\(\int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx\) [191]
Optimal result
Integrand size = 23, antiderivative size = 116 \[
\int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx=\frac {3 \text {arctanh}(\sin (c+d x)) \sqrt {\cos (c+d x)}}{8 b d \sqrt {b \cos (c+d x)}}+\frac {\sin (c+d x)}{4 b d \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {3 \sin (c+d x)}{8 b d \cos ^{\frac {3}{2}}(c+d x) \sqrt {b \cos (c+d x)}}
\]
[Out]
1/4*sin(d*x+c)/b/d/cos(d*x+c)^(7/2)/(b*cos(d*x+c))^(1/2)+3/8*sin(d*x+c)/b/d/cos(d*x+c)^(3/2)/(b*cos(d*x+c))^(1
/2)+3/8*arctanh(sin(d*x+c))*cos(d*x+c)^(1/2)/b/d/(b*cos(d*x+c))^(1/2)
Rubi [A] (verified)
Time = 0.04 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.00, number of
steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {18, 3853, 3855}
\[
\int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx=\frac {3 \sqrt {\cos (c+d x)} \text {arctanh}(\sin (c+d x))}{8 b d \sqrt {b \cos (c+d x)}}+\frac {3 \sin (c+d x)}{8 b d \cos ^{\frac {3}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {\sin (c+d x)}{4 b d \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)}}
\]
[In]
Int[1/(Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^(3/2)),x]
[Out]
(3*ArcTanh[Sin[c + d*x]]*Sqrt[Cos[c + d*x]])/(8*b*d*Sqrt[b*Cos[c + d*x]]) + Sin[c + d*x]/(4*b*d*Cos[c + d*x]^(
7/2)*Sqrt[b*Cos[c + d*x]]) + (3*Sin[c + d*x])/(8*b*d*Cos[c + d*x]^(3/2)*Sqrt[b*Cos[c + d*x]])
Rule 18
Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[a^(m - 1/2)*b^(n + 1/2)*(Sqrt[a*v]/Sqrt[b*v])
, Int[u*v^(m + n), x], x] /; FreeQ[{a, b, m}, x] && !IntegerQ[m] && ILtQ[n - 1/2, 0] && IntegerQ[m + n]
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]
Rubi steps \begin{align*}
\text {integral}& = \frac {\sqrt {\cos (c+d x)} \int \sec ^5(c+d x) \, dx}{b \sqrt {b \cos (c+d x)}} \\ & = \frac {\sin (c+d x)}{4 b d \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {\left (3 \sqrt {\cos (c+d x)}\right ) \int \sec ^3(c+d x) \, dx}{4 b \sqrt {b \cos (c+d x)}} \\ & = \frac {\sin (c+d x)}{4 b d \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {3 \sin (c+d x)}{8 b d \cos ^{\frac {3}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {\left (3 \sqrt {\cos (c+d x)}\right ) \int \sec (c+d x) \, dx}{8 b \sqrt {b \cos (c+d x)}} \\ & = \frac {3 \text {arctanh}(\sin (c+d x)) \sqrt {\cos (c+d x)}}{8 b d \sqrt {b \cos (c+d x)}}+\frac {\sin (c+d x)}{4 b d \cos ^{\frac {7}{2}}(c+d x) \sqrt {b \cos (c+d x)}}+\frac {3 \sin (c+d x)}{8 b d \cos ^{\frac {3}{2}}(c+d x) \sqrt {b \cos (c+d x)}} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.05 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.57
\[
\int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx=\frac {3 \text {arctanh}(\sin (c+d x)) \cos ^4(c+d x)+\left (2+3 \cos ^2(c+d x)\right ) \sin (c+d x)}{8 d \cos ^{\frac {5}{2}}(c+d x) (b \cos (c+d x))^{3/2}}
\]
[In]
Integrate[1/(Cos[c + d*x]^(7/2)*(b*Cos[c + d*x])^(3/2)),x]
[Out]
(3*ArcTanh[Sin[c + d*x]]*Cos[c + d*x]^4 + (2 + 3*Cos[c + d*x]^2)*Sin[c + d*x])/(8*d*Cos[c + d*x]^(5/2)*(b*Cos[
c + d*x])^(3/2))
Maple [A] (verified)
Time = 2.94 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.91
| | |
| method | result | size |
| | |
| default |
