3.337 \(\int \frac {(1-c^2 x^2)^{5/2}}{x (a+b \sin ^{-1}(c x))} \, dx\)

Optimal. Leaf size=196 \[ \text {Int}\left (\frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )},x\right )+\frac {11 \sin \left (\frac {a}{b}\right ) \text {Ci}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{8 b}+\frac {7 \sin \left (\frac {3 a}{b}\right ) \text {Ci}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b}+\frac {\sin \left (\frac {5 a}{b}\right ) \text {Ci}\left (\frac {5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b}-\frac {11 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a+b \sin ^{-1}(c x)}{b}\right )}{8 b}-\frac {7 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b}-\frac {\cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 \left (a+b \sin ^{-1}(c x)\right )}{b}\right )}{16 b} \]

[Out]

-11/8*cos(a/b)*Si((a+b*arcsin(c*x))/b)/b-7/16*cos(3*a/b)*Si(3*(a+b*arcsin(c*x))/b)/b-1/16*cos(5*a/b)*Si(5*(a+b
*arcsin(c*x))/b)/b+11/8*Ci((a+b*arcsin(c*x))/b)*sin(a/b)/b+7/16*Ci(3*(a+b*arcsin(c*x))/b)*sin(3*a/b)/b+1/16*Ci
(5*(a+b*arcsin(c*x))/b)*sin(5*a/b)/b+Unintegrable(1/x/(a+b*arcsin(c*x))/(-c^2*x^2+1)^(1/2),x)

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Rubi [A]  time = 1.15, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x \left (a+b \sin ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Int[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcSin[c*x])),x]

[Out]

(11*CosIntegral[a/b + ArcSin[c*x]]*Sin[a/b])/(8*b) + (7*CosIntegral[(3*a)/b + 3*ArcSin[c*x]]*Sin[(3*a)/b])/(16
*b) + (CosIntegral[(5*a)/b + 5*ArcSin[c*x]]*Sin[(5*a)/b])/(16*b) - (11*Cos[a/b]*SinIntegral[a/b + ArcSin[c*x]]
)/(8*b) - (7*Cos[(3*a)/b]*SinIntegral[(3*a)/b + 3*ArcSin[c*x]])/(16*b) - (Cos[(5*a)/b]*SinIntegral[(5*a)/b + 5
*ArcSin[c*x]])/(16*b) + Defer[Int][1/(x*Sqrt[1 - c^2*x^2]*(a + b*ArcSin[c*x])), x]

Rubi steps

\begin {align*} \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x \left (a+b \sin ^{-1}(c x)\right )} \, dx &=\int \left (\frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}-\frac {3 c^2 x}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}+\frac {3 c^4 x^3}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}-\frac {c^6 x^5}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )}\right ) \, dx\\ &=-\left (\left (3 c^2\right ) \int \frac {x}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\right )+\left (3 c^4\right ) \int \frac {x^3}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx-c^6 \int \frac {x^5}{\sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx+\int \frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\\ &=-\left (3 \operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )\right )+3 \operatorname {Subst}\left (\int \frac {\sin ^3(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\int \frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx-\operatorname {Subst}\left (\int \frac {\sin ^5(x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )\\ &=3 \operatorname {Subst}\left (\int \left (\frac {3 \sin (x)}{4 (a+b x)}-\frac {\sin (3 x)}{4 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )-\left (3 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\left (3 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\int \frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx-\operatorname {Subst}\left (\int \left (\frac {5 \sin (x)}{8 (a+b x)}-\frac {5 \sin (3 x)}{16 (a+b x)}+\frac {\sin (5 x)}{16 (a+b x)}\right ) \, dx,x,\sin ^{-1}(c x)\right )\\ &=\frac {3 \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{b}-\frac {3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{b}-\frac {1}{16} \operatorname {Subst}\left (\int \frac {\sin (5 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\frac {5}{16} \operatorname {Subst}\left (\int \frac {\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )-\frac {5}{8} \operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )-\frac {3}{4} \operatorname {Subst}\left (\int \frac {\sin (3 x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\frac {9}{4} \operatorname {Subst}\left (\int \frac {\sin (x)}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\int \frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\\ &=\frac {3 \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{b}-\frac {3 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{b}-\frac {1}{8} \left (5 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\frac {1}{4} \left (9 \cos \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\frac {1}{16} \left (5 \cos \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{4} \left (3 \cos \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{16} \cos \left (\frac {5 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\sin \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\frac {1}{8} \left (5 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{4} \left (9 \sin \left (\frac {a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {a}{b}+x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )-\frac {1}{16} \left (5 \sin \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\frac {1}{4} \left (3 \sin \left (\frac {3 a}{b}\right )\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {3 a}{b}+3 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\frac {1}{16} \sin \left (\frac {5 a}{b}\right ) \operatorname {Subst}\left (\int \frac {\cos \left (\frac {5 a}{b}+5 x\right )}{a+b x} \, dx,x,\sin ^{-1}(c x)\right )+\int \frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\\ &=\frac {11 \text {Ci}\left (\frac {a}{b}+\sin ^{-1}(c x)\right ) \sin \left (\frac {a}{b}\right )}{8 b}+\frac {7 \text {Ci}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right ) \sin \left (\frac {3 a}{b}\right )}{16 b}+\frac {\text {Ci}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right ) \sin \left (\frac {5 a}{b}\right )}{16 b}-\frac {11 \cos \left (\frac {a}{b}\right ) \text {Si}\left (\frac {a}{b}+\sin ^{-1}(c x)\right )}{8 b}-\frac {7 \cos \left (\frac {3 a}{b}\right ) \text {Si}\left (\frac {3 a}{b}+3 \sin ^{-1}(c x)\right )}{16 b}-\frac {\cos \left (\frac {5 a}{b}\right ) \text {Si}\left (\frac {5 a}{b}+5 \sin ^{-1}(c x)\right )}{16 b}+\int \frac {1}{x \sqrt {1-c^2 x^2} \left (a+b \sin ^{-1}(c x)\right )} \, dx\\ \end {align*}

