Optimal. Leaf size=83 \[ -\frac {2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^{7/2}}{7 b^3 d}-\frac {2 (a+b \sin (c+d x))^{11/2}}{11 b^3 d}+\frac {4 a (a+b \sin (c+d x))^{9/2}}{9 b^3 d} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.09, antiderivative size = 83, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2668, 697} \[ -\frac {2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^{7/2}}{7 b^3 d}-\frac {2 (a+b \sin (c+d x))^{11/2}}{11 b^3 d}+\frac {4 a (a+b \sin (c+d x))^{9/2}}{9 b^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 697
Rule 2668
Rubi steps
\begin {align*} \int \cos ^3(c+d x) (a+b \sin (c+d x))^{5/2} \, dx &=\frac {\operatorname {Subst}\left (\int (a+x)^{5/2} \left (b^2-x^2\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=\frac {\operatorname {Subst}\left (\int \left (\left (-a^2+b^2\right ) (a+x)^{5/2}+2 a (a+x)^{7/2}-(a+x)^{9/2}\right ) \, dx,x,b \sin (c+d x)\right )}{b^3 d}\\ &=-\frac {2 \left (a^2-b^2\right ) (a+b \sin (c+d x))^{7/2}}{7 b^3 d}+\frac {4 a (a+b \sin (c+d x))^{9/2}}{9 b^3 d}-\frac {2 (a+b \sin (c+d x))^{11/2}}{11 b^3 d}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.10, size = 58, normalized size = 0.70 \[ -\frac {2 (a+b \sin (c+d x))^{7/2} \left (8 a^2-28 a b \sin (c+d x)+63 b^2 \sin ^2(c+d x)-99 b^2\right )}{693 b^3 d} \]
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
fricas [B] time = 1.08, size = 143, normalized size = 1.72 \[ -\frac {2 \, {\left (161 \, a b^{4} \cos \left (d x + c\right )^{4} + 8 \, a^{5} - 96 \, a^{3} b^{2} - 136 \, a b^{4} - {\left (3 \, a^{3} b^{2} + 25 \, a b^{4}\right )} \cos \left (d x + c\right )^{2} + {\left (63 \, b^{5} \cos \left (d x + c\right )^{4} - 4 \, a^{4} b - 184 \, a^{2} b^{3} - 36 \, b^{5} - {\left (113 \, a^{2} b^{3} + 27 \, b^{5}\right )} \cos \left (d x + c\right )^{2}\right )} \sin \left (d x + c\right )\right )} \sqrt {b \sin \left (d x + c\right ) + a}}{693 \, b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {5}{2}} \cos \left (d x + c\right )^{3}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [A] time = 0.48, size = 55, normalized size = 0.66 \[ -\frac {2 \left (a +b \sin \left (d x +c \right )\right )^{\frac {7}{2}} \left (-63 b^{2} \left (\cos ^{2}\left (d x +c \right )\right )-28 a b \sin \left (d x +c \right )+8 a^{2}-36 b^{2}\right )}{693 b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [A] time = 0.36, size = 61, normalized size = 0.73 \[ -\frac {2 \, {\left (63 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {11}{2}} - 154 \, {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {9}{2}} a + 99 \, {\left (a^{2} - b^{2}\right )} {\left (b \sin \left (d x + c\right ) + a\right )}^{\frac {7}{2}}\right )}}{693 \, b^{3} d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int {\cos \left (c+d\,x\right )}^3\,{\left (a+b\,\sin \left (c+d\,x\right )\right )}^{5/2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________