Optimal. Leaf size=193 \[ -\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {128 \sqrt {a+a \sin (c+d x)}}{195 a^3 d e (e \cos (c+d x))^{3/2}}+\frac {256 (a+a \sin (c+d x))^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.26, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 2, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.074, Rules used = {2751, 2750}
\begin {gather*} \frac {256 (a \sin (c+d x)+a)^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}}-\frac {128 \sqrt {a \sin (c+d x)+a}}{195 a^3 d e (e \cos (c+d x))^{3/2}}-\frac {32}{195 a^2 d e \sqrt {a \sin (c+d x)+a} (e \cos (c+d x))^{3/2}}-\frac {16}{117 a d e (a \sin (c+d x)+a)^{3/2} (e \cos (c+d x))^{3/2}}-\frac {2}{13 d e (a \sin (c+d x)+a)^{5/2} (e \cos (c+d x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2750
Rule 2751
Rubi steps
\begin {align*} \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{5/2}} \, dx &=-\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}+\frac {8 \int \frac {1}{(e \cos (c+d x))^{5/2} (a+a \sin (c+d x))^{3/2}} \, dx}{13 a}\\ &=-\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}+\frac {16 \int \frac {1}{(e \cos (c+d x))^{5/2} \sqrt {a+a \sin (c+d x)}} \, dx}{39 a^2}\\ &=-\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}+\frac {64 \int \frac {\sqrt {a+a \sin (c+d x)}}{(e \cos (c+d x))^{5/2}} \, dx}{195 a^3}\\ &=-\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {128 \sqrt {a+a \sin (c+d x)}}{195 a^3 d e (e \cos (c+d x))^{3/2}}+\frac {128 \int \frac {(a+a \sin (c+d x))^{3/2}}{(e \cos (c+d x))^{5/2}} \, dx}{195 a^4}\\ &=-\frac {2}{13 d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{5/2}}-\frac {16}{117 a d e (e \cos (c+d x))^{3/2} (a+a \sin (c+d x))^{3/2}}-\frac {32}{195 a^2 d e (e \cos (c+d x))^{3/2} \sqrt {a+a \sin (c+d x)}}-\frac {128 \sqrt {a+a \sin (c+d x)}}{195 a^3 d e (e \cos (c+d x))^{3/2}}+\frac {256 (a+a \sin (c+d x))^{3/2}}{585 a^4 d e (e \cos (c+d x))^{3/2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.28, size = 76, normalized size = 0.39 \begin {gather*} -\frac {2 (77+136 \cos (2 (c+d x))-16 \cos (4 (c+d x))-40 \sin (c+d x)+80 \sin (3 (c+d x)))}{585 d e (e \cos (c+d x))^{3/2} (a (1+\sin (c+d x)))^{5/2}} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [A]
time = 0.16, size = 80, normalized size = 0.41
method | result | size |
default | \(\frac {2 \left (128 \left (\cos ^{4}\left (d x +c \right )\right )-320 \left (\cos ^{2}\left (d x +c \right )\right ) \sin \left (d x +c \right )-400 \left (\cos ^{2}\left (d x +c \right )\right )+120 \sin \left (d x +c \right )+75\right ) \cos \left (d x +c \right )}{585 d \left (e \cos \left (d x +c \right )\right )^{\frac {5}{2}} \left (a \left (1+\sin \left (d x +c \right )\right )\right )^{\frac {5}{2}}}\) | \(80\) |
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 404 vs.
\(2 (148) = 296\).
time = 0.55, size = 404, normalized size = 2.09 \begin {gather*} -\frac {2 \, {\left (197 \, \sqrt {a} + \frac {400 \, \sqrt {a} \sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + \frac {15 \, \sqrt {a} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} - \frac {1760 \, \sqrt {a} \sin \left (d x + c\right )^{3}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{3}} - \frac {2230 \, \sqrt {a} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {2230 \, \sqrt {a} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {1760 \, \sqrt {a} \sin \left (d x + c\right )^{7}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{7}} - \frac {15 \, \sqrt {a} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} - \frac {400 \, \sqrt {a} \sin \left (d x + c\right )^{9}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{9}} - \frac {197 \, \sqrt {a} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} {\left (\frac {\sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + 1\right )}^{5} e^{\left (-\frac {5}{2}\right )}}{585 \, {\left (a^{3} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{2}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{2}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{4}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{4}} + \frac {10 \, a^{3} \sin \left (d x + c\right )^{6}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{6}} + \frac {5 \, a^{3} \sin \left (d x + c\right )^{8}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{8}} + \frac {a^{3} \sin \left (d x + c\right )^{10}}{{\left (\cos \left (d x + c\right ) + 1\right )}^{10}}\right )} d {\left (\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {15}{2}} {\left (-\frac {\sin \left (d x + c\right )}{\cos \left (d x + c\right ) + 1} + 1\right )}^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [A]
time = 0.36, size = 138, normalized size = 0.72 \begin {gather*} -\frac {2 \, {\left (128 \, \cos \left (d x + c\right )^{4} - 400 \, \cos \left (d x + c\right )^{2} - 40 \, {\left (8 \, \cos \left (d x + c\right )^{2} - 3\right )} \sin \left (d x + c\right ) + 75\right )} \sqrt {a \sin \left (d x + c\right ) + a} \sqrt {\cos \left (d x + c\right )}}{585 \, {\left (3 \, a^{3} d \cos \left (d x + c\right )^{4} e^{\frac {5}{2}} - 4 \, a^{3} d \cos \left (d x + c\right )^{2} e^{\frac {5}{2}} + {\left (a^{3} d \cos \left (d x + c\right )^{4} e^{\frac {5}{2}} - 4 \, a^{3} d \cos \left (d x + c\right )^{2} e^{\frac {5}{2}}\right )} \sin \left (d x + c\right )\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [B]
time = 11.54, size = 379, normalized size = 1.96 \begin {gather*} -\frac {\sqrt {a+a\,\sin \left (c+d\,x\right )}\,\left (\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,2464{}\mathrm {i}}{585\,a^3\,d\,e^2}+\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (2\,c+2\,d\,x\right )\,4352{}\mathrm {i}}{585\,a^3\,d\,e^2}-\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (4\,c+4\,d\,x\right )\,512{}\mathrm {i}}{585\,a^3\,d\,e^2}+\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (3\,c+3\,d\,x\right )\,512{}\mathrm {i}}{117\,a^3\,d\,e^2}-\frac {{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (c+d\,x\right )\,256{}\mathrm {i}}{117\,a^3\,d\,e^2}\right )}{\cos \left (c+d\,x\right )\,{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}\,28{}\mathrm {i}-{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\cos \left (3\,c+3\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}\,12{}\mathrm {i}+{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (2\,c+2\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}\,28{}\mathrm {i}-{\mathrm {e}}^{c\,4{}\mathrm {i}+d\,x\,4{}\mathrm {i}}\,\sin \left (4\,c+4\,d\,x\right )\,\sqrt {e\,\left (\frac {{\mathrm {e}}^{-c\,1{}\mathrm {i}-d\,x\,1{}\mathrm {i}}}{2}+\frac {{\mathrm {e}}^{c\,1{}\mathrm {i}+d\,x\,1{}\mathrm {i}}}{2}\right )}\,2{}\mathrm {i}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________