\(-\frac {3 \left (\cos ^{4}\left (d x +c \right )\right ) \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )-1\right )-3 \left (\cos ^{4}\left (d x +c \right )\right ) \ln \left (-\cot \left (d x +c \right )+\csc \left (d x +c \right )+1\right )-3 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-2 \sin \left (d x +c \right )}{8 d b \sqrt {\cos \left (d x +c \right ) b}\, \cos \left (d x +c \right )^{\frac {7}{2}}}\) |
\(106\) |
| risch |
\(-\frac {i \left (3 \,{\mathrm e}^{6 i \left (d x +c \right )}+11 \,{\mathrm e}^{4 i \left (d x +c \right )}-11 \,{\mathrm e}^{2 i \left (d x +c \right )}-3\right )}{8 b \sqrt {\cos \left (d x +c \right ) b}\, \sqrt {\cos \left (d x +c \right )}\, \left ({\mathrm e}^{2 i \left (d x +c \right )}+1\right )^{3} d}+\frac {3 \left (\sqrt {\cos }\left (d x +c \right )\right ) \ln \left ({\mathrm e}^{i \left (d x +c \right )}+i\right )}{8 b \sqrt {\cos \left (d x +c \right ) b}\, d}-\frac {3 \left (\sqrt {\cos }\left (d x +c \right )\right ) \ln \left ({\mathrm e}^{i \left (d x +c \right )}-i\right )}{8 b \sqrt {\cos \left (d x +c \right ) b}\, d}\) |
\(155\) |
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[In]
int(1/cos(d*x+c)^(7/2)/(cos(d*x+c)*b)^(3/2),x,method=_RETURNVERBOSE)
[Out]
-1/8/d*(3*cos(d*x+c)^4*ln(-cot(d*x+c)+csc(d*x+c)-1)-3*cos(d*x+c)^4*ln(-cot(d*x+c)+csc(d*x+c)+1)-3*cos(d*x+c)^2
*sin(d*x+c)-2*sin(d*x+c))/b/(cos(d*x+c)*b)^(1/2)/cos(d*x+c)^(7/2)
Fricas [A] (verification not implemented)
none
Time = 0.33 (sec) , antiderivative size = 233, normalized size of antiderivative = 2.01
\[
\int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx=\left [\frac {3 \, \sqrt {b} \cos \left (d x + c\right )^{5} \log \left (-\frac {b \cos \left (d x + c\right )^{3} - 2 \, \sqrt {b \cos \left (d x + c\right )} \sqrt {b} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right ) - 2 \, b \cos \left (d x + c\right )}{\cos \left (d x + c\right )^{3}}\right ) + 2 \, \sqrt {b \cos \left (d x + c\right )} {\left (3 \, \cos \left (d x + c\right )^{2} + 2\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{16 \, b^{2} d \cos \left (d x + c\right )^{5}}, -\frac {3 \, \sqrt {-b} \arctan \left (\frac {\sqrt {b \cos \left (d x + c\right )} \sqrt {-b} \sin \left (d x + c\right )}{b \sqrt {\cos \left (d x + c\right )}}\right ) \cos \left (d x + c\right )^{5} - \sqrt {b \cos \left (d x + c\right )} {\left (3 \, \cos \left (d x + c\right )^{2} + 2\right )} \sqrt {\cos \left (d x + c\right )} \sin \left (d x + c\right )}{8 \, b^{2} d \cos \left (d x + c\right )^{5}}\right ]
\]
[In]
integrate(1/cos(d*x+c)^(7/2)/(b*cos(d*x+c))^(3/2),x, algorithm="fricas")
[Out]
[1/16*(3*sqrt(b)*cos(d*x + c)^5*log(-(b*cos(d*x + c)^3 - 2*sqrt(b*cos(d*x + c))*sqrt(b)*sqrt(cos(d*x + c))*sin
(d*x + c) - 2*b*cos(d*x + c))/cos(d*x + c)^3) + 2*sqrt(b*cos(d*x + c))*(3*cos(d*x + c)^2 + 2)*sqrt(cos(d*x + c
))*sin(d*x + c))/(b^2*d*cos(d*x + c)^5), -1/8*(3*sqrt(-b)*arctan(sqrt(b*cos(d*x + c))*sqrt(-b)*sin(d*x + c)/(b
*sqrt(cos(d*x + c))))*cos(d*x + c)^5 - sqrt(b*cos(d*x + c))*(3*cos(d*x + c)^2 + 2)*sqrt(cos(d*x + c))*sin(d*x
+ c))/(b^2*d*cos(d*x + c)^5)]
Sympy [F(-1)]
Timed out. \[
\int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx=\text {Timed out}
\]
[In]
integrate(1/cos(d*x+c)**(7/2)/(b*cos(d*x+c))**(3/2),x)
[Out]
Timed out
Maxima [B] (verification not implemented)
Leaf count of result is larger than twice the leaf count of optimal. 1679 vs. \(2 (98) = 196\).
