1)*Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f* 
x]^2, x], x], x] /; FreeQ[{a, b, d, e, f, A, B, m}, x] && NeQ[A*b - a*B, 0] 
 && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
 

rule 4546

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))/Sqrt[csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_)], x_Symbol] :> Int[(A + (B - C 
)*Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]], x] + Simp[C   Int[Csc[e + f*x]*(( 
1 + Csc[e + f*x])/Sqrt[a + b*Csc[e + f*x]]), x], x] /; FreeQ[{a, b, e, f, A 
, B, C}, x] && NeQ[a^2 - b^2, 0]
 

rule 4548

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a_))^(m_), x_Symbol] :> Simp[(A*b^2 - 
a*b*B + a^2*C)*Cot[e + f*x]*((a + b*Csc[e + f*x])^(m + 1)/(a*f*(m + 1)*(a^2 
 - b^2))), x] + Simp[1/(a*(m + 1)*(a^2 - b^2))   Int[(a + b*Csc[e + f*x])^( 
m + 1)*Simp[A*(a^2 - b^2)*(m + 1) - a*(A*b - a*B + b*C)*(m + 1)*Csc[e + f*x 
] + (A*b^2 - a*b*B + a^2*C)*(m + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, 
 b, e, f, A, B, C}, x] && NeQ[a^2 - b^2, 0] && LtQ[m, -1]
 

rule 4592

Int[((A_.) + csc[(e_.) + (f_.)*(x_)]*(B_.) + csc[(e_.) + (f_.)*(x_)]^2*(C_. 
))*(csc[(e_.) + (f_.)*(x_)]*(d_.))^(n_)*(csc[(e_.) + (f_.)*(x_)]*(b_.) + (a 
_))^(m_), x_Symbol] :> Simp[A*Cot[e + f*x]*(a + b*Csc[e + f*x])^(m + 1)*((d 
*Csc[e + f*x])^n/(a*f*n)), x] + Simp[1/(a*d*n)   Int[(a + b*Csc[e + f*x])^m 
*(d*Csc[e + f*x])^(n + 1)*Simp[a*B*n - A*b*(m + n + 1) + a*(A + A*n + C*n)* 
Csc[e + f*x] + A*b*(m + n + 2)*Csc[e + f*x]^2, x], x], x] /; FreeQ[{a, b, d 
, e, f, A, B, C, m}, x] && NeQ[a^2 - b^2, 0] && LeQ[n, -1]
 

3.4.84.4 Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(6976\) vs. \(2(581)=1162\).

Time = 12.85 (sec) , antiderivative size = 6977, normalized size of antiderivative = 11.07

method	result	size
default	\(\text {Expression too large to display}\)	\(6977\)

input

int(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(3/2),x,method=_RETURNV 
ERBOSE)
 

output

result too large to display
 

3.4.84.5 Fricas [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx=\text {Timed out} \]

input

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(3/2),x, algorith 
m="fricas")
 

output

Timed out
 

3.4.84.6 Sympy [F]

\[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {\left (A + B \sec {\left (c + d x \right )}\right ) \cos ^{3}{\left (c + d x \right )}}{\left (a + b \sec {\left (c + d x \right )}\right )^{\frac {3}{2}}}\, dx \]

input

integrate(cos(d*x+c)**3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))**(3/2),x)
 

output

Integral((A + B*sec(c + d*x))*cos(c + d*x)**3/(a + b*sec(c + d*x))**(3/2), 
 x)
 

3.4.84.7 Maxima [F]

\[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

input

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(3/2),x, algorith 
m="maxima")
 

output

integrate((B*sec(d*x + c) + A)*cos(d*x + c)^3/(b*sec(d*x + c) + a)^(3/2), 
x)
 

3.4.84.8 Giac [F]

\[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx=\int { \frac {{\left (B \sec \left (d x + c\right ) + A\right )} \cos \left (d x + c\right )^{3}}{{\left (b \sec \left (d x + c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]

input

integrate(cos(d*x+c)^3*(A+B*sec(d*x+c))/(a+b*sec(d*x+c))^(3/2),x, algorith 
m="giac")
 

output

integrate((B*sec(d*x + c) + A)*cos(d*x + c)^3/(b*sec(d*x + c) + a)^(3/2), 
x)
 

3.4.84.9 Mupad [F(-1)]

Timed out. \[ \int \frac {\cos ^3(c+d x) (A+B \sec (c+d x))}{(a+b \sec (c+d x))^{3/2}} \, dx=\int \frac {{\cos \left (c+d\,x\right )}^3\,\left (A+\frac {B}{\cos \left (c+d\,x\right )}\right )}{{\left (a+\frac {b}{\cos \left (c+d\,x\right )}\right )}^{3/2}} \,d x \]

input

int((cos(c + d*x)^3*(A + B/cos(c + d*x)))/(a + b/cos(c + d*x))^(3/2),x)
 

output

int((cos(c + d*x)^3*(A + B/cos(c + d*x)))/(a + b/cos(c + d*x))^(3/2), x)