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Mathematica [A]  time = 3.57, size = 0, normalized size = 0.00 \[ \int \frac {\left (1-c^2 x^2\right )^{5/2}}{x \left (a+b \sin ^{-1}(c x)\right )} \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcSin[c*x])),x]

[Out]

Integrate[(1 - c^2*x^2)^(5/2)/(x*(a + b*ArcSin[c*x])), x]

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fricas [A]  time = 1.00, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (c^{4} x^{4} - 2 \, c^{2} x^{2} + 1\right )} \sqrt {-c^{2} x^{2} + 1}}{b x \arcsin \left (c x\right ) + a x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(5/2)/x/(a+b*arcsin(c*x)),x, algorithm="fricas")

[Out]

integral((c^4*x^4 - 2*c^2*x^2 + 1)*sqrt(-c^2*x^2 + 1)/(b*x*arcsin(c*x) + a*x), x)

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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]

integrate((-c^2*x^2+1)^(5/2)/x/(a+b*arcsin(c*x)),x, algorithm="giac")

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:sym2
poly/r2sym(const gen & e,const index_m & i,const vecteur & l) Error: Bad Argument Value

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maple [A]  time = 0.72, size = 0, normalized size = 0.00 \[ \int \frac {\left (-c^{2} x^{2}+1\right )^{\frac {5}{2}}}{x \left (a +b \arcsin \left (c x \right )\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((-c^2*x^2+1)^(5/2)/x/(a+b*arcsin(c*x)),x)

[Out]

int((-c^2*x^2+1)^(5/2)/x/(a+b*arcsin(c*x)),x)

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maxima [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {{\left (-c^{2} x^{2} + 1\right )}^{\frac {5}{2}}}{{\left (b \arcsin \left (c x\right ) + a\right )} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c^2*x^2+1)^(5/2)/x/(a+b*arcsin(c*x)),x, algorithm="maxima")

[Out]

integrate((-c^2*x^2 + 1)^(5/2)/((b*arcsin(c*x) + a)*x), x)

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mupad [A]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (1-c^2\,x^2\right )}^{5/2}}{x\,\left (a+b\,\mathrm {asin}\left (c\,x\right )\right )} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((1 - c^2*x^2)^(5/2)/(x*(a + b*asin(c*x))),x)

[Out]

int((1 - c^2*x^2)^(5/2)/(x*(a + b*asin(c*x))), x)

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sympy [A]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\left (- \left (c x - 1\right ) \left (c x + 1\right )\right )^{\frac {5}{2}}}{x \left (a + b \operatorname {asin}{\left (c x \right )}\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((-c**2*x**2+1)**(5/2)/x/(a+b*asin(c*x)),x)

[Out]

Integral((-(c*x - 1)*(c*x + 1))**(5/2)/(x*(a + b*asin(c*x))), x)

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