Time = 0.42 (sec) , antiderivative size = 1679, normalized size of antiderivative = 14.47
\[
\int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx=\text {Too large to display}
\]
[In]
integrate(1/cos(d*x+c)^(7/2)/(b*cos(d*x+c))^(3/2),x, algorithm="maxima")
[Out]
-1/16*(12*(sin(8*d*x + 8*c) + 4*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + 4*sin(2*d*x + 2*c))*cos(7/2*arctan2(si
n(2*d*x + 2*c), cos(2*d*x + 2*c))) + 44*(sin(8*d*x + 8*c) + 4*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + 4*sin(2*
d*x + 2*c))*cos(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 44*(sin(8*d*x + 8*c) + 4*sin(6*d*x + 6*c) +
6*sin(4*d*x + 4*c) + 4*sin(2*d*x + 2*c))*cos(3/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 12*(sin(8*d*x
+ 8*c) + 4*sin(6*d*x + 6*c) + 6*sin(4*d*x + 4*c) + 4*sin(2*d*x + 2*c))*cos(1/2*arctan2(sin(2*d*x + 2*c), cos(
2*d*x + 2*c))) - 3*(2*(4*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + 4*cos(2*d*x + 2*c) + 1)*cos(8*d*x + 8*c) + co
s(8*d*x + 8*c)^2 + 8*(6*cos(4*d*x + 4*c) + 4*cos(2*d*x + 2*c) + 1)*cos(6*d*x + 6*c) + 16*cos(6*d*x + 6*c)^2 +
12*(4*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) + 36*cos(4*d*x + 4*c)^2 + 16*cos(2*d*x + 2*c)^2 + 4*(2*sin(6*d*x
+ 6*c) + 3*sin(4*d*x + 4*c) + 2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + sin(8*d*x + 8*c)^2 + 16*(3*sin(4*d*x + 4*
c) + 2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 16*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 48*sin(4*d*x + 4*c
)*sin(2*d*x + 2*c) + 16*sin(2*d*x + 2*c)^2 + 8*cos(2*d*x + 2*c) + 1)*log(cos(1/2*arctan2(sin(2*d*x + 2*c), cos
(2*d*x + 2*c)))^2 + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 + 2*sin(1/2*arctan2(sin(2*d*x + 2*c
), cos(2*d*x + 2*c))) + 1) + 3*(2*(4*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + 4*cos(2*d*x + 2*c) + 1)*cos(8*d*x
+ 8*c) + cos(8*d*x + 8*c)^2 + 8*(6*cos(4*d*x + 4*c) + 4*cos(2*d*x + 2*c) + 1)*cos(6*d*x + 6*c) + 16*cos(6*d*x
+ 6*c)^2 + 12*(4*cos(2*d*x + 2*c) + 1)*cos(4*d*x + 4*c) + 36*cos(4*d*x + 4*c)^2 + 16*cos(2*d*x + 2*c)^2 + 4*(
2*sin(6*d*x + 6*c) + 3*sin(4*d*x + 4*c) + 2*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + sin(8*d*x + 8*c)^2 + 16*(3*si
n(4*d*x + 4*c) + 2*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + 16*sin(6*d*x + 6*c)^2 + 36*sin(4*d*x + 4*c)^2 + 48*sin
