Optimal. Leaf size=113 \[ \frac {\tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}-\frac {1}{4 a^2 d \sqrt {a \sin (c+d x)+a}}-\frac {1}{6 a d (a \sin (c+d x)+a)^{3/2}}-\frac {1}{5 d (a \sin (c+d x)+a)^{5/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2667, 51, 63, 206} \[ -\frac {1}{4 a^2 d \sqrt {a \sin (c+d x)+a}}+\frac {\tanh ^{-1}\left (\frac {\sqrt {a \sin (c+d x)+a}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}-\frac {1}{6 a d (a \sin (c+d x)+a)^{3/2}}-\frac {1}{5 d (a \sin (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 51
Rule 63
Rule 206
Rule 2667
Rubi steps
\begin {align*} \int \frac {\sec (c+d x)}{(a+a \sin (c+d x))^{5/2}} \, dx &=\frac {a \operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{7/2}} \, dx,x,a \sin (c+d x)\right )}{d}\\ &=-\frac {1}{5 d (a+a \sin (c+d x))^{5/2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{5/2}} \, dx,x,a \sin (c+d x)\right )}{2 d}\\ &=-\frac {1}{5 d (a+a \sin (c+d x))^{5/2}}-\frac {1}{6 a d (a+a \sin (c+d x))^{3/2}}+\frac {\operatorname {Subst}\left (\int \frac {1}{(a-x) (a+x)^{3/2}} \, dx,x,a \sin (c+d x)\right )}{4 a d}\\ &=-\frac {1}{5 d (a+a \sin (c+d x))^{5/2}}-\frac {1}{6 a d (a+a \sin (c+d x))^{3/2}}-\frac {1}{4 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{(a-x) \sqrt {a+x}} \, dx,x,a \sin (c+d x)\right )}{8 a^2 d}\\ &=-\frac {1}{5 d (a+a \sin (c+d x))^{5/2}}-\frac {1}{6 a d (a+a \sin (c+d x))^{3/2}}-\frac {1}{4 a^2 d \sqrt {a+a \sin (c+d x)}}+\frac {\operatorname {Subst}\left (\int \frac {1}{2 a-x^2} \, dx,x,\sqrt {a+a \sin (c+d x)}\right )}{4 a^2 d}\\ &=\frac {\tanh ^{-1}\left (\frac {\sqrt {a+a \sin (c+d x)}}{\sqrt {2} \sqrt {a}}\right )}{4 \sqrt {2} a^{5/2} d}-\frac {1}{5 d (a+a \sin (c+d x))^{5/2}}-\frac {1}{6 a d (a+a \sin (c+d x))^{3/2}}-\frac {1}{4 a^2 d \sqrt {a+a \sin (c+d x)}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.09, size = 41, normalized size = 0.36 \[ -\frac {\, _2F_1\left (-\frac {5}{2},1;-\frac {3}{2};\frac {1}{2} (\sin (c+d x)+1)\right )}{5 d (a \sin (c+d x)+a)^{5/2}} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.74, size = 169, normalized size = 1.50 \[ \frac {15 \, \sqrt {2} {\left (3 \, \cos \left (d x + c\right )^{2} + {\left (\cos \left (d x + c\right )^{2} - 4\right )} \sin \left (d x + c\right ) - 4\right )} \sqrt {a} \log \left (-\frac {a \sin \left (d x + c\right ) + 2 \, \sqrt {2} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {a} + 3 \, a}{\sin \left (d x + c\right ) - 1}\right ) - 4 \, {\left (15 \, \cos \left (d x + c\right )^{2} - 40 \, \sin \left (d x + c\right ) - 52\right )} \sqrt {a \sin \left (d x + c\right ) + a}}{240 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{2} - 4 \, a^{3} d + {\left (a^{3} d \cos \left (d x + c\right )^{2} - 4 \, a^{3} d\right )} \sin \left (d x + c\right )\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.17, size = 88, normalized size = 0.78 \[ -\frac {2 a \left (\frac {1}{8 a^{3} \sqrt {a +a \sin \left (d x +c \right )}}+\frac {1}{12 a^{2} \left (a +a \sin \left (d x +c \right )\right )^{\frac {3}{2}}}+\frac {1}{10 a \left (a +a \sin \left (d x +c \right )\right )^{\frac {5}{2}}}-\frac {\sqrt {2}\, \arctanh \left (\frac {\sqrt {a +a \sin \left (d x +c \right )}\, \sqrt {2}}{2 \sqrt {a}}\right )}{16 a^{\frac {7}{2}}}\right )}{d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 1.43, size = 114, normalized size = 1.01 \[ -\frac {\frac {15 \, \sqrt {2} \log \left (-\frac {\sqrt {2} \sqrt {a} - \sqrt {a \sin \left (d x + c\right ) + a}}{\sqrt {2} \sqrt {a} + \sqrt {a \sin \left (d x + c\right ) + a}}\right )}{a^{\frac {3}{2}}} + \frac {4 \, {\left (15 \, {\left (a \sin \left (d x + c\right ) + a\right )}^{2} + 10 \, {\left (a \sin \left (d x + c\right ) + a\right )} a + 12 \, a^{2}\right )}}{{\left (a \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} a}}{240 \, a d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {1}{\cos \left (c+d\,x\right )\,{\left (a+a\,\sin \left (c+d\,x\right )\right )}^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sec {\left (c + d x \right )}}{\left (a \left (\sin {\left (c + d x \right )} + 1\right )\right )^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________