(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*sin(2*d*x + 2*c)^2 + 8*cos(2*d*x + 2*c) + 1)*log(cos(1/2*arctan2(sin(2*d*x
+ 2*c), cos(2*d*x + 2*c)))^2 + sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c)))^2 - 2*sin(1/2*arctan2(sin
(2*d*x + 2*c), cos(2*d*x + 2*c))) + 1) - 12*(cos(8*d*x + 8*c) + 4*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + 4*co
s(2*d*x + 2*c) + 1)*sin(7/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) - 44*(cos(8*d*x + 8*c) + 4*cos(6*d*x
+ 6*c) + 6*cos(4*d*x + 4*c) + 4*cos(2*d*x + 2*c) + 1)*sin(5/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))) + 4
4*(cos(8*d*x + 8*c) + 4*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + 4*cos(2*d*x + 2*c) + 1)*sin(3/2*arctan2(sin(2*
d*x + 2*c), cos(2*d*x + 2*c))) + 12*(cos(8*d*x + 8*c) + 4*cos(6*d*x + 6*c) + 6*cos(4*d*x + 4*c) + 4*cos(2*d*x
+ 2*c) + 1)*sin(1/2*arctan2(sin(2*d*x + 2*c), cos(2*d*x + 2*c))))/((b*cos(8*d*x + 8*c)^2 + 16*b*cos(6*d*x + 6*
c)^2 + 36*b*cos(4*d*x + 4*c)^2 + 16*b*cos(2*d*x + 2*c)^2 + b*sin(8*d*x + 8*c)^2 + 16*b*sin(6*d*x + 6*c)^2 + 36
*b*sin(4*d*x + 4*c)^2 + 48*b*sin(4*d*x + 4*c)*sin(2*d*x + 2*c) + 16*b*sin(2*d*x + 2*c)^2 + 2*(4*b*cos(6*d*x +
6*c) + 6*b*cos(4*d*x + 4*c) + 4*b*cos(2*d*x + 2*c) + b)*cos(8*d*x + 8*c) + 8*(6*b*cos(4*d*x + 4*c) + 4*b*cos(2
*d*x + 2*c) + b)*cos(6*d*x + 6*c) + 12*(4*b*cos(2*d*x + 2*c) + b)*cos(4*d*x + 4*c) + 8*b*cos(2*d*x + 2*c) + 4*
(2*b*sin(6*d*x + 6*c) + 3*b*sin(4*d*x + 4*c) + 2*b*sin(2*d*x + 2*c))*sin(8*d*x + 8*c) + 16*(3*b*sin(4*d*x + 4*
c) + 2*b*sin(2*d*x + 2*c))*sin(6*d*x + 6*c) + b)*sqrt(b)*d)
Giac [F]
\[
\int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx=\int { \frac {1}{\left (b \cos \left (d x + c\right )\right )^{\frac {3}{2}} \cos \left (d x + c\right )^{\frac {7}{2}}} \,d x }
\]
[In]
integrate(1/cos(d*x+c)^(7/2)/(b*cos(d*x+c))^(3/2),x, algorithm="giac")
[Out]
integrate(1/((b*cos(d*x + c))^(3/2)*cos(d*x + c)^(7/2)), x)
Mupad [F(-1)]
Timed out. \[
\int \frac {1}{\cos ^{\frac {7}{2}}(c+d x) (b \cos (c+d x))^{3/2}} \, dx=\int \frac {1}{{\cos \left (c+d\,x\right )}^{7/2}\,{\left (b\,\cos \left (c+d\,x\right )\right )}^{3/2}} \,d x
\]
[In]
int(1/(cos(c + d*x)^(7/2)*(b*cos(c + d*x))^(3/2)),x)
[Out]
int(1/(cos(c + d*x)^(7/2)*(b*cos(c + d*x))^(3/2